, the authors proved that in the focusing mass critical case, if the initial data u 0 has the mass near the mass of ground state prole Q, and u 0 has negative energy and zero momentum then the associated solution blows up in nite time. We have the following result.

). Let N = 1 or N ⩾ 2 with a spectral assumption. Then there exists a number a > 0 and a constant C > 0 such that the following is true. Let u 0 ∈ H 1 (R N ) be such that

Let u(t) be the corresponding solution of (1.1). Then u blows up in nite time T > 0 and for t close to T :

, and

, for some constant C 1 , C 2 > 0.

Standing waves and stability theory

The equation (1.1) is invariant by Galilean transform. More precisely, if u solves (1.1) then for any v ∈ R N , the following function solves (1.1)

In this section, we would like to introduce the general theory on classical nonlinear Schrödinger equations. We consider the following power type nonlinear Schrödinger equation:

iu t + ∆u + λ|u| α u = 0, u(0) = φ. (1.1)
where u : R × R N → C, λ = ±1 and 0 < α < 4 N -2 (0 < α < ∞ if N = 1, 2). The equation (1.1) is called focusing if λ = 1 and defocusing if λ = -1.

Let I be a open subset of R with 0 ∈ I. We observe that u ∈ L ∞ (I, H 1 (R N )) is a solution of (1.1) if and only if u satises the following integral equation (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Proposition 3.1.3]):

u(t) = S(t)φ + iλ t 0 S(t -s)|u| α u(s) ds, (1.2)
where S(t) is the Schrödinger group.

It is well known that (1.2) is locally well posed on H 1 (R N ). More precisely, for any φ ∈ H 1 (R N ), there exists a unique maximal solution u ∈ C(I, H 1 (R N )) of (1.2). This solution u satises a blow up alternative and depends continuously on the initial data (see Section 1.3.1 for details). Finally, u satises the following conservation laws:

M (u(t)) := 1 2 ∥u(t)∥ 2 L 2 = 1 2 ∥φ∥ 2 L 2 , (1.3)
E(u(t)) := 1 2 ∥∇u(t)∥ 2 L 2 - λ α + 2 ∥u(t)∥ α+2 L α+2 = E(φ), (1.4)
P (u(t)) := Im R N u(t, x)∇u(t, x) dx = P (φ).

(1.5)

Strichartz estimates

Strichartz estimates are an important tool to study the local well posedness of dispersive equations. In this section, we introduce the Strichartz estimates for the Schrödinger group.

The following well known result is the fundamental estimate for Schrödinger group.

Proposition 1.1. If p ∈ [2, ∞] and t ̸ = 0, then S(t) maps L p ′ (R N ) continuously to L p (R N ) and

∥S(t)φ∥ L p ⩽ (4π|t|) -N ( 1 2 -1 p ) ∥φ∥ L p ′ , for all φ ∈ L p ′ (R N ).
Before stating the Strichartz estimates, we need the following denition. Denition 1.2 (Admissible pairs). We say that a pair (q, r)

∈ [2, ∞] × [2, ∞] is admissible if 2 q = N 1 2 - 1 r , (q, r, N ) ̸ = (2, ∞, 2).
We say that the pair is a strictly admissible pair if (q, r) ̸ = 2, 2N N -2 . The point 2, 2N N -2 is called endpoint. Theorem 1.3 (Strichartz estimates). For any admissible pairs (q 1 , r 1 ), (q 2 , r 2 ) there exist C > 0 such that the following holds: Homogeneous estimate. For any φ ∈ L 2 (R N ) we have

∥S(t)φ∥ L q 1 t L r 1 x ⩽ C∥φ∥ L 2 .

In homogeneous estimate. For

F ∈ L q ′ 2 t L r ′ 2 x (R × R N ), we have t 0 S(t -s)F (s) ds L q 1 t L r 1 x ⩽ C∥F ∥ L q ′ 2 t L r ′ 2 x .
Using Strichartz estimates, one can prove the local well posedness of (1.2) in H 1 (R N ) (see e.g [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Theorem 4.4.1]).

Abstract local theory

In this section, we would like to introduce the general method to establish the local theory for evolution equations. For a deeper discussion of the local well posedness, we refer to [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF]. Let X be a Banach space and A be a linear operator in X with D(A) the domain of A. We assume that A is the generator of a bounded continuous group (S(t)) t∈R in X. We consider the following Cauchy problem:

u t = Au + f (u), u(t = 0) = u 0 , (1.6)
where u : R → X. We see that S(t)u 0 is the unique solution of (1.6) ). Thus, we reduce the study of the local theory of (1.6) to the study of the local theory of (1.7). Local well posedness of (1.7) is usually established by using contraction mapping theorem.

In our case, we are interested in the Schrödinger equations i.e A = i∆. The denition of a strong and weak solution to nonlinear Schrödinger equations is given in [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Denition 3.1.1]. The denition of locally well posed is given in [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Denition 3.1.5]. We would like to recall and give some comments on this. We say that the problem (1.7) is locally well-posed in H 1 (R N ) if the following properties hold:

(1) Let u 0 ∈ H 1 (R N ). Then there exists a unique solution in H 1 (R N ) for the problem (1.7). Moreover, the solution is dened on a maximal interval (T min , T max ), with T max and T min depending on u 0 . In some cases, it is useful to prove the existence of blow up solutions.

(2) There is the blowup alternative: If T max < ∞ then lim t→Tmax ∥u(t)∥ H 1 = ∞. A similar statement holds for T min . This blowup alternative is useful to prove the existence of global solutions. Indeed, if we can show that ∥u(t)∥ H 1 is bounded when t is close to T max then T max = ∞.

(3) The solution depends continuously on the initial value i.e if u n0 → u 0 in H 1 (R N ) and if I ⊂ (T min , T max ) is a closed interval, then the corresponding solution u n with initial data u n0 is dened on I for n large enough and satisfy ∥u n -u∥ L ∞ (I,H 1 ) → 0. This property is useful to verify the conservation laws in H 1 (R N ) of (1.7). Indeed, the conservation laws are obtained for an smooth enough and decaying solution of (1.6). We know that under some conditions of f , a solution of (1.6) also solves (1.7). By an approximation argument and using the continuous dependence of the solution on the initial value, we may show rigorously the conservation law for a solution of (1.7).

Global well posedness and blow up

Consider the equation (1.1). As in the previous section, (1.1) is locally well posed on H 1 (R N ) in the energy sub-critical case i.e 0 < α < 4 N -2 . Moreover, the conservation laws are satised. Let u ∈ C((T min , T max ), H 1 (R N )) be the maximal solution of (1.1) corresponding with the initial data u(0) = u 0 . In this section, we present the well known results of global well posedness (T max = ∞ and T min = -∞) and blow up of this solution (T max < ∞ or T min > -∞).

Global well posedness

In the case λ < 0, using the conservation of mass and energy we may prove that the H 1 -norm of u is uniformly bounded in time. This implies that the solution exists globally in time. In the case λ > 0, the situation is more complex. If 0 < α < 4 N , or α = 4 N and ∥u 0 ∥ L 2 small enough then the solution is global. We may expect the existence of blow up solutions for α ⩾ 4 N . Thus, in the focusing case, α = 4 N is a threshold between global existence an blow up. These are the most complex cases to study the long time dynamic of (1.1). In this section, we focus on introducing the well known results on global existence of solutions of (1.1).

First, in the case of small initial data, the solution is global. More precisely, we have the following result.

Theorem 1.4. Let 0 ⩽ α < 4 N -2 . There exists a number a > 0 such that if ∥u 0 ∥ H 1 < a then the associated solution u of (1.1) is global. This theorem is proved by using Sobolev-embedding theorem and a boostrap argument.

In the case α suciently large, the solution is global for oscillating data. We have the following result.

Theorem 1.5. Assume 4

N -2 > α > α 0 = 2-N + √ N 2 +12N +4 2N
and a = 2α(α+2) 4-α(N -2) . Let

u 0 ∈ H 1 (R N ) be such that | • |u 0 (•) ∈ L 2 (R N ). Given b ∈ R, set u 0b (x) = e ib|x| 2 4
u 0 (x),

and let u b be the maximal solution of (1.1) with the initial data u 0b . There exists a number b 0 such that if b ⩾ b 0 then u b is global. Moreover, u b ∈ L a (R, L α+2 (R N )) ∩ L γ (R, W 1,ρ (R N )) for every admissible pair (γ, ρ).

Moreover, if α is given as in Theorem 1.5 and the space time norm of initial data is small in some space, we also obtain the global existence of solution.

Theorem 1.6. Let u 0 ∈ H 1 (R N ), α 0 be as in Theorem 1.5 and u be the associated solution of (1.1) with initial data u 0 . There exists a number ε 0 such that if u 0 satises

sup t∈R t 4-(N -2)α 2α(α+2) ∥S(t)u 0 ∥ L α+2 < ε 0 then u is global and satises ess sup t∈R t 4-(N -2)α 2α(α+2) ∥u(t)∥ L α+2 < ∞.
For the proof of the above theorems, we refer to [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and references therein.

Blow up of solution

As said in the previous section, the existence of blow solution for (1.1) only occurs in the focusing case for α ⩾ 4 N . These assumptions are made throughout this section.

In [START_REF] Glassey | On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations[END_REF], the author used the following functional

f (t) = R N |x| 2 |u(t, x)| 2 dx.
Assume that the initial data belongs to weighted space

Σ = {φ ∈ H 1 (R N ) : | • |φ(•) ∈ L 2 (R N )}.
Then the associated solution u of (1.1) satises u ∈ C((T min , T max ), Σ). Thus, the function f is well dened. Moreover, f ∈ C 2 (T min , T max ) and we have the following virial identity

f ′′ (t) = 16E(u 0 ) - 4(N α -4) α + 2 ∥u(t)∥ α+2 L α+2 ,
where E(u 0 ) is the energy. Since N α ⩾ 4, if we assume E(u 0 ) < 0 then f ′′ (t) < δ < 0 for some constant δ independent in time. This implies that the time of existence of the solution is nite in both directions. More precisely, we have the following result.

Theorem 1.7 (Glassey [45]). Let u 0 ∈ Σ be such that E(u 0 ) < 0. Then the corresponding solution of (1.1) blows up in nite time.

In the radial setting, the condition x|u 0 (x)| ∈ L 2 (R N ) can be removed. We have the following result.

Theorem 1.8 ). Let N ⩾ 2 and

4 N ⩽ α < 4 N -2 (2 ⩽ α ⩽ 4 if N = 2).
If u 0 ∈ H 1 (R N ) is such that E(u 0 ) < 0 and u 0 is radial, then the corresponding solution of (1.1) blows up in nite time in both directions.

In the case N = 1, α = 4, Ogawa-Tsutsumi [START_REF] Ogawa | Blow-up of H 1 solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity[END_REF] proved that any solution with negative energy blows up in nite time. More precisely, we have the following result. Theorem 1.9 ). Let N = 1, α = 4, u 0 ∈ H 1 (R) be such that E(u 0 ) < 0. Then the corresponding solution of (1.1) blows up in nite time.

In the mass critical case α = 4 N , the existence and uniqueness of blow up solution with critical mass was obtained in [START_REF] Merle | Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power[END_REF]. More precisely, we have the following result.

Theorem 1.10 (Merle [START_REF] Merle | Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power[END_REF]). Let u 0 ∈ H 1 (R N ) be such that the associated solution of (1.1) blows up in nite time T > 0. Moreover, assume that ∥u 0 ∥ L 2 = ∥Q∥ L 2 , where Q is the unique radial positive solution of the elliptic equation

∆u + |u| 4 N u = u.
There exist θ ∈ R, ω > 0, x 0 ∈ R N , x 1 ∈ R N such that

u 0 = ω T N 2 e iθ-i|x-x 1 |/4T +iω 2 /T Q ω x -x 1 T -x 0 ,
and for t < T ,

u(t, x) = ω T -t N 2 e iθ+i|x-x 1 | 2 /4(-T +t)-iω 2 /(-T +t) Q ω T -t ((x -x 1 ) -(T -t)x 0 ) . u(t, x -vt).
The equation (1.1) admits a special type of solution called solitary waves. A solitary wave of (1.1) is a solution of the form e iωt φ(x -vt), where φ ∈ H 1 (R N ). In the case v = 0, this solution is called standing wave.

In the defocusing case λ = -1, there is no standing wave of (1.1). In the focusing case λ = 1, there is no standing wave in the case ω ⩽ 0. Throughout this section, we only consider the focusing case i.e λ = 1. Assume ω > 0, the standing waves of (1.1) are of the form e iωt φ(x), where φ solves:

-∆φ + ωφ -|φ| α φ = 0 φ ∈ H 1 (R N ) \ {0} (1.8)
The function φ is called ground state if it solves the following variational problem inf{S ω (v); v is a solution of (1.8)}, where S ω is dened by

S ω (u) = 1 2 ∥∇u∥ 2 L 2 + ω 2 ∥u∥ 2 L 2 - 1 α + 2 ∥u∥ α+2 L α+2 .
Existence of a radial positive ground state φ can be shown by using variational techniques (see [START_REF] Coz | Standing waves in nonlinear Schrödinger equations[END_REF] and the references therein). Moreover, the set of ground state is the following G = {e iθ φ(• -y); θ ∈ R, y ∈ R N }.

It turns out that in some cases the solution is close to the orbit of the standing wave if the initial data is enough close to the standing wave prole. Before stating the main results, we need the following denition. Denition 1.12. Let φ be a solution of (1.8). The standing wave e iωt φ(x) is said to be orbitally stable in H 1 (R N ) if for all ε > 0 there exists δ > 0 such that if u 0 ∈ H 1 (R N ) satises ∥u 0 -φ∥ H 1 < δ then the maximal solution u(t) of (1.1) with u(0) = u 0 exists for all t ∈ R and

sup t∈R inf θ∈R,y∈R N ∥u(t) -e iθ φ(• -y)∥ H 1 < ε.
Otherwise, the standing wave is said to be unstable.

In addition, If there exists a sequence φ n → φ in H 1 (R N ) as n → ∞ such that the associated solution u n of (1.1) with initial data φ n blows up in nite time for all n, then the standing wave is said to be strongly unstable or unstable by blow up in nite time. The strongly instability of standing waves implies its instability.

We have the following result. Theorem 1.13. Let φ be a ground state of (1.8). If 0 < α < 4 N then the standing wave e iωt φ(x) is orbitally stable.

There are many methods to prove the stability of standing waves. One of them is the variational method introduced by Cazenave and Lions [START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF][START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF]. This method relies on the following compactness result. Proposition 1.14. Let 0 < α < 4 N . Fix ρ > 0. Consider the following minimization problem

d ρ := inf{E(v) : v ∈ H 1 (R N ), ∥v∥ 2 L 2 = ρ}, (1.9)
where E is the functional energy of (1.1) in the focusing case. Let v n ∈ H 1 (R N ) satisfy the following condition:

E(v n ) → d ρ , and ∥v n ∥ 2 L 2 → ρ.
Then there exist a sequence (y n ) ∈ R N and a function v ∈ H 1 (R N ) such that up to a subsequence we have

v n (• -y n ) → v strongly in H 1 (R N ).
In particular, E(v) = d ρ and ∥v∥ 2 L 2 = ρ. By using Proposition 1.14, Cazenave and Lions [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF] proved Theorem 1.13. See also Le Coz [START_REF] Coz | Standing waves in nonlinear Schrödinger equations[END_REF].

The method of Cazenave and Lions relies on the variational characterization and the uniqueness of ground state under phase shift and translation. In the general case, for standing wave which is not a ground state this method may be not applicable. However, in [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF], Grillakis-Shatah-Strauss introduced a famous theory which can treat for larger class of bound state. This theory especially treat to the evolution equation with Hamilton structure.

Let E and M be the functional of the energy and the mass of (1.1). Let φ ω be a solution of (1.8), where the subscript is to exhibit the dependence of solution with the parameter ω. Set H ω = E ′′ (φ ω ) -ωM ′′ (φ ω ) and d(ω) = E(φ ω ) -ωM (φ ω ). It turns out that the stability of bound state depends on the convexity or concavity of function d : ω → d(ω). Before stating the main result, we need the following important assumption.

Assumption A1. Assume that H ω has exactly one simple negative eigenvalue and ker (H

ω ) = iφ ω , ∂ ∂ 1 φ ω , ..., ∂ ∂ N φ ω ,
and the rest of its spectrum is positive and bounded away from zero. The main result is the following.

Theorem 1.15 ( The main ingredient in the proof the stability of the about theorem is a coercivity property of operator H ω . Consider the case φ ω is a ground state. Assumption A1 is veried by the work in [START_REF] Coz | Standing waves in nonlinear Schrödinger equations[END_REF]. The condition d ′′ (ω) > 0 is equivalent to α < 4 N . Thus, using Theorem 1.15 we obtain the conclusion of Theorem 1. [START_REF] Biagioni | Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations[END_REF].

It turn out that the stability of ground states depend on the nonlinear exponent α. Indeed, in the case α > 4 N , d ′′ (ω) < 0 then using Theorem 1.15 we obtain that the ground state is unstable. Moreover, in this case and the case α = 4 N , ground states are strongly unstable. More precisely, we have the following result. Theorem 1.16. Let φ be a ground state of (1.8). If α ⩾ 4 N then the standing wave e iωt φ(x) is unstable by blow up in nite time.

For the proof of the above theorem, we refer the reader to [START_REF] Coz | Standing waves in nonlinear Schrödinger equations[END_REF]Theorem 5.3].

Gross-Pitaevskii equation

In this section, we would like to introduce the following equation:

iu t + ∆u + u(1 -|u| 2 ) = 0, (1.10)
where u : R × R N → C satises the nonvanishing boundary condition |u| → 1 as |x| → ∞. The equation (1.10) is called Gross-Pitaevskii equation. Its energy is given by

E(u) = 1 2 ∥∇u∥ 2 L 2 + 1 4 R N (|u| 2 -1) 2 dx,
which is dened on the energy space

E = u ∈ H 1 loc (R N ) : ∇u ∈ L 2 (R N ), |u| 2 -1 ∈ L 2 (R N ) .
Consider the Madelung transform u = √ ρe iθ , for u ̸ = 0.

The hydrodynamical variables (ρ, v = 2∇θ) satisfy the hydrodynamical system ρ t + div(ρv) = 0,

v t + v • ∇v + 2∇ρ = 2∇ ∆ √ ρ √ ρ
.

(1.11)

Cauchy problem

First, we recall the denition of Zhidkov spaces which were introduced in [START_REF] Zhidkov | Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory[END_REF]:

X k (R N ) = u ∈ L ∞ (R N ), ∂ α u ∈ L 2 (R N ), 1 ⩽ |α| ⩽ k , (1.12)
equipped with the natural norm

∥u∥ X k = ∥u∥ L ∞ + Σ 1⩽|α|⩽k ∥∂ α u∥ L 2 .
The global well-posedness of (1.10) in one dimension in the energy space E was proved in [START_REF] Zhidkov | The Cauchy problem for the nonlinear Schrödinger equation[END_REF][START_REF] Zhidkov | Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory[END_REF]. In higher dimensions, the situation is more complex.

As shown in [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF], the space E ⊂ X 1 (R N ) + H 1 (R N ) is a complete metric space with the following distance metric:

d E (u, ũ) = ∥u -ũ∥ X 1 +H 1 + ∥|u| 2 -|ũ| 2 ∥ L 2 .
In [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF], the author established the local and global theory of (1.10) in the energy space E.

Theorem 1.17 (Gérard [42]). Let N = 2, 3. For each u 0 ∈ E, there exists a unique solution u ∈ C(R, E) of (1.10) with the initial data u(0) = u 0 . Moreover, u satises the following properties:

Regularity: If ∆u 0 ∈ L 2 (R N ) then ∆u ∈ C(R, L 2 (R N )).
Conservation energy: for all t ∈ R, we have E(u(t)) = E(u 0 ).

For each R > 0, T > 0, there exists C > 0 such that for each u 0 , ũ0 ∈ E such that E(u 0 ) ⩽ R and E(ũ 0 ) ⩽ R, the corresponding solutions u, ũ satisfy sup |t|⩽T d E (u(t), ũ(t)) ⩽ Cd E (u 0 , ũ0 ).

In dimension N = 4, in [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF], Gérard proved that (1.10) is globally well-posed in the case of small energy of the initial data. The proof uses the contraction mapping theorem.

Theorem 1.18 (Gérard [42]). Let N = 4. There exists δ > 0 such that, for every u 0 ∈ E such that E(u 0 ) ⩽ δ, there exists a unique solution of (1.10) u ∈ C(R, E) with ∇u ∈ L 2 loc (R, L 4 (R 4 )) and u(0) = u 0 . Moreover, the energy is conserved and the solution satises the regularity property and Lipschitz continuity stated in Theorem 1.17.

In [START_REF] Killip | Global well-posedness of the Gross-Pitaevskii and cubic-quintic nonlinear Schrödinger equations with nonvanishing boundary conditions[END_REF], the authors improved the result of [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF] in the case N = 4 for arbitrary large energy of the initial data.

Theorem 1.19 (Killip-Oh-Pocovnicu-Visan [START_REF] Killip | Global well-posedness of the Gross-Pitaevskii and cubic-quintic nonlinear Schrödinger equations with nonvanishing boundary conditions[END_REF]). Let N = 4 and u 0 ∈ E. There exists a unique solution u ∈ C(R, E) of (1.10) with the initial data u(t = 0) = u 0 .

We also mention the work of Gallo [START_REF] Gallo | The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at innity[END_REF], in which the author proves the local theory on energy space E for general nonlinearity.

Travelling waves

Travelling waves of (1.10) are special solutions of the form (up to a space rotation)

u(t, x) = U c (x 1 -ct, ..., x N ), (1.13)
for a speed c ∈ R and the prole U c solves the equation

-ic∂ 1 U c + ∆U c + U c (1 -|U c | 2 ) = 0. (1.14)
In dimension N = 1, travelling waves for (1.10) are uniquely (up to translation and phase shift) given by

U c (x) = 2 -c 2 2 tanh √ 2 -c 2 2 x + i c √ 2 ,
for |c| < √ 2. In this case, travelling waves are called dark solitons. In the case of higher dimensions, the situation is more complex.

In the case N ⩾ 2, the situation is more complex. In many cases, the travelling waves are constant functions. We have the following result. Theorem 1.20 (Gravejat [46,[START_REF] Gravejat | Limit at innity and nonexistence results for sonic travelling waves in the Gross-Pitaevskii equation[END_REF], Bethuel-Saut [START_REF] Bethuel | Travelling waves for the Gross-Pitaevskii equation[END_REF]). Consider (1.10) and a travelling wave prole U c solving (1.14). Assume c = 0 for N ⩾ 2 or c > √ 2 for N ⩾ 2 or c = √ 2 for N = 2. Then U c is a constant function.

In [START_REF] Mari³ | Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with nonzero conditions at innity[END_REF], Maris developed the above result for general cases of (1.10). Non-existence of non-constant travelling waves also holds in the case of high dimensions with small energy. More precisely, we have the following result.

Theorem 1.21 , de Laire [START_REF] De Laire | Non-existence for travelling waves with small energy for the Gross-Pitaevskii equation in dimension N ⩾ 3[END_REF]). Let N ⩾ 3. For (1.10), there exists a number ε > 0 such that a travelling wave prole U c with energy

E(U ) ⩽ ε, is constant.
We consider the following minimization problem

d ρ = inf{E(u), u ∈ W (R N ), p(u) = ρ}, (1.15)
where ρ ∈ C and W (R N ) = {1} + V (R N ) with V (R N ) is dened by

V (R N ) = {v : R N → C, s.t. (∇v, Re(v)) ∈ L 2 (R N ) 2 , Im(v) ∈ L 4 (R N ), and ∇Re(v) ∈ L 4 3 (R N )},
and p is the rst component of momentum function dened by

p(u) = 1 2 R N ⟨i∂ 1 u, u -1⟩ dx,
where ⟨f, g⟩ = Ref Reg + Imf Img. We have the following result.

Theorem 1.22 ). The following holds:

(i) For N = 2 and ρ > 0, there exists a minimizing travelling wave U ρ for (1.15).

(ii) For N = 3, there exists ρ * > 0 such that there exists a minimizing travelling wave U ρ for (1.15) if and only if ρ ⩾ ρ * .

For the general nonlinearity case, see Chiron-Maris [START_REF] Chiron | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at innity[END_REF].

The uniqueness of solutions to the minimization problem was proved in the case of large momentum. More precisely, we have the following result. Theorem 1.23 [START_REF] Chiron | A uniqueness result for the two vortex travelling wave in the nonlinear schrodinger equation[END_REF]). Let N = 2. There exists a number ρ 0 > 0 such that, for each ρ ⩾ ρ 0 , there exists a unique (up to phase shift and translation) minimizer U ρ of (1.15). Moreover, they form a smooth branch of travelling waves.

In the case 0 < c < √ 2 (Subsonic travelling waves), the existence of non constant travelling wave is proved in dimensions N ⩾ 3. [START_REF] Mari³ | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at innity[END_REF], Bellazzini-Ruiz [START_REF] Bellazzini | Finite energy traveling waves for the gross-pitaevskii equation in the subsonic regime[END_REF]). Let N ⩾ 3. There exists a non constant travelling wave U c of (1.10) for each 0 < c < √ 2.

Theorem 1.24 (Maris

Orbital and asymptotic stability of travelling waves

In dimension N = 1

Before presenting the well known results, we introduce the following distance in the energy space E

d c (φ 1 , φ 2 ) 2 = R |φ ′ 1 -φ ′ 2 | 2 + (1 -|U c | 2 )|φ 2 -φ 1 | 2 + |φ 1 | 2 -|φ 2 | 2 2 dx.
We have the following stability result.

Theorem 1.25 , Bethuel-Gravejat-Saut-Smets [START_REF] Béthuel | Orbital stability of the black soliton for the Gross-Pitaevskii equation[END_REF]). Let c be such that c 2 < 2. There exist δ c > 0 and K c > 0 such that, for each u 0 ∈ E satisfying the condition

δ := d c (u 0 , U c ) < δ c ,
then the corresponding solution u of (1.10) is such that there exist two functions

a ∈ C 1 (R, R) and θ ∈ C 1 (R, R) with sup t∈R |a ′ (t) -c| < K c δ,
such that the following holds:

sup t∈R d c e -iθ(t) u(• + a(t), t), U c < K c δ.
The asymptotic stability of dark solitons is as follows.

Theorem 1.26 (Bethuel-Gravejat-Smets [START_REF] Bethuel | Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation[END_REF], Gravejat-Smets [START_REF] Gravejat | Asymptotic stability of the black soliton for the Gross-Pitaevskii equation[END_REF], Cuccagna-Jenkins [START_REF] Cuccagna | On the asymptotic stability of N -soliton solutions of the defocusing nonlinear Schrödinger equation[END_REF]). Let c ∈ (-

√ 2, √ 2
). There exists δ c > 0 such that for each u 0 ∈ E satises

d c (u 0 , U c ) < δ c ,
then there exist a number c ∞ ∈ (-

√ 2, √ 2) and two functions a ∈ C 1 (R, R) and θ ∈ C 1 (R, R) with a ′ (t) → c ∞ ,
and θ ′ (t) → 0, as t → ∞ such that the correspoding solution u of (1.10) satises e -iθ(t) u(• + a(t), t) → U c∞ locally uniformly on R.

In [START_REF] Lin | Stability and instability of traveling solitonic bubbles[END_REF], Lin used the abstract theory of Grillakis-Shatah-Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] to prove the stability and instability of travelling waves in the case of general nonlinearity. More precisely, we have the following result.

Theorem 1.27 (Lin [78]). Consider the following equation:

iu t + u xx + f (|u| 2 )u = 0, (1.16)
where f ∈ C 2 (R + ) and f (ρ 0 ) = 0 for some ρ 0 > 0 and satises other conditions. Let c ⩾ 0 be small enough. Then there exists a travelling wave U c (x -ct) = a c e iθc (x -ct) solution to (1.16). This solution is stable when dPc dc < 0 and unstable when dPc dc > 0,

where

P c = Im R U c ′ U c 1 - ρ 0 |U c | 2 dx.
Here the stability means that for all ε > 0, there exists δ > 0 such that if the initial data u 0 = a 0 e iθ 0 satises s∈R ) is the solution of (1.16) with a(0) = a 0 , θ(0) = θ 0 . Instability means that the travelling wave is not stable.

(∥a 2 0 (• + s) -a 2 c ∥ H 1 + ∥θ ′ 0 (• + s) -θ ′ c ∥ L 2 ) < δ, then inf s∈R (∥a(t) 2 (• + s) -a 2 c ∥ H 1 + ∥θ(t) ′ (• + s) -θ ′ c ∥ L 2 ) < ε, for t ∈ R + . Here u(t) = a(t)e iθ(t

In higher dimensions

In the case N = 2, 3, we equip the energy set E the following metric distance

d(f, g) = ∥f -g∥ L 2 (B(0,1)) + ∥∇f -∇g∥ L 2 + ∥|f | 2 -|g| 2 ∥ L 2 .
We have the following result.

Theorem 1.28 ). Let M ρ be the set of minimizing travelling waves U ρ with scalar momentum ρ. Fix U ρ ∈ M ρ . For all ε > 0, there exists δ > 0 such that for each u 0 ∈ E such that

d(u 0 , U ρ ) < δ,
then the corresponding solution u of (1.10) satises

sup t∈R inf U ∈Mρ d(u(•, t), U ) < ε.
The proof of the above theorem used the variational problem of minimizing the energy with xed momentum.

1.3

The derivative nonlinear Schrödinger equations This thesis is devoted to the study of Schrödinger-type equations, especially derivative nonlinear Schrödinger equations i.e the equations of the following form:

iu t + u xx + iλ|u| 2 u x + iµu 2 u x + b|u| 4 u = 0, u(0) = u 0 , (1.17)
where u : R × R → C, b ∈ R and λ, µ ∈ R.

The equation (1.17) is invariant under the scaling transformation:

u κ (t, x) := κ 1 2 u(κ 2 t, κx).
Moreover, in the case b = 0, the equation (1.17) has a complete integral structure. We may use inverse scattering techniques to study the long time behaviour of this equation. In [START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF], by using this techniques, Bahouri and Perelman proved the global existence of solution in

H 1 2 (R).
This was an open problem in long time. Let u be a H 1 (R) solution of (1.17). We consider the Gauge transform

v(t, x) = u(t, x) exp ia x -∞ |u(t, y)| 2 dy . (1.18)
It is easy to check that v is a H 1 (R) solution of the following equation

iv t + v xx + ic 1 |v| 2 v x + ic 2 v 2 v x + c 3 |v| 4 v = 0, (1.19)
where c 1 , c 2 , c 3 are the constants which depend on a, λ, µ, b. The dynamics of solutions of (1.17) is equivalent to the dynamics of solutions of (1.19). For each of choice of the value of a, we have another equation equivalent to (1.17). In some cases, if we choose a suitable value of a then studying the long time dynamics of solutions of (1.19) is easier than for (1.17). This is one of advantage of this transform. Specially, let u be a solution of the Chen-Liu-Lee equation [START_REF] Chen | Integrability of nonlinear Hamiltonian systems by inverse scattering method[END_REF]:

iu t + u xx + i|u| 2 u x = 0. (1.20)
Let v be the Gauge transform of u given by (1.18) with a = -1 2 . Then v is a solution of the Kaup-Newell equation [START_REF] Kaup | An exact solution for a derivative nonlinear Schrödinger equation[END_REF]:

iv t + v xx + i(|v| 2 v) x = 0.
(1.21)

Let w be the Gauge transform of u given by (1.18) with a = 1 2 . Then w is a solution of the Gerdzhikov-Ivanov equation [START_REF] Gerdzhikov | A quadratic pencil of general type and nonlinear evolution equations. II. Hierarchies of Hamiltonian structures[END_REF]:

iw t + w xx -iw 2 w x + 1 2 |w| 4 w = 0. (1.22)
Moreover, (1.17) has some conservation laws in the energy space:

∥u(t)∥ 2 L 2 = ∥u 0 ∥ 2 L 2 , (1.23)
E(u(t)) = E(u 0 ), (1.24)
P (u(t)) = P (u 0 ), (1.25)
where

E(φ) = ∥φ x ∥ 2 L 2 - λ + µ 2 Im |φ| 2 φ, φ x - (λ + µ)µ 6 ∥φ∥ 6 L 6 - b 6 R |φ| 6 dx, and
P (u(t)) = Im R u x u dx + µ 2 R |u| 4 dx,

Local theory

In this section, we present some well-known results for the local theory of (1.17), some method used and our main goal on establishing local well-posedness of this kind equation.

Local theory of (1.17) has attracted a lot of interests in several years (see e.g [START_REF] Colliander | Global wellposedness for Schrödinger equations with derivative[END_REF][START_REF] Colliander | A rened global well-posedness result for Schrödinger equations with derivative[END_REF][START_REF] Hayashi | Well-posedness for a generalized derivative nonlinear Schrödinger equation[END_REF][START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF][START_REF] Hayashi | Finite energy solutions of nonlinear Schrödinger equations of derivative type[END_REF][START_REF] Takaoka | Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity[END_REF][START_REF] Takaoka | Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces[END_REF][START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem[END_REF][START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation[END_REF] and references therein). The main diculty is the appearance of the derivative term. We cannot use the classical contraction method for this type of nonlinear Schrödinger equations. Some methods were used to overcome this diculty. In [START_REF] Colliander | Global wellposedness for Schrödinger equations with derivative[END_REF][START_REF] Colliander | A rened global well-posedness result for Schrödinger equations with derivative[END_REF][START_REF] Takaoka | Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity[END_REF][START_REF] Takaoka | Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces[END_REF], the authors used the Fourier restriction method to established local well-posedness and global well-posedness results for (1.17). By using this method, we can directly use the contraction mapping theorem for the Duhamel form of equation (1.17) to obtain existence results. Another approach was used in [START_REF] Hayashi | Well-posedness for a generalized derivative nonlinear Schrödinger equation[END_REF][START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem[END_REF][START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation[END_REF] where the authors used an approximation argument to prove the existence of solutions. Another method was used in [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF][START_REF] Hayashi | Finite energy solutions of nonlinear Schrödinger equations of derivative type[END_REF][START_REF] Ozawa | On the nonlinear Schrödinger equations of derivative type[END_REF], where the authors used a Gauge transform to obtain a system of two equations without derivative nonlinearities from the original equation (1.17). More precisely, we set

φ(t, x) = exp i λ 2 x -∞ |u(t, y)| 2 dy u(t, x), ψ(t, x) = exp i λ 2 x -∞ |u(t, y)| 2 dy u x (t, x) + i µ 2 |u| 2 u(t, x) .
We observe that if u solves (1.17) then (φ, ψ) solves the following system

       iφ t + φ xx = i(λ -µ)φ 2 ψ -b|φ| 4 φ, iψ t + ψ xx = -i(λ -µ)ψ 2 φ -(λ-2µ)µ 4 (3|φ| 4 ψ + 2φ 3 φψ) -3b|φ| 4 ψ -2b|φ| 2 φ 2 ψ.
(1.26)

By denition, the functions φ and ψ satisfy the following relation

ψ = φ x -i λ -µ 2 |φ| 2 φ. (1.27)
The Cauchy problem of the system (1.26) is established by classical arguments. The main diculty in this method is to prove that the relation (1.27) is conserved under the ow of the system (1.26). When we prove this relation, the existence of solutions of (1.17) is implied by the existence of solutions of the system (1.26). The uniqueness and continuous dependence on initial data of solutions of (1.17) is obtained by the corresponding properties of solutions of the system. Recently, the inverse scattering transform (IST) was used to proved global well posed result in the case b = 0 of (1.17). In [START_REF] Jenkins | Global existence for the derivative nonlinear Schrödinger equation with arbitrary spectral singularities[END_REF], Jenkins-Liu-Perry-Sulem proved that for any initial data [START_REF] Pelinovsky | Existence of global solutions to the derivative NLS equation with the inverse scattering transform method[END_REF], Pelinovsky and Shimabukuro proved the global existence result of solutions of (1.21) in the space H 2 (R) ∩ H 1,1 (R). Finally, in [START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF], Bahouri and Perelman proved that the equation (1.21) is globally well posed in H 1 2 (R). Moreover, the authors proved that for any initial data in H 1 2 (R), the associated solution is uniformly bounded in time. This solves an open problem in long time.

u 0 ∈ H 2,2 (R) = {u ∈ H 2 (R), | • | 2 u(•) ∈ L 2 (R)} then the associated solution of (1.20) is global existence in H 2,2 (R). Moreover, in

Stability theory

In this section, we introduce the well known results on stability and instability of solitons of the equation (1.17).

Solitons

In the case λ ̸ = 0 or µ ̸ = 0, (1.17) has no Galilean invariance as in the case of simple power nonlinearity. Thus, the family of solitary waves has two parameters (frequency and speed) which make the studying of stability and instability is more dicult than the usual cases. Consider (1.17) in the case λ = 1 and µ = 0. The solitons of (1.17) are solutions of the form

R ω,c (t, x) = e iωt ϕ ω,c (x -ct), where ϕ ω,c ∈ H 1 (R). It is clear that ϕ ω,c solves -ϕ ′′ + ωϕ + icϕ ′ -i|ϕ| 2 ϕ ′ -b|ϕ| 4 ϕ = 0, x ∈ R.
As in [START_REF] Hayashi | Potential well theory for the derivative nonlinear Schrödinger equation[END_REF], we use the gauge transformation

ϕ ω,c (x) = Φ ω,c (x) exp ic 2 - i 4 x -∞
|Φ ω,c (y)| 2 dy .

Let γ = 1 + 16 3 b. The positive radial prole Φ ω,c obtained as follows: if γ > 0, Φ 2 ω,c (x) = 2(4ω-c 2 ) √ c 2 +γ(4ω-c 2 ) cosh( √ 4ω-c 2 x)-c if -2 √ ω < c < 2 √ ω, 4c (cx) 2 +γ if c = 2 √ ω, (1.28) if γ ⩽ 0, Φ 2 ω,c (x) = 2(4ω -c 2 ) c 2 + γ(4ω -c 2 ) cosh( √ 4ω -c 2 x) -c if -2 √ ω < c < -2s * √ ω, (1.29)
where

s * = -γ 1-γ .

On stability/instability of solitons

As we know, (1.17) has no Galilean invariance. We know that (1.17) has a two parameter family of solitary waves. In [START_REF] Colin | Stability of solitary waves for derivative nonlinear Schrödinger equation[END_REF], in the case b = 0, µ = 0, Colin and Ohta proved that the solitons are orbitally stable in the whole range of parameters values by using variational methods. In this case, in [START_REF] Kwon | Orbital stability of solitary waves for derivative nonlinear Schrödinger equation[END_REF], Kwon and Wu showed that the algebraic soliton u ω,2 √ ω is orbitally stable (up to scaling symmetry). In the case µ = 0, λ = 1 there are many works on the stability/instability of solitons of (1.17). In [START_REF] Ohta | Instability of solitary waves for nonlinear Schrödinger equations of derivative type[END_REF], in the case b > 0, Ohta proved there exists k = k(b) ∈ (0, 1) such that the solitons u ω,c of (1.17

) is stable if -2 √ ω < c < 2k √ ω and unstable if 2k √ ω < c < 2 √ ω.
The stability/instability of solitons in the case c = 2k √ ω is an open problem. In [START_REF] Hayashi | Potential well theory for the derivative nonlinear Schrödinger equation[END_REF], Hayashi showed a relation between the stability/instability of solitons and the positivity of momentum of the solitons. More precisely, if the momentum is positive then the solitons are stable and if the momentum is negative then the solitons are unstable. Moreover, the author proved that in the case b < 0, the momentum of solitons is positive, hence solitons are orbitally stable. Specially, in [START_REF] Ning | Instability of solitary wave solutions for derivative nonlinear Schrödinger equation in endpoint case[END_REF], Ning-Ohta-Wu showed that the algebraic soliton u ω,2 √ ω is unstable in the case b > 0 sucient small.

Multi-solitons theory

In this section, we present the multi-solitons theory.

A multi-soliton of a dispersive equation is a solution which behaves at large time like a nite or innite sum of solitons. Usually, in the Cauchy problem theory, when the mass of the initial data is small, the solution exists globally in time. The existence of multi-solitons shows that there also exists a global solution with arbitrary large mass. The main motivation of multi-solitons theory comes from the conjecture called soliton resolution conjecture. This conjecture states that all global solutions of a dispersive equation behave at large time like a sum of a radiative term and solitons. Thus, multi-solitons theory gives us more information about the long time behaviour of solutions.

In classical nonlinear Schrödinger equations, the existence of multi-solitons was showed in [START_REF] Côte | High-speed excited multi-solitons in nonlinear Schrödinger equations[END_REF][START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF][START_REF] Ferriere | Existence of multi-solitons for the focusing logarithmic non-linear Schrödinger equation[END_REF][START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF][START_REF] Martel | Multi solitary waves for nonlinear Schrödinger equations[END_REF][START_REF] Zakharov | Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear media[END_REF]. For focusing energy-critical nonlinear Schrödinger equation, Jendrej [START_REF] Jendrej | Construction of two-bubble solutions for the energy-critical NLS[END_REF] proved existence of pure two-bubles in space dimension N ⩾ 7. The main ingredient in [START_REF] Jendrej | Construction of two-bubble solutions for the energy-critical NLS[END_REF] is a uniformly bounded of a sequence of solutions and by taking a weak limit to obtain the desired solution. This argument goes back to the works Martel [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF], Merle [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF], Bellazzini-Ghimenti-Le Coz [START_REF] Bellazzini | Multi-solitary waves for the nonlinear Klein-Gordon equation[END_REF]. A similar argument was used to obtain the existence of two buble solutions for energy critical equations in dimension N = 6, see Jendrej [START_REF] Jendrej | Construction of two-bubble solutions for energy-critical wave equations[END_REF]. For the energy-critical focusing wave equation with spatial dimension N = 5, [START_REF] Jendrej | Construction of multi-bubble solutions for the energy-critical wave equation in dimension 5[END_REF] proved existence of multi-bubble solutions which blows up in innite time at any K given points, K ⩾ 2. For Klein-Gordon equations, see the works [START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF][START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF]. The stability of multi-solitons was shown for generalized Korteweg-de Vries equations and L 2 -subcritical nonlinear Schrödinger equations in [START_REF] Martel | Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations[END_REF][START_REF] Martel | Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations[END_REF]. In [START_REF] Coz | Stability of multisolitons for the derivative nonlinear Schrödinger equation[END_REF], Le Coz and Wu proved a stability result of multi-solitons of (1.17) in the case b = 0. In this thesis, we prove the existence of multi solitons of (1.17) for any value of b. First, we recall the denition of multi solitons.

Let K ∈ N * and (θ j , ω j , c j ) j=1,K be given parameters such that

ω j > c 2 j 4 , c j ̸ = c k for j ̸ = k. Let R ω j ,
c j be the soliton associated with the parameters ω j , c j for each j. A multi-soliton prole is dened by

R(t, x) = K j=1 e iθ j R ω j ,c j (t, x).
(1.30) Denition 1.29. A solution u of (1.17) is called a multi-soliton if it behaves like a multi-soliton prole at large time, i.e:

∥u(t) -R(t)∥ H 1 → 0 as t → ∞.
In the next part, we consider the following equation

iu t + u xx + iu 2 u x + b|u| 4 u = 0. (1.31)
Let R ω,c (t, x) be a solution of (1.31) of form e iωt ϕ ω,c (x -ct). Let Φ ω,c be the associated function dened by

Φ ω,c = exp -i c 2 x + i 4 x ∞ |ϕ ω,c (y)| 2 dy ϕ ω,c . (1.32)
We note that the prole Φ ω,c is well dened when Φ ω,c restricted on R -belongs to L 2 (R -). Thus, Φ ω,c does not need to belong to L 2 (R). In this thesis, we prove the existence of multi kink solitons of (1.31). Our motivation comes from the works [START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF][START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF] for classical nonlinear Schrödinger equations. Before stating the next result, we need the denition of multi kink solitons of (1.31).

Denition 1.30. The half kink solution R ω,c of (1.31) is a solution of (1.31) of the type e iωt ϕ ω,c (x -ct) where ϕ ω,c is such that the associated function

Φ ω,c dened in (1.32) veries            -Φ ′′ + ω -c 2 4 Φ -c 2 Φ 3 + 3 16 γΦ 5 = 0, lim x→∞ Φ(x) = 0, lim x→-∞ Φ(x) > 0, Φ is a real valued function. (1.33)
We have the following denition of multi-kink-soliton. Denition 1.31. Let K ∈ N * and R ω 0 ,c 0 be a half kink solution of (1.31). Let (θ j , c j , ω j ) j=0,..,K be given parameters. The multi-kink-soliton prole is dened by

V = K j=0 e iθ j R ω j ,c j .
(1.34)

A multi-kink-soliton of (1.31) is a solution u of (1.31) such that

∥u -V ∥ H 1 → 0, as t → ∞.
In the next part, we consider the following general derivative nonlinear Schrödinger equation:

iu t + u xx + i|u| 2σ u x = 0, u(0) = u 0 . (1.35)
The local existence and global existence of solutions of (1.35) were studied in many works (see e.g [START_REF] Hayashi | Well-posedness for a generalized derivative nonlinear Schrödinger equation[END_REF][START_REF] Santos | Existence and uniqueness of solution for a generalized nonlinear derivative Schrödinger equation[END_REF] and reference therein).

The equation (1.35) has a two parameters family of solitary waves dened as follows

R ω,c = φ ω,c (x -ct) exp i ωt + c 2 (x -ct) - 1 2σ + 2 x-ct -∞ φ 2σ ω,c (η) dη
where ω ⩾ c 2 4 and

φ ω,c (y) 2σ = (σ + 1)(4ω -c 2 ) 2 √ ω(cosh(σ √ 4ω -c 2 y) -c 2 √ ω )
.

In [START_REF] Liu | Stability of solitary waves for a generalized derivative nonlinear Schrödinger equation[END_REF], Liu-Simpson-Sulem showed that in the case σ ⩾ 2, the solitons of (1.35) are orbitally unstable; in the case 0 < σ < 1 they are orbitally stable. In the case σ ∈ (1, 2), the situation is more complex. The authors proved that there exists z 0 ∈ (-1, 1) such that if c < 2z 0 √ ω then the soliton is orbitally stable and if c > 2z 0 √ ω the soliton is orbitally unstable. In [START_REF] Guo | Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the critical frequency case[END_REF], in the case 1 < σ < 2, the authors proved that the soliton is unstable in the critical frequency case i.e c = 2z 0 √ ω. The multi-soliton prole and multi-solitons of (1.35) are dened similarly as the ones of (1.17). The stability of multi-solitons of (1.35) was obtained in [START_REF] Tang | Stability of the sum of two solitary waves for (gDNLS) in the energy space[END_REF] in the case 1 < σ < 2.

Our main results

In this section, we present the main results of this thesis.

Local theory

All well known results on the Cauchy problem of (1.17) are established on the usual Sobolev spaces H s (R). To our knowledge, there is no result for a local theory of (1.17) under nonvanishing boundary conditions. One of our goals in this thesis is to study the Cauchy problem of (1.17) under nonvanishing boundary conditions. Our main results are the following.

Theorem 1.32. Let X k (R) be the Zhidkov space dened in (1.12). Consider the following special case of (1.17)

iu t + u xx = -iu 2 u x .
(1.36)

The equation (1.36) is locally well-posed in X 4 (R) and ϕ+H 2 (R) for any ϕ ∈ X 4 (R). Moreover, if ∥ϕ x ∥ L 2 and ∥u 0 -ϕ∥ H 1 are small enough then there exist T > 0 and unique solution (1.36). Moreover, all non-vanishing stationary solutions of (1.36) in X 1 (R) are constant functions or functions of form e iθ √ k, where

u ∈ ϕ+C([-T, T ], H 1 (R))∩L 4 ([-T, T ], W 1,∞ ) of
k(x) = 2 √ B + -1 5 72B cosh(2 √ B(x -x 0 )) + 5 12 √ B , θ = θ 0 - ∞ x B k(y) - k(y) 4 dy,
for some constants θ 0 , x 0 ∈ R, B > 0.

To study the Cauchy problem of (1.36), we use the idea in Hayashi-Ozawa [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF][START_REF] Hayashi | Finite energy solutions of nonlinear Schrödinger equations of derivative type[END_REF][START_REF] Ozawa | On the nonlinear Schrödinger equations of derivative type[END_REF]. Set

v = u x + i 2 |u| 2 u. (1.37)
If u solves (1.36) then (u, v) solves the following system

Lu = -iu 2 v + 1 2 |u| 4 u, Lv = iuv 2 + 3 2 |u| 4 v + u 2 |u| 2 v, (1.38)
where L = i∂ t +∂ xx is the Schrödinger operator. We establish the local well posedness of solutions of (1.38) in spaces X 4 (R), ϕ + H 2 (R) and ϕ + H 1 (R) with restrictions on ϕ. Moreover, we prove that the relation (1.37) is conserved. The existence of solutions of (1.36) in X 4 (R) is obtained by the following argument. Let

u 0 ∈ X 4 . Set v 0 = u 0x + i 2 |u 0 | 2 u 0 . Let (u, v)
be the corresponding solution of (1.38) with the initial data (u 0 , v 0 ). We may prove that v = u x + i 2 |u| 2 u in the time interval of existence of solutions. Thus, u solves

Lu = -iu 2 u x - i 2 |u| 2 u + 1 2 |u| 4 u = -iu 2 u x .
This implies the existence in X 4 (R) of solutions of (1.36). The uniqueness and other properties of X 4 (R) solutions of (1.36) follow from the corresponding properties of the associated solutions of (1.38). The proof of local well posedness of (1.36) is completed. Similarly, we have the local well posedness of (1.36) in the cases ϕ + H 2 (R) and ϕ + H 1 (R).

To prove the uniqueness of stationary solution of (1.36) in X 1 (R), we use the suitable changes of variables as in [START_REF] Murai | Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition[END_REF]. More precisely, let ϕ be a nonvanishing stationary solution of (1.36) in X 1 (R). Then we may write ϕ as

ϕ(x) = k(x)e iθ(x) ,
where k, θ ∈ C 2 (R) and k > 0. We prove that θ and k satisfy the following

θ x = B k - k 4 , (1.39
)

0 = k xx 2 - 5 12 k 3 + 3Bk -2a, (1.40)
for some B > 0, a ∈ R. Since the relation (1.39), we may obtain the formulation of θ by the formulation of R. Moreover,

k satises k -2 √ B ∈ H 3 (R). Combining to (1.40), we have a = 4B √ B 3 . Setting h = k -2 √ B, we have h ∈ H 3 (R) and 0 = h xx - 5 6 h 3 -5 √ Bh 2 -4Bh.
By a classical argument, we may obtain the explicit formulation of h and thus the explicit formulation of k. This completes the proof of Theorem 1.32. For detail discussion, we refer reader to Chap 2.

Stability theory

In this section, we consider (1.17) on the half line with Robin boundary condition at 0:

     iv t + v xx = i 2 |v| 2 v x -i 2 v 2 v x -3 16 |v| 4 v, ∀x ∈ R + , v(0, x) = v 0 (x), v x (t, 0) = αv(t, 0), ∀t ∈ R, (1.41)
where α ∈ R is a given constant.

The equation (1.41) has a standing wave of the form e iωt φ ω (x), where ω > α 2 and

φ ω (x) = 2 4 √ ω sech 1 2 2 √ ω|x| + tanh -1 -α √ ω . (1.42)
The linear part of (1.41) can be written as follows

iv t + Hα v = 0, v(0) = v 0 ,
where Hα is the self adjoint operator which is dened by

Hα : D( Hα ) ⊂ L 2 (R + ) → L 2 (R + ), Hα v = v xx , D( Hα ) = v ∈ H 2 (R + ) : v x (0) = αv(0) .
The equation (1.41) in Duhamel form is the following

v(t) = e i Hαt v 0 -i t 0 e i Hα(t-s) g(v(s)) ds, (1.43)
where

g(v) = i 2 |v| 2 v x - i 2 v 2 v x - 3 16 |v| 4 v.
It turns out that the self adjoint operator Hα has a relation with the following delta potential Schrödinger operator on the whole line

H γ : D(H γ ) ⊂ L 2 (R) → L 2 (R), H γ u = u xx , D(H γ ) = u ∈ H 2 (R \ 0) ∩ H 1 (R), u x (0 + ) -u x (0 -) = γu(0) .
More precisely, the operator Hα can be seen as the restriction of the operator H 2α on even functions and we have

e i Hαt φ = e iH 2α t φ | R + , (1.44)
where φ ∈ D( Hα ) and φ is the even function on R whose restriction on R + is φ. It is well known that the operator e iH 2α t is bounded on H 1 (R) (see e.g [START_REF] Ianni | On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation[END_REF]). It implies that the operator e i Hαt is bounded on H 1 (R + ). We assume that (1.41) is locally well posed on H 1 (R + ). By formal calculation, we show that (1.41) has two conservation laws: conservation of the mass and the energy. In this thesis, we use these tools to study the dynamics of (1.41). Our main results are the following.

Theorem 1.33. Let α > 0 and v 0 ∈ Σ = v ∈ D( Hα ), xv ∈ L 2 (R + ) . If the energy of v 0 be negative, then the associated H 1 (R + ) solution of (1.36) blows up in nite time.

Let ω > α 2 and e iωt φ ω be the standing wave of (1.36). If α < 0 then the standing wave is orbitally stable. If α ⩾ 0 then the standing wave is unstable by blow up.

To prove the existence of blow up solutions, we use a similar arguments as in Glassey [START_REF] Glassey | On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations[END_REF]. Dene

u(t, x) = v(t, x) exp - i 4 ∞ x |v(t, y)| 2 , I(t) = R + x 2 |v(t)| 2 dx = R + x 2 |u(t)| 2 dx, J(t) = Im R + xu x u dx.
By a direct calculation, we have

∂ t I(t) = 4J(t) - R + x|u(t)| 4 dx ⩽ 4J(t), ∂ t J(t) = 2 R + |u x | 2 -Im R + |u| 2 u x u dx + α|u(t, 0)| 2 = 4E(v) -α|v(t, 0)| 2 ⩽ 4E(v) = 4E(v 0 ).
Here, we use the condition α ⩾ 0. Thus, it is easy to show that

I(t) ⩽ I(0) + 4J(0)t + 8E(v 0 )t 2 .
This implies that the time of existence must be nite. Let e iωt φ ω be the standing wave of (1.41) dened in (1.42). To prove the stability of standing waves in the case α < 0, we use variational techniques. Set

S ω (v) = 1 2 ∥v x ∥ 2 L 2 (R + ) + ω∥v∥ 2 L 2 (R + ) + α|v(0)| 2 - 1 32 ∥v∥ 6 L 6 (R + ) , (1.45)
K ω (v) = ∥v x ∥ 2 L 2 (R + ) + ω∥v∥ 2 L 2 (R + ) + α|v(0)| 2 - 3 16 ∥v∥ 6 L 6 (R + ) , (1.46)
N (v) = 3S ω (v) - 3 2 K ω (v), (1.47)
d(ω) = inf{S ω (v)|v ∈ H 1 (R + ) \ 0, K ω (v) = 0}. (1.48)
First, we prove the following compactness result:

If (v n ) ⊂ H 1 (R + ) satisfy S ω (v n ) → d(ω), K ω (v n ) → 0,
then there exists a constant θ 0 ∈ R such that v n → e iθ 0 φ ω , where φ ω is the standing wave prole. Next, we prove that under the assumption α < 0, if

(v n ) ⊂ H 1 (R + ) satises ∥v n -φ ω ∥ H 1 (R + ) → 0 as n → ∞,
then the corresponding solution v(t) of (1.41) satises N (v n (t)) → 3d(ω) as n → ∞.

Combining the above tools, we conclude the stability of standing waves in the case α < 0 by a contradiction argument.

To prove the instability by blow up of the standing waves in the case α ⩾ 0, we may use a similar argument as in [START_REF] Coz | Standing waves in nonlinear Schrödinger equations[END_REF]. The case α = 0 is not dicult. We consider the case α > 0. Dene

V = {v ∈ H 1 (R + ) : K ω (v) < 0, S ω (v) < d(ω), P (v) < 0}, where P (v) = ∂ ∂ λ S(v λ )| λ=1 = ∥v x ∥ 2 L 2 (R + ) -1 16 ∥v∥ 6 L 6 (R + ) + α 2 |v(0)| 2 .
We prove that V is invariant under ow of (1.41). Moreover, we prove that if φ ∈ V is such that | • |φ(•) ∈ L 2 (R + ) then the corresponding solution v of (1.41) blows up in nite time on H 1 (R + ). Thus, to conclude the instability by blow up of standing waves, we only need to construct a sequence

φ n → φ in H 1 (R + ) such that φ n ∈ V and | • |φ n (•) ∈ L 2 (R +
) for each n. This sequence can be obtained from a scaling of φ ω . Thus, we complete the proof of Theorem 1.33. For more details, we refer the reader to Chapter 3.

Multi solitons theory

In [START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF], Le Coz-Li-Tsai proved the existence and uniqueness of multi solitons for classical nonlinear Schrödinger equations by using xed point arguments around the desired prole. We cannot directly apply this argument to obtain similar results for derivative monlinear Schrödinger equations, because of the presence of derivatives in the nonlinearities. In this thesis, we improve the method of Le Coz-Li-Tsai [START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF] to obtain the existence of multi-solitons solutions of derivative nonlinear Schrödinger equations under an implicit condition on the parameters. Our rst main result is the following. Theorem 1.34. Considering (1.17), we assume that λ = 1, µ = 0. Let (θ j , c j , ω j ) be sequence of parameters such that -2 √ ω j < c j < 2 √ ω j if γ > 0 and -2 √ ω j < c j < -2s * √ ω j if γ ⩽ 0, where γ = 1 + 16 3 b and s * = -γ 1-γ . Let R be the multi-soliton prole dened in (1.30). Then there exists a certain positive constant C * such that if the parameters (ω j , c j ) satisfy

C * (1 + ∥R x ∥ L ∞ t L ∞ x )(1 + ∥R∥ L ∞ t L ∞ x ) + ∥R∥ 4 L ∞ t L ∞ x ⩽ v * := inf j̸ =k h j |c j -c k |, (1.49)
where h j = 4ω j -c 2 j , then there exist T 0 > 0 depending on ω 1 , ..., ω K , c 1 , ..., c K and a solution u of (1.17) on [T 0 , ∞) such that

∥u -R∥ H 1 ⩽ Ce -v * 16 t , ∀t ⩾ T 0 ,
where λ = v * 16 and C is a positive constant depending on the parameters ω 1 , ..., ω K , c 1 , ..., c K .

We note that the condition (1.49) ensures that c i ̸ = c j for i ̸ = j. Thus, the solitons are separated at large time.

Let us sketch the proof of the above theorem. Dene

φ(t, x) = exp i 2 -∞ |u(t, y)| 2 dy u(t, x), ψ = φ x - i 2 |φ| 2 φ.
We observe that if u solves (1.17) then (φ, ψ) solves a system of the form

     Lφ = P (φ, ψ), Lψ = Q(φ, ψ), ψ = ∂ x φ -i 2 (|φ| 2 φ), (1.50)
where L = i∂ t + ∂ xx and P, Q are polynomials of variables φ, ψ and their conjugates.

Let R be the multi-soliton prole and q = u -R. Then R solves

LR + i|R| 2 R x + b|R| 4 R = e -λt v(t, x),
where λ = v * 16 and ∥v∥

L ∞ t H 2 x is bounded. Dene h(t, x) = exp i 2 x -∞ |R(t, y)| 2 dy R(t, x), k = h x - i 2 |h| 2 h.
We prove that (h, k) solves

Lh = P (h, k) + e -λt m(t, x), Lk = Q(h, k) + e -λt n(t, x),
where m, n satisfy ∥m∥

L ∞ t H 1 x + ∥n∥ L ∞ t H 1 x bounded. Let φ = φ -h and ψ = ψ -k. Then ψ = φx - i 2 (| φ + h| 2 ( φ + h) -|h| 2 h), (1.51)
and ( φ, ψ) solves

L φ = P ( φ, ψ) -P (h, k) -e -λt m(t, x), L ψ = Q( φ, ψ) -Q(h, k) -e -λt n(t, x). (1.52)
We construct the solution of (1.52) by similar arguments as in [START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF]Proposition 3.1].

The relation (1.51) is proved by using the exponential decay in time of solutions of (1.52) and the assumption (1.49). This implies that the prole (φ, ψ) solves (1.50). Then, by setting

u(t, x) = exp - i 2 x -∞ |φ(t, y)| 2 dy φ(t, x).
we obtain a solution u of (1.17) which satises the desired property. Next, consider the equation (1.31). In [START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF], Le Coz-Li-Tsai have successfully proved the existence of multi-kink-soliton solutions of classical nonlinear Schrödinger equations. In this thesis, we use a similar method as in the proof of Theorem 1.34 to prove the existence of multi-kink-soliton solutions for the equation (1.31). Our result is the following. Theorem 1.35. Consider (1.31). We assume that b < 5 16 . Let (θ j , ω j , c j ) j=0,...,K be parameters such that 2

√ γ < c 0 < 2 √ ω 0 , 2 √ ω j < c j > 2s * √ ω j , where γ = 5 3 -16 3 b
and s * = γ 1+γ . Let V be given as in (1.34). There exists a certain positive constant C * such that if the parameters (ω j , c j ) satisfy

C * 1 + ∥V x ∥ L ∞ t L ∞ x 1 + ∥V ∥ L ∞ t L ∞ x + ∥V ∥ 4 L ∞ t L ∞ x ⩽ v * := min inf j̸ =k h j |c j -c k |, inf j̸ =0 |c j -c 0 | ,
where h j = 4ω j -c 2 j , then there exist a solution u to (1.31) such that

∥u -V ∥ H 1 ⩽ Ce -λt . ∀t ⩾ T 0 ,
where λ = v * 16 and C, T 0 are positive constants depending on the parameters ω 0 , ..., ω K , c 0 , ..., c K .

For details, we refer to Chapter 4. Consider the equation (1.35). The stability of multi solitons of (1.35) has been studied in [START_REF] Coz | Stability of multisolitons for the derivative nonlinear Schrödinger equation[END_REF] for σ = 1 and in [START_REF] Tang | Stability of the sum of two solitary waves for (gDNLS) in the energy space[END_REF] for σ ∈ (1, 2). In this thesis, we give the proof of existence of multi solitons in the cases σ = 1 or σ = 2 or σ ⩾ 5 2 . We have the following result. Theorem 1.36. Let σ ⩾ 5 2 or σ = 1 or σ = 2. Let (θ j , ω j , c j ) be parameters such that θ j ∈ R and ω j > c 2 j 4 and R be the multi-soliton prole of (1.35) dened similarly to the one of the equation (1.17). There exists a certain positive constant C * such that if the parameters (ω j , c j ) satisfy

C * (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 )(1 + ∥R x ∥ L ∞ L ∞ + ∥R∥ 2σ+1 L ∞ L ∞ ) ⩽ v * := inf j̸ =k h j |c j -c k |,
where h j = 4ω j -c 2 j , then there exist a solution u to (1.31) such that

∥u -R∥ H 1 ⩽ Ce -λt . ∀t ⩾ T 0 ,
where λ = v * 16 and C, T 0 are positive constants depending on the parameters ω 0 , ..., ω K , c 0 , ..., c K .

Our method is similar to the one in the case of equation (1.17). In the proof of Theorem 1.36, we use the following inequality

(a + b) 2(σ-2) -a 2(σ-2) ≲ b 2(σ-2) + ba 2(σ-2)-1 , ∀a, b ⩾ 0.
The condition σ ⩾ 5 2 ensures that the order of b on the right hand sight of the above inequality is larger than 1. This is an important point to close xed point argument. For details, we refer to Chapter 5.

Instability of algebraic standing waves

This section is to present our work on the triple power nonlinear Schrödinger equation:

iu t + ∆u + a 1 |u|u + a 2 |u| 2 u + a 3 |u| 3 u = 0, (t, x) ∈ R × R n , (1.53)
where a 1 , a 2 , a 3 ∈ R and n ∈ {1, 2, 3}.

In [START_REF] Liu | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF], Liu-Tsai-Zwiers studied a 1D triple power nonlinear Schrödinger equation. In particular, the authors presented a picture which shows regions of existence, stability and instability of standing waves with positive frequency. In [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF], the authors proved the instability of standing waves with zero frequency i.e algebraic standing waves. Our goal is to study the existence and stability of algebraic standing waves of triple power nonlinear Schrödinger equations. Before stating our main result, we recall the denition of algebraic standing waves for (1.53).

A standing wave of (1.53) is a solution of form e iωt ϕ ω (x). In this thesis, we are interested in the case of frequency equals to zero i.e ω = 0. The prole ϕ 0 , which we prefer to denote by ϕ, solves the following equation:

∆ϕ + a 1 |ϕ|ϕ + a 2 |ϕ| 2 ϕ + a 3 |ϕ| 3 ϕ = 0.
(1.54)

The equation (1.54) admits a unique radial positive solution. This solution is algebraically decaying in space. Moreover, it is a minimizer of a variational problem. Our main goal is to prove orbital instability of this solution. We have the following result.

Theorem 1.37. Let ϕ be the radial positive solution of (1.54) and a 1 = -1, a 3 = 1 and a 2 small when a 2 > 0. The algebraic standing wave ϕ of (1.53) is orbitally unstable in H 1 (R).

The above theorem is a direct consequence of the following result.

Proposition 1.38. Assume the assumptions of Theorem 1.37 and

∂ 2 λ S(ϕ λ )| λ=1 < 0, where v λ (x) = λ N 2 v(λx).
Then the algebraic standing wave ϕ is unstable.

Dene

N ε = v ∈ H 1 (R N ) : inf (θ,y)∈R×R N ∥v -e iθ ϕ(• -y)∥ H 1 < ε .
Let u 0 ∈ N ε and u(t) be the corresponding solution of (1.53). We dene the exit time from the tube N ε by

T ± ε (u 0 ) := inf{t > 0 : u(±t) / ∈ N ε }. Set I ε := (-T - ε (u 0 ), T + ε (u 0 )).
Then, I ε is the maximal interval for which the solution stays in N ε . Thus, to prove the instability of ϕ, we show that there exists ε > 0 such that there exists a sequence (ϕ n ) satisfying ∥ϕ n -ϕ∥ H 1 → 0 as n → ∞ and |I ε (ϕ n )| < ∞ for all n. The conclusion of Theorem 1.37 is proved by the following result.

Proposition 1.39. There exists ε > 0 such that for all u 0 ∈ N ε such that P (u 0 ) < 0,

S(u 0 ) < µ and | • |u 0 (•) ∈ L 2 (R), we have |I ε (u 0 )| < ∞, where S(v) = 1 2 ∥∇v∥ 2 L 2 + 1 3 ∥v 3 ∥ L 3 - a 2 4 ∥v∥ 4 L 4 - 1 5 ∥v∥ 5 L 5 , P (v) = ∂ λ S(v λ )| λ=1 = ∥∇v∥ 2 L 2 + N 6 ∥v∥ 3 L 3 - N a 2 4 ∥v∥ 4 L 4 - 3N 10 ∥v∥ 5 L 5 K(v) = ∥∇v∥ 2 L 2 + ∥v∥ 3 L 3 -a 2 ∥v∥ 4 L 4 -∥v∥ 5 L 5 µ = inf S(v) : v ∈ Ḣ3 (R N ) ∩ L 3 (R N ) \ {0}, K(v) = 0 .
The existence of the desired sequence follows by using a suitable scaling of ϕ. For details, we refer to Chapter 6. In this chapter, we are interested in the Cauchy problem for the following derivative nonlinear Schödinger equation with nonvanishing boundary conditions:

i∂ t u + ∂ 2 u = -iu 2 ∂u, u(0) = u 0 , (2.1)
where u : R t × R x → C, ∂ = ∂ x denotes derivative in space and ∂ t denotes derivative in time.

Our attention was drawn to this equation by the work of Hayashi and Ozawa [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF] concerning the more general nonlinear Schrödinger equation

i∂ t u + ∂ 2 u = iλ|u| 2 ∂u + iµu 2 ∂u + f (u), u(0) = u 0 .
(2.2)

When λ = 0, µ = -1, f ≡ 0, then (2.
2) reduces to (2.1). This type of equation is usually refered to as derivative nonlinear Schrödinger equations. It may appear in various areas of physics, e.g. in Plasma Physics for the propagation of Alfvén waves [START_REF] Mjølhus | On the modulational instability of hydromagnetic waves parallel to the magnetic eld[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation[END_REF]. Under Dirichlet boundary conditions in space, the Cauchy problem for (2.1) has been solved in [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF]: local well-posedness holds in H 1 (R), i.e. for any u 0 ∈ H 1 (R) there exists a unique solution u ∈ C(I, H 1 (R)) of (2.1) on a maximal interval of time I. Moreover, we have continuous dependence with respect to the initial data, blowup at the ends of the time interval of existence I if I is bounded and conservation of energy, mass and momentum.

The main diculty is the appearance of the derivative term -iu 2 u x . We cannot use the classical contraction method for this type of nonlinear Schrödinger equations.

In [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF] Hayashi and Ozawa use the Gauge transform to establish the equivalence of the local well-posedness between the equation (2.2) and a system of equations without derivative terms. By studying the Cauchy problem for this system, they obtain the associated results for (2.2). In [START_REF] Hayashi | Well-posedness for a generalized derivative nonlinear Schrödinger equation[END_REF], Hayashi and Ozawa construct a sequence of solutions of approximated equations and prove that this sequence is converging to a solution of (2.2), obtaining this way the local well-posedness of (2.2). The approximation method has also been used by Tsutsumi and Fukuda in [START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem[END_REF][START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation[END_REF]. The dierence between [START_REF] Hayashi | Well-posedness for a generalized derivative nonlinear Schrödinger equation[END_REF] and [START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem[END_REF][START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation[END_REF] lies in the way of constructing the approximate equation. In [START_REF] Hayashi | Well-posedness for a generalized derivative nonlinear Schrödinger equation[END_REF], the authors use approximation on the non-linear term, whereas in [START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem[END_REF][START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation[END_REF] the authors use approximation on the linear operator.

To our knowledge, the Cauchy problem for (2.1) has not been studied under non-zero boundary conditions, and our goal in this paper is to initiate this study. Note that non-zero boundary conditions on the whole space are much rarely considered in the literature around nonlinear dispersive equations than Dirichlet boundary conditions. In the case of the nonlinear Schrödinger equation with power-type nonlinearity, we refer to the works of Gérard [START_REF] Gérard | The Cauchy problem for the Gross-Pitaevskii equation[END_REF][START_REF] Gérard | The Gross-Pitaevskii equation in the energy space[END_REF] for local well-posedness in the energy space and to the works of Gallo [START_REF] Gallo | Schrödinger group on Zhidkov spaces[END_REF] and Zhidkov [START_REF] Zhidkov | Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory[END_REF] for local well-posedness in Zhidkov spaces (see Section 2.2.1 for the denition of Zhidkov spaces) and Gallo [START_REF] Gallo | The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at innity[END_REF] for local well-posedness in u 0 + H 1 (R). In this paper, using the method of Hayashi and Ozawa as in [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF] on the Zhidkov-space X k (R), (k ⩾ 4) and in the space ϕ + H k (R) (k = 1, 2) for ϕ in a Zhidkov space, we obtain the existence, uniqueness and continuous dependence on the initial data of solutions of (2.1) in these spaces. Using the transform

v = ∂u + i 2 |u| 2 u, (2.3)
we see that if u is a solution of (2.1) then (u, v) is a solution of a system of two equations without derivative terms. It is easy to obtain the local wellposedness of this system on Zhidkov spaces. The main diculty is how to obtain a solution of (2.1) from a solution of the system. Actually, we must prove that the relation (2.3) is conserved in time. The main dierence in our setting with the setting in [START_REF] Hayashi | Finite energy solutions of nonlinear Schrödinger equations of derivative type[END_REF] is that we work on Zhidkov spaces instead of the space of localized functions H 1 (R).

Our rst main result is the following.

Theorem 2.1. Let u 0 ∈ X 4 (R). Then there exists a unique maximal solution of

(2.1) u ∈ C((T min , T max ), X 4 (R)) ∩ C 1 ((T min , T max ), X 2 (R)).
Moreover, u satises the two following properties.

Blow-up alternative. If

T max < ∞ (resp. T min > -∞) then lim t→Tmax(resp. T min ) ∥u(t)∥ X 2 = ∞.
Continuity with respect to the initial data. If

u n 0 ∈ X 4 (R) is such that u n 0 → u 0 in X 4 (R) then for any subinterval [T 1 , T 2 ] ⊂ (T min , T max ) the associated solutions of equation (2.1) (u n ) satisfy lim n→∞ ∥u n -u∥ L ∞ ([T 1 ,T 2 ],X 4 ) = 0.
To obtain the local wellposedness on ϕ + H k (R) for ϕ in Zhidkov spaces X l (R). First, we use the transform

v = ∂u + i 2 |u| 2 u. We see that if u ∈ ϕ + H k (R) then v ∈ i 2 |ϕ| 2 ϕ + H k-1 (R)
. This motivates us to dene ũ = u -ϕ and ṽ = v -i 2 |ϕ| 2 ϕ. We have

ṽ = ∂ ũ + i 2 (|ũ + ϕ| 2 (ũ + ϕ) -|ϕ| 2 ϕ) + ∂ϕ. (2.4)
We see that if u is a solution of (2.1) then (ũ, ṽ) is a solution of a system of two equations without the derivative terms. For technical reasons, we will need some regularity on ϕ. With a solution of the system in hand, we want to obtain a solution of (2.1). In practice, we need to prove that the relation (2.4) is conserved in time.

Our main second result is the following. Moreover u satises the following properties.

(1) Blow-up alternative:

If T max < ∞ (resp. T min > -∞ then lim t→Tmax(resp. T min ) (∥u(t) -ϕ∥ H 2 (R) ) = ∞.
(2) Continuous dependence on initial data:

If (u n 0 ) ⊂ ϕ + H 2 (R) is such that ∥u n 0 - u 0 ∥ H 2 → 0 as n → ∞ then for all [T 1 , T 2 ] ⊂ (T min , T max ) the associated solutions (u n ) of (2.1) satisfy lim n→∞ ∥u n -u∥ L ∞ ([T 1 ,T 2 ],H 2 ) = 0.
In the less regular space ϕ + H 1 (R), we obtain the local well posedness under a smallness condition on the initial data. Our third main result is the following.

Theorem 2.3. Let ϕ ∈ X 4 (R) such that ∥∂ϕ∥ L 2 is small enough, u 0 ∈ ϕ + H 1 (R)
such that ∥u 0 -ϕ∥ H 1 (R) is small enough. There exist T > 0 and a unique solution u of (2.1) such that

u -ϕ ∈ C([-T, T ], H 1 (R)) ∩ L 4 ([-T, T ], W 1,∞ (R)).
In the proof of Theorem 2.3, the main dierence with the case ϕ+H 2 (R) is that we use Strichartz estimates to prove the contractivity of a map on

L ∞ ([-T, T ], L 2 (R))∩ L 4 ([-T, T ], L ∞ (R)).
In the case of a general nonlinear term (as in (2.2)), our method is not working. The main reason is that we do not have a proper transform to give a system without derivative terms. Moreover, our method is not working if the initial data lies on X 1 (R). It is because when we study the system of equations, we would have to study it on L ∞ (R), but we know that the Schrödinger group is not bounded from L ∞ (R) to L ∞ (R). Thus, the local wellposedness on less regular spaces is a dicult problem for nonlinear derivative Schrödinger equations.

To prove the conservation laws of (2.1), we need to use a localizing function, which is necessary for integrals to be well dened. Indeed, to obtain the conservation of the energy, using (2.1), at least formally, we have

∂ t (|∂u| 2 ) = ∂ x (F (u)) + ∂ t (G(u)),
for functions F and G which will be dened later. The important thing is that when u is not in H 1 (R), there are some terms in G(u) which do not belong to L 1 (R), hence, it is impossible to integrate the two sides as in the usual case. However, we can use a localizing function to deal with this problem. Similarly, we use the localizing function to prove the conservation of the mass and the momentum. The localizing function χ is dened as follows χ ∈ C 1 (R) and even , suppχ ⊂ [-2, 2], and χ = 1 on [-1, 1].

(2.5)

For all a ∈ R and R > 0, we dene

χ a,R (x) = χ x -a R = χ |x -a| R . (2.6)
To prove the conservation of mass, we use the similar notations as in [34, section 7]

m + (u) = inf a∈R lim sup R→∞ R (|u| 2 -q 2 0 )χ a,R dx, m -(u) = sup a∈R lim inf R→∞ R (|u| 2 -q 2 0 )χ a,R dx.
If u is such that m + (u) = m -(u) we dene generalized mass as

m(u) ≡ m + (u) = m -(u).
Especially, for a = 0 we dene

χ R (x) = χ x R . (2.7)
Our fourth main result is the following.

Theorem 2.4. Let q 0 ∈ R be a constant and u 0 ∈ q 0 +H 2 (R) and u ∈ C((T min , T max ), q 0 + H 2 (R)) be the associated solution of (2.1) given by Theorem 2.2. Then, we have

E(u) := R |∂u| 2 dx + 1 2 Im R (|u| 2 u -q 3 0 )∂u dx + 1 6 R (|u| 2 -|q 0 | 2 ) 2 (|u| 2 + 2|q 0 | 2 ) dx = E(u 0 ), (2.8)
P (u) := 1 2 Im R (u -q 0 )∂u dx - R 1 4 (|u| 2 -|q 0 | 2 ) 2 dx = P (u 0 ), (2.9)
for all t ∈ (T min , T max ). Moreover,

u satises m + (u(t)) = m + (u 0 ) (respectively m -(u(t)) = m -(u 0 )).
In particular, if u 0 has nite generalized mass then the generalized mass is conserved by the ow, that is m(u(t)) = m(u 0 ).

Remark 2.5. When q 0 = 0, we recover the classical conservation of mass, energy and momentum as usually dened.

In the classical Schrödinger equation, there are special solutions which are called standing waves. There are many works on standing waves (see e.g [START_REF] Coz | Standing waves in nonlinear Schrödinger equations[END_REF], [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and the references therein). In [START_REF] Zhidkov | Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory[END_REF], Zhidkov shows that there are two types of bounded solitary waves possessing limits as x → ±∞. These are monotone solutions and solutions which have precisely one extreme point. They are called kinks and solitonlike solutions, respectively. In [START_REF] Zhidkov | Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory[END_REF], Zhidkov studied the stability of kinks of classical Schrödinger equations. In [START_REF] Béthuel | Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation[END_REF], the authors have studied the stability of kinks in the energy space. To our knowledge, all these solitary waves are in Zhidkov spaces i.e the Zhidkov space is largest space we know to nd special solutions. We want to investigate stationary solutions of (2.1) in Zhidkov spaces. Before stating the next main result, we need the following denition:

Denition 2.6. The stationary solutions of (2.1) are functions ϕ ∈ X 2 (R) satisfying

ϕ xx + iϕ 2 ϕ x = 0. (2.10)
In [START_REF] Murai | Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition[END_REF], the authors proved the existence of periodic traveling waves of a derivative nonlinear Schrödinger equation using a skillful changes of variables. In this paper, we use a similar changes of variables as in [START_REF] Murai | Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition[END_REF] to prove the existence and uniqueness of stationary solution of (2.80) on X 2 (R). Our fth main result is the following. Theorem 2.7. Let ϕ be a stationary solution of (2.1) (see Denition 2.6). The followings is true:

(1) If ϕ is not a constant function and satises

inf x∈R |ϕ(x)| > 0 then ϕ is of the form e iθ √ k where k(x) = 2 √ B + -1 5 72B cosh(2 √ B(x -x 0 )) + 5 12 √ B , θ = θ 0 - ∞ x B k(y) - k(y) 4 dy,
for some constants θ 0 , x 0 ∈ R, B > 0.

(2) If ϕ is a stationary solution of (2.1) such that ϕ(∞) = 0 then ϕ ≡ 0 on R.

Remark 2.8. We have classied stationary solutions of (2.1) for the functions which are vanishing at innity, and for the functions which are not vanishing on R. One question still unanswered is the class of stationary solutions of (2.1) vanishing at a point in R.

This paper is organized as follows. In Section 2.2, we give the proof of local well posedness of solution of (2.1) on Zhidkov spaces. In Section 2.3, we prove the local well posedness on ϕ + H 2 (R) and ϕ + H 1 (R), for ϕ ∈ X 4 (R) a given function. In Section 2.4, we give the proof of conservation laws when the initial data is in q 0 + H 2 (R), for a given constant q 0 ∈ R. Finally, in Section 2.5, we have some results on stationary solutions of (2.1) on Zhidkov spaces.

Notation. In this paper, we will use in the following notation L for the linear part of the Schrödinger equation, that is

L = i∂ t + ∂ 2 .
Moreover, C denotes various positive constants and C(R) denotes the constant depending on R.

Local existence in Zhidkov spaces

In this section, we give the proof of Theorem 2.1.

Preliminaries on Zhidkov spaces

Before presenting our main results, we give some preliminaries. We start by recalling the denition of Zhidkov spaces, which were introduced by Peter Zhidkov in his pioneering works on Schrödinger equations with non-zero boundary conditions (see [START_REF] Zhidkov | Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory[END_REF] and the references therein). Denition 2.9.

Let k ∈ N, k ⩾ 1. The Zhidkov space X k (R) is dened by X k (R) = {u ∈ L ∞ (R) : ∂u ∈ H k-1 (R)}.
It is a Banach space when endowed with the norm

∥•∥ X k = ∥•∥ L ∞ + k α=1 ∥∂ α •∥ L 2 .
It was proved by Gallo [40, Theorem 3.1 and Theorem 3.2] that the Schrödinger operator denes a group on Zhidkov spaces. More precisely, we have the following result.

Proposition 2.10. Let k ⩾ 1 and u 0 ∈ X k (R). For t ∈ R and x ∈ R, the quantity

S(t)u 0 (x) :=        e -iπ/4 π -1/2 lim ε→0 R e (i-ε)z 2 u 0 (x + 2 √ tz)dz if t ⩾ 0, e iπ/4 π -1/2 lim ε→0 R e (-i-ε)z 2 u 0 (x + 2 √ -tz)dz if t ⩽ 0. (2.11)
is well-dened and S denes a strongly continuous group on X k (R). For all u 0 ∈ X k (R) and t ∈ R we have

∥S(t)u 0 ∥ X k ⩽ C(k)(1 + |t| 1/4 )∥u 0 ∥ X k .
The generator of the group

(S(t))| t∈R on X k (R) is i∂ 2 and its domain is X k+2 (R). Remark 2.11. Since, for all ϕ ∈ X k (R), we have ϕ + H k (R) ⊂ X k (R), the uniqueness of solution in X k (R) implies the uniqueness of solution in ϕ + H k (R)
, and the existence of solution in ϕ + H k (R) implies the existence of solution in X k (R).

From the equation to the system

The equation (2.1) contains a spatial derivative of u in the nonlinear part, which makes it dicult to work with. In the following proposition, we indicate how to eliminate the derivative in the nonlinearity by introducing an auxiliary function and converting the equation into a system. Proposition 2.12.

Let k ⩾ 2. Given u ∈ X k (R), we dene v by v = ∂u + i 2 |u| 2 u. (2.12)
Hence, v ∈ X k-1 (R). Furthermore, if u satises the equation (2.1), then the couple (u, v) veries the system

Lu = P 1 (u, v), Lv = P 2 (u, v), (2.13)
where P 1 and P 2 are given by

P 1 (u, v) = -iu 2 v + 1 2 |u| 4 u, P 2 (u, v) = iuv 2 + 3 2 |u| 4 v + u 2 |u| 2 v.
(2.14)

Proof. Let u be a solution of (2.1) and v be dened by (2.12). Then we have

Lu = -iu 2 ∂u = -iu 2 v + i 2 (|u| 2 u) = -iu 2 v + 1 2 |u| 4 u,
which gives us the rst equation in (2.13).

On the other hand, since L and ∂ commute and u solves (2.1), we have

Lv = ∂(Lu)+ i 2 L(|u| 2 u) = ∂(-iu 2 ∂u)+ i 2 L(|u| 2 u) = -i(u 2 ∂ 2 u+2u|∂u| 2 )+ i 2 L(|u| 2 u).
(2.15)

Using L(uv) = L(u)v + uL(v) + 2∂u∂v, L(u) = -Lu + 2∂ 2 u, (2.16)
we have

L(|u| 2 u) = L(u 2 u) = L(u 2 )u + u 2 L(u) + 2∂(u 2 )∂u = 2L(u)u + 2(∂u) 2 ) u + u 2 (-Lu + 2∂ 2 u) + 4u|∂u| 2 = 2L(u)|u| 2 + 2u(∂u) 2 + 2u 2 ∂ 2 u -u 2 Lu + 4u|∂u| 2 . (2.17)
We now recall that u veries (2.1) to obtain

i 2 L(|u| 2 u) = u 2 ∂u|u| 2 + iu(∂u) 2 + iu 2 ∂ 2 u + 1 2 ∂u|u| 4 + 2iu|∂u| 2 . (2.18)
Subsituting in (2.15), we get

Lv = -i(u 2 ∂ 2 u + 2u|∂u| 2 ) + u 2 ∂u|u| 2 + iu(∂u) 2 + iu 2 ∂ 2 u + 1 2 ∂u|u| 4 + 2iu|∂u| 2 , = u 2 ∂u|u| 2 + iu(∂u) 2 + 1 2 ∂u|u| 4 .
Observe here that the second order derivatives of u have vanished and only rst order derivatives remain. Therefore, using the expression of v given in (2.12) to subsitute ∂u, we obtain by direct calculations

Lv = iuv 2 + 3 2 |u| 4 v + u 2 |u| 2 v,
which gives us the second equation in (2.13).

Resolution of the system

We now establish the local well-posedness of the system (2.13) in Zhidkov spaces.

Proposition 2.13. Let k ⩾ 3, and

(u 0 , v 0 ) ∈ X k (R) × X k (R).
There exist T min < 0, T max > 0 and a unique maximal solution (u, v) of system (2.13) such that

(u, v) ∈ C((T min , T max ), X k (R)) ∩ C 1 ((T min , T max ), X k-2 (R)).
Furthermore the following properties are satised.

Blow-up alternative. If T max < ∞ (resp. T min > -∞ then lim t→Tmax(resp. T min ) (∥u(t)∥ X 1 + ∥v(t)∥ X 1 ) = ∞.
Continuity with respect to the initial data. If

(u n 0 , v n 0 ) ∈ X k × X k is such that ∥u n 0 -u 0 ∥ X k + ∥v n 0 -v 0 ∥ X k → 0 then for any subinterval [T 1 , T 2 ] ⊂ (T min , T max ) the associated solution (u n , v n ) of (2.13) satises lim n→∞ ∥u n -u∥ L ∞ ([T 1 ,T 2 ],X k ) + ∥v n -v∥ L ∞ ([T 1 ,T 2 ],X k ) = 0.

Proof. Consider the operator

A : D(A) ⊂ X k-2 (R) → X k-2 (R) dened by A = i∂ 2 with domain D(A) = X k (R).
From Proposition 2.10 we know that the operator A is the generator of the Schrödinger group S(t) on X k-2 (R). From classical arguments (see [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF]Lemma 4.1.1 and Corollary 4.

1.8]) the couple (u, v) ∈ C((T min , T max ), X k (R)) ∩ C 1 ((T min , T max ), X k-2 (R)) solves (2.13) if and only if the couple (u, v) ∈ C((T min , T max ), X k (R)) solves (u, v) = S(t)(u 0 , v 0 ) -i t 0 S(t -s)P (u, v)(s)ds, u(0) = u 0 ∈ X k (R), v(0) = v 0 ∈ X k (R), (2.19)
where S(t)(u, v) := (S(t)u, S(t)v), P (u, v) = (P 1 (u, v), P 2 (u, v)) and P 1 and P 2 are dened in (2.14). Consider P as a map from

X k (R) × X k (R) into X k (R) × X k (R).
Since P 1 and P 2 are polynomial in u and v, the map P is Lipschitz continuous on bounded sets of

X k (R) × X k (R). Since (
(u, v) ∈ C((T min , T max ), X k (R) × X k (R)) ∩ C 1 ((T min , T max ), X k-2 (R)×X k-2 (R)) of system (2.13). Moreover, (u, v) satisfy blow- up alternative continuous dependence on initial data in X k (R) × X k (R). It remains to prove the blow-up alternative in X 1 (R) × X 1 (R).
We use the similar arguments as in [120, Proof of Theorem 1.2.4]. For each 1 ⩽ s ⩽ k -1, since the map P is Lipschitz continuous on bounded sets of X s (R) × X s (R), there exists T smin and

T smax such that (u, v) is the maximal X s (R) × X s (R) solution of system (2.19) on (T smin , T smax ) and (u, v) satisfy: lim t→Tsmax(resp. T smin ) (∥u(t)∥ X s + ∥v(t)∥ X s ) = ∞.
It is sucient to prove that T 1max = T max and T 1min = T min . We have

T 1max ⩾ T 2max ⩾ .. ⩾ T (k-1)max ⩾ T max .
We rst prove

T 1max = T 2max . Assume T 1max > T 2max . For t ∈ [0, T 2max ], since (2.19) we have ∥u∥ X 2 + ∥v∥ X 2 ⩽ ∥u 0 ∥ X 2 + ∥v 0 ∥ X 2 + max t∈[0,T 2max ] (∥u∥ X 1 + ∥v∥ X 1 + 1) 4 t 0 (∥u(s)∥ X 2 + ∥v(s)∥ X 2 ) ds.
By Gronwall's inequality in integral form we obtain

sup t∈[0,T 2max ] (∥u∥ X 2 + ∥v∥ X 2 ) < ∞.
This contradicts to blow-up alternative of

(u, v) in X 2 (R) × X 2 (R). Thus, T 1max = T 2max
. By apply many times this arguments we obtain T 1max = T max and by similar arguments we have T 1min = T min . This completes the proof of Proposition 2.13.

Preservation of the dierential identity

The following proposition establishes the link from (2.13) to (2.1) by showing preservation along the time evolution of the dierential identity

v 0 = ∂u 0 + i 2 |u 0 | 2 u 0 .
Proposition 2.14. Let u 0 , v 0 ∈ X 3 (R) be such that

v 0 = ∂u 0 + i 2 u 0 |u 0 | 2 .
Then the associated solution

(u, v) ∈ C((-T min , T max ), X 3 (R) × X 3 (R)) obtained in Proposition 2.
13 satises for all t ∈ (-T min , T max ) the dierential identity

v = ∂u + i 2 |u| 2 u. Proof. Given (u, v) ∈ C((-T min , T max ), X 3 (R) × X 3 (R)
) the solution of (2.13) obtained in Proposition 2.13, we dene

w = ∂u + i 2 |u| 2 u.
Our goal will be to show that w = v. We rst have

Lu = -iu 2 v + 1 2 |u| 4 u = -iu 2 (v -w) -iu 2 w + 1 2 |u| 4 u = -iu 2 (v -w) -iu 2 ∂u.
Applying L to w and using (2.17) and the expression previously obtained for Lu, we get

Lw = ∂(Lu) + i 2 L(|u| 2 u) = ∂(Lu) + i 2 2Lu|u| 2 + 2u(∂u) 2 + 2u 2 ∂ 2 u -u 2 Lu + 4u|∂u| 2 = ∂(-iu 2 (v -w) -iu 2 ∂u) + i 2 2(-iu 2 ∂u)|u| 2 + 2u(∂u) 2 -u 2 (-iu 2 ∂u) + 2u 2 ∂ 2 u + 4u|∂u| 2 + i 2 2(-iu 2 (v -w))|u| 2 -u 2 (-iu 2 (v -w)) = -i∂(u 2 (v -w)) + u 2 |u| 2 (v -w) + 1 2 |u| 4 (v -w) + -i∂(u 2 ∂u) + u 2 ∂u|u| 2 + iu(∂u) 2 + 1 2 |u| 4 ∂u + iu 2 ∂ 2 u + 2iu|∂u| 2 =: I 1 + I 2 .
As in the proof of Proposition 2.12, we obtain

I 2 = iuw 2 + 3 2 |u| 4 w + |u| 2 u 2 w.
Furthermore

I 1 = ∂(-iu 2 (v -w)) + u 2 |u| 2 (v -w) + 1 2 |u| 4 (v -w) = -iu 2 ∂(v -w) -2iu∂u(v -w) + u 2 |u| 2 (v -w) + 1 2 |u| 4 (v -w).

It follows that

Lw

-Lv = I 1 + (I 2 -Lv) (2.20) = I 1 + iu(w -v)(w + v) + 3 2 |u| 4 (w -v) + |u| 2 u 2 (w -v) (2.21) = (w -v)A 1 + (w -v)A 2 -iu 2 ∂(v -w), (2.22)
where A 1 and A 2 are polynomials of degree at most 4 in u, ∂u, v, ∂v and their complex conjugates. Hence,

(Lw -Lv)(w -v) = |w -v| 2 A 1 + (w -v) 2 A 2 -iu 2 ∂(v -w) 2 2 := K, (2.23)
where K is a polynomial of degree at most 6 in u, v, w, ∂u, ∂v, ∂w and their complex conjugates. Remembering that L = i∂ t + ∂ 2 , and taking imaginary part in the two sides of (2.23) we obtain

1 2 ∂ t |w -v| 2 + Im(∂ ((∂w -∂v)(w -v))) = Im(K). (2.24) Let χ : R → R be a cut-o function such that χ ∈ C 1 (R), supp(χ) ⊂ [-2, 2], χ ≡ 1 on (-1, 1), 0 ⩽ χ ⩽ 1, |χ ′ (x)| 2 ≲ χ(x) for all x ∈ R. For each n ∈ N, dene χ n (x) = χ x n .
Multiplying both sides of (2.24) by χ n and integrating in space we obtain

1 2 ∂ t ∥(w -v) √ χ n ∥ 2 L 2 + R Im (∂ ((∂w -∂v)(w -v))) χ n dx = R Im(K)χ n dx. (2.25)
For the right hand side, we have

R Im(K)χ n dx = Im R |w-v| 2 A 1 χ n dx+Im R (w-v) 2 A 2 χ n dx-Im R iu 2 ∂((v -w) 2 ) 2 χ n dx,
and therefore

R Im(K)χ n dx ⩽ ∥(w-v) √ χ n ∥ 2 L 2 (∥A 1 ∥ L ∞ + ∥A 2 ∥ L ∞ )+ 1 2 R u 2 ∂((v -w) 2 )χ n dx .
We now x some arbitrary interval

[-T 1 , T 2 ] such that 0 ∈ [-T 1 , T 2 ] ⊂ (-T min , T max )
in which we will be working from now on, and we set

R = ∥u∥ L ∞ ([T 1 ,T 2 ],X 3 ) + ∥v∥ L ∞ ([T 1 ,T 2 ],X 3 ) .
From the fact that A 1 and A 2 are polynomials in u, ∂u, v, ∂v of degree at most 4, for all t ∈ [T 1 , T 2 ] we have

∥A 1 ∥ L ∞ + ∥A 2 ∥ L ∞ ⩽ C(R). It follows that R Im(K)χ n dx ⩽ ∥(w -v) √ χ n ∥ 2 L 2 C(R) + 1 2 R (v -w) 2 ∂(u 2 )χ n + u 2 ∂χ n )dx .
By denition of χ we have

∂(u 2 )χ n ⩽ C(R)χ n , u 2 ∂χ n ⩽ |u 2 | 1 n χ ′ • n ⩽ 1 n C(R) χ • n ⩽ C(R) 1 n χ n (.). Hence, R Im(K)χ n dx ⩽ ∥(w -v) √ χ n ∥ 2 L 2 C(R) + C(R) n R (v -w) 2 √ χ n dx ⩽ C(R)∥(w -v) √ χ n ∥ 2 L 2 + C(R) 2 n R |v -w| √ χ n dx ⩽ C(R)∥(w -v) √ χ n ∥ 2 L 2 + C(R) 2 n 2n -2n |v -w| √ χ n dx ⩽ C(R)∥(w -v) √ χ n ∥ 2 L 2 + C(R) 2 n 2n -2n (|v -w| √ χ n ) 2 dx 1 2 2n -2n dx 1 2 ⩽ C(R)∥(w -v) √ χ n ∥ 2 L 2 + 2C(R) 2 √ n ∥(w -v) √ χ n ∥ L 2 .
(2.26)

In addition, we have

R Im(∂ ((∂w -∂v)(w -v)) χ n )dx = R Im(((∂w -∂v)(w -v)) χ ′ n )dx = R Im (∂w -∂v)(w -v) 1 n χ ′ x n dx ⩽ R |∂w -∂v||w -v| 1 n √ χ n dx ⩽ 1 n ∥∂w -∂v∥ L 2 ∥(w -v) √ χ n ∥ L 2 ⩽ C(R) n ∥(w -v) √ χ n ∥ L 2 . (2.27)
From (2.25), (2.26), (2.27) we obtain that

∂ t ∥(w -v) √ χ n ∥ 2 L 2 ⩽ C(R)∥(w -v) √ χ n ∥ 2 L 2 + C(R) √ n ∥(w -v) √ χ n ∥ L 2 (2.28) ⩽ C(R)∥(w -v) √ χ n ∥ 2 L 2 + C(R) √ n (2.29)
where we have used the Cauchy inequality |x|

⩽ |x| 2 +1 2 . Dene the function g : [-T 1 , T 2 ] by g = ∥(w -v) √ χ n ∥ 2 L 2 .
Then by denition of w we have g(t = 0) = 0. Furthermore, from (2.29) we have

∂ t g ⩽ C(R)g + C(R) √ n .
By Gronwall inequality for all t ∈ [-T 1 , T 2 ] we have

g ⩽ C(R) √ n exp(C(R)(T 2 + T 1 )) ⩽ C(R) √ n . (2.30)
Assume by contradiction that there exist t and x such that

w(t, x) ̸ = v(t, x).
By continuity of v and w, there exists ε > 0 such that (for n > |x|) we have

g(t) = ∥(w -v) √ χ n ∥ 2 L 2 > ε.
Since ε > 0 is independant of n, we obtain a contradiction with (2.30) when n is large enough. Therefore for all t and x, we have

v(t, x) = w(t, x),
which concludes the proof.

From the system to the equation

With Proposition 2.14 in hand, we give the proof of Theorem 2.1.

Proof of Theorem 2.1. We start by dening v 0 by

v 0 = ∂u 0 + i 2 |u 0 | 2 u 0 ∈ X 3 (R).
From Proposition 2.13 there exists a unique maximal solution

(u, v) ∈ C((T min , T max ), X 3 (R)× X 3 (R)) ∩ C 1 ((T min , T max ), X 1 (R) × X 1 (R))
of the system (2.13) associated with (u 0 , v 0 ). From Proposition 2.14, for all t ∈ (T min , T max ) we have

v = ∂u + i 2 |u| 2 u. (2.31) It follows that Lu = -iu 2 v + 1 2 |u| 4 u = -iu 2 ∂u,
and therefore u is a solution of (2.1) on (T min , T max ). Furthermore

u ∈ C((T min , T max ), X 3 (R)) ∩ C 1 ((T min , T max ), X 1 (R)).
To obtain the desired regularity on u, we observe that, since v has the same regularity as u, and veries (2.31), we have

∂u = v - i 2 |u| 2 u ∈ C((T min , T max ), X 3 (R)) ∩ C 1 ((T min , T max ), X 1 (R))
This implies that

u ∈ C((T min , T max ), X 4 (R)) ∩ C 1 ((T min , T max ), X 2 (R)).
This proves the existence part of the result. Uniqueness is a direct consequence from Proposition 2.12 and Proposition 2.13.

To prove the blow-up alternative, assume that T max < ∞. Then from Proposition 2.13 we have lim

t→Tmax (∥u(t)∥ X 1 (R) + ∥v(t)∥ X 1 (R) ) = ∞
On the other hand, since (2.31) we obtain

lim t→Tmax (∥u(t)∥ X 1 (R) + ∥∂u(t)∥ X 1 (R) ) = ∞. It follows that lim t→Tmax ∥u(t)∥ X 2 (R) = ∞.
Finally, we establish the continuity with respect to the initial data. Take a subinterval [T 1 , T 2 ] ⊂ (T min , T max ), and a sequence (u n 0 ) ∈ X 4 (R) such that u n 0 → u 0 in X 4 . Let u n be the solution of (2.1) associated with u n 0 and dene v n by

v n = ∂u n + i 2 |u n | 2 u n . (2.32)
By Proposition 2.13 the couple (u n , v n ) is the unique maximal solution of system (2.13) in

C((T min , T max ), X 3 (R) × X 3 (R)) ∩ C 1 ((T min , T max ), X 1 (R) × X 1 (R)).
Moreover, we have

lim n→+∞ ∥u n -u∥ L ∞ ([T 1 ,T 2 ],X 3 ) + ∥v n -v∥ L ∞ ([T 1 ,T 2 ],X 3 ) = 0 (2.33)
Since v and v n verify the dierential identity (2.32), we have

∂(u n -u) = (v n -v) - i 2 |u n | 2 u n -|u| 2 u .
Therefore we have lim

n→+∞ ∥u n -u∥ L ∞ ([T 1 ,T 2 ],X 4 ) = 0,
which completes the proof.

2.3

Results on the space ϕ

+ H k-2 (R) for ϕ ∈ X k (R)
In this section, we give the proof of Theorem 2.2 and Theorem 2.3.

The local well posedness on

ϕ + H 2 (R)
From the equation to the system

Dene v = ∂u + i 2 |u| 2 u. (2.34)
Since Proposition 2.12, if u solves (2.1) then (u, v) solves the following system:

         Lu = -iu 2 v + 1 2 |u| 4 u, Lv = iuv 2 + 3 2 |u| 4 v + u 2 |u| 2 v, u(0) = u 0 , v(0) = v 0 := ∂u 0 + i 2 |u 0 | 2 u 0 . (2.35) Let ϕ ∈ X 4 (R). Dene ũ = u -ϕ, ṽ = v -i 2 |ϕ| 2 ϕ. We have if u solves (2.1) then (ũ, ṽ) solves:          Lũ = Q 1 (ũ, ṽ, ϕ), Lṽ = Q 2 (ũ, ṽ, ϕ), ũ(0) = ũ0 := u 0 -ϕ, ṽ(0) = ṽ0 := v 0 -i 2 |ϕ| 2 ϕ, (2.36)
where

Q 1 (ũ, ṽ, ϕ) = -i(ũ + ϕ) 2 ṽ - i 2 |ϕ| 2 ϕ + 1 2 |ũ + ϕ| 4 (ũ + ϕ) -L(ϕ), (2.37)
Q 2 (ũ, ṽ, ϕ) = i(ũ + ϕ) ṽ + i 2 |ϕ| 2 ϕ 2 + 3 2 |ũ + ϕ| 4 ṽ + i 2 |ϕ| 2 ϕ (2.38) + (ũ + ϕ) 2 |ũ + ϕ| 2 ṽ - i 2 |ϕ| 2 ϕ - i 2 L(|ϕ| 2 ϕ). (2.39) Resolution of the system Let k ⩾ 1. We note that if ϕ ∈ X k+2 then Q 1 : (ũ, ṽ) → Q 1 (ũ, ṽ, ϕ) and Q 2 : (ũ, ṽ) → Q 2 (ũ, ṽ, ϕ) dened
H k (R) × H k (R)
. By similar arguments to the one used for the proof of Proposition 2.13, we obtain the following local well-posedness result:

Proposition 2.15. Let k ⩾ 1, ϕ ∈ X k+2 , ũ0 , ṽ0 ∈ H k (R).
There exist T min < 0, T max > 0 and a unique maximal solution (ũ, ṽ) of the system (2.36

) such that ũ, ṽ ∈ C((T min , T max ), H k (R)) ∩ C 1 ((T min , T max ), H k-2 (R)).
Furthermore the following properties are satised.

Blow-up alternative. If T max < ∞ (resp. T min > -∞ then lim t→Tmax(resp. T min ) (∥ũ∥ H k + ∥ṽ∥ H k ) = ∞.
Continuity with respect to the initial data. If ũn 0 , ṽn

0 ∈ H k (R) are such that ∥ũ n 0 -ũ0 ∥ H k + ∥ṽ n 0 -ṽ0 ∥ H k → 0 then for any subinterval [T 1 , T 2 ] ⊂ (T min , T max ) the associated solution (ũ n , ṽn ) of (2.36) satises lim n→+∞ ∥ũ n -ũ∥ L ∞ ([T 1 ,T 2 ],H k ) + ∥ṽ n -ṽ∥ L ∞ ([T 1 ,T 2 ],H k ) = 0.

Preservation of a dierential identity

Let (ũ 0 , ṽ0 ) be dened as in section 2.3.1. By an elementary calculation, we have

ṽ0 = ∂ ũ0 + i 2 (|ũ 0 + ϕ| 2 ( ũ0 + ϕ) -|ϕ| 2 ϕ) + ∂ϕ. (2.40)
We have the following results:

Proposition 2.16. Let ϕ ∈ X 4 (R) and ũ0 , ṽ0 ∈ H 2 (R) satisfy (2.40). Then the associated solution (ũ, ṽ) obtained in Proposition 2.15 also satisfy (2.40) for all t ∈ (T min , T max ).

Proof. We dene

w = ∂ ũ + i 2 (|ũ + ϕ| 2 (ũ + ϕ) -|ϕ| 2 ϕ) + ∂ϕ. (2.41) Set u = ũ + ϕ, v = ṽ + i 2 |ϕ| 2 ϕ, w = w + i 2 |ϕ| 2 ϕ. We have w = ∂u + i 2 |u| 2 u. (2.42)
Since (ũ, ṽ) is a solution of (2.36), we have (u, v) is a solution of (2.35). We have

Lu = -iu 2 (v -w) + H,
where H dened by

H = -iu 2 w + 1 2 |u| 4 u.
By using (2.17) and the previously expression obtained for Lu, we get

Lw = ∂(Lu) + i 2 L(|u| 2 u) = ∂(Lu) + i 2 2L(u)|u| 2 + 2u(∂u) 2 + 2u 2 ∂ 2 u -u 2 L(u) + 4u|∂u| 2 = ∂ -iu 2 (v -w) + ∂H + i H|u| 2 -iu 2 |u| 2 (v -w) + u(∂u) 2 + u 2 ∂ 2 u - 1 2 u 2 iu 2 (v -w) + H + 2u|∂u| 2 = -i∂ u 2 (v -w) + u 2 |u| 2 (v -w) + 1 2 |u| 4 (v -w) + K,
where K is dened by

K = ∂H + iH|u| 2 + iu(∂u) 2 + iu 2 ∂ 2 u - i 2 u 2 H + 2iu|∂u| 2 .
Using (2.42) to replace the term ∂u in K and remark that the role of w is the same the one of v as in Proposition 2.12, we have

K = iuw 2 + 3 2 |u| 4 w + u 2 |u| 2 w.
Thus,

Lw -Lv = -i∂ u 2 (v -w) + u 2 |u| 2 (v -w) + 1 2 |u| 4 (v -w) + (K -L(v)) = -i∂ u 2 (v -w) + u 2 |u| 2 (v -w) + 1 2 |u| 4 (v -w) + iu(w 2 -v 2 ) + 3 2 |u| 4 (w -v) + u 2 |u| 2 (w -v) = -iu 2 ∂(v -w) + A(v -w) + B(v -w),
where

A := -|u| 4 -iu(v + w), B := -2iu∂u = -2iu w - i 2 |u| 2 u = -2iuw -|u| 2 u 2 .
This implies that

L( w -ṽ) = -i(ũ + ϕ) 2 ∂(ṽ -w) + A(ṽ -w) + B(ṽ -w). (2.43)
Multiplying both sides of (2.43) by w -ṽ, taking the imaginary part, and integrating over space with integration by part for the rst term of right hand side of (2.43), we obtain

d dt ∥ w -ṽ∥ 2 L 2 ≲ (∥ũ + ϕ∥ L ∞ ∥∂ ũ + ∂ϕ∥ L ∞ + ∥A∥ L ∞ + ∥B∥ L ∞ )∥ w -ṽ∥ 2 L 2 .
By Grönwall's inequality we obtain

∥ w-ṽ∥ 2 L 2 ⩽ ∥ w(0)-ṽ(0)∥ 2 L 2 ×exp(C t 0 (∥ũ+ϕ∥ L ∞ ∥∂ ũ+∂ϕ∥ L ∞ +∥A∥ L ∞ +∥B∥ L ∞ ) ds).
Using the fact that w(0) = ṽ(0), we obtain w = ṽ, for all t. This implies that

ṽ = ∂ ũ + i 2 (|ũ + ϕ| 2 (ũ + ϕ) -|ϕ| 2 ϕ) + ∂ϕ.
This completes the proof of Proposition 2.16.

From the system to the equation Now, we nish the proof of Theorem 2.2.

Proof of Theorem 2.2. Let ϕ ∈ X 4 (R) and

u 0 ∈ ϕ + H 2 (R). We dene v 0 ∈ X 1 (R),
ũ0 ∈ H 2 (R) and ṽ0 ∈ H 1 (R) in the following way:

v 0 = ∂u 0 + i 2 u 0 |u 0 | 2 , ũ0 = u 0 -ϕ, and ṽ0 = v 0 - i 2 |ϕ| 2 ϕ.
We have

ṽ0 = ∂ ũ0 + i 2 (|ũ 0 + ϕ| 2 (ũ 0 + ϕ) -|ϕ| 2 ϕ) + ∂ϕ.
From Proposition 2.15 there exists a unique maximal solution (ũ, ṽ)

∈ C((T min , T max ), H 1 (R))∩ C 1 ((T min , T max ), H -1 (R)) of (2.36). Let ũn 0 ∈ H 3 (R) be such that ∥ũ n 0 -ũ0 ∥ H 2 (R) → 0 as n → ∞. Dene ṽn 0 ∈ H 2 (R) by ṽn 0 = ∂ ũn 0 + i 2 (|ũ n 0 + ϕ| 2 (ũ n 0 + ϕ) -|ϕ| 2 ϕ) + ∂ϕ.
From Proposition 2.15, there exists a unique solution maximal solution.

ũn , ṽn ∈ C((T n min , T n max ), H 2 (R)) ∩ C 1 ((T n min , T n max ), L 2 (R)) of the system (2.36). Let [T 1 , T 2 ] ⊂ (T min , T max ) be any closed interval. From [17, proposition 4.3.7], for n ⩾ N 0 large enough, we have [T 1 , T 2 ] ⊂ (T n min , T n max ). By Proposition 2.16, for n ⩾ N 0 , t ∈ [T 1 , T 2 ], we have ṽn = ∂ ũn + i 2 (|ũ n + ϕ| 2 (ũ n + ϕ) -|ϕ| 2 ϕ) + ∂ϕ.
By Proposition 2.15, we have

lim n→∞ sup t∈[T 1 ,T 2 ] (∥ũ n (t) -ũ(t)∥ H 1 (R) + ∥ṽ n (t) -ṽ(t)∥ H 1 (R) ) → 0.
We obtain that for all t ∈ [T 1 , T 2 ], and then for all t ∈ (T min , T max ):

ṽ = ∂ ũ + i 2 (|ũ + ϕ| 2 (ũ + ϕ) -|ϕ| 2 ϕ) + ∂ϕ.
This follows that

∂ ũ ∈ C((T min , T max ), H 1 (R)) ∩ C 1 ((T min , T max ), H -1 (R)).
Hence we have

ũ ∈ C((T min , T max ), H 2 (R)) ∩ C 1 ((T min , T max ), L 2 (R)).
Dene u = ϕ + ũ and dene v by

v = ṽ + i 2 |ϕ| 2 ϕ = ∂u + i 2 |u| 2 u.
Since (ũ, ṽ) solves (2.36), we have (u, v) solves (2.35). Therefore, u ∈ ϕ+C((T min , T max ),

H 2 (R))∩ C 1 ((T min , T max ), L 2 (R)) solves: Lu = -iu 2 v + 1 2 |u| 4 u = -iu 2 ∂u.
This establishes the existence of a solution to (2.1). To prove uniqueness, assume that (2.36). By the uniqueness statement in Proposition 2.15, we obtain Ũ = ũ. Hence, u = U , which proves uniqueness. The blow-up alternative and continuity with respect to the initial data are proved using similar arguments as in the proof of Theorem 2.1. This completes the proof of Theorem 2.2.

U ∈ ϕ + C((T min , T max ), H 2 (R)) ∩ C 1 ((T min , T max ), L 2 (R)) is another solution of (2.1). Set V = ∂U + i 2 |U | 2 U , and Ũ = U -ϕ, Ṽ = V -i 2 |ϕ| 2 ϕ. Thus, ( Ũ , Ṽ ) ∈ C((T min , T max ), H 1 (R)) ∩ C 1 ((T min , T max ), H -1 (R)) is a solution of

The local well posedness on ϕ + H 1 (R)

In this section, we give the proof of Theorem 2.3, using the method of Hayashi and Ozawa [START_REF] Hayashi | Finite energy solutions of nonlinear Schrödinger equations of derivative type[END_REF]. As in Section 2.3.1, we work with the system (2.36).

Since we are working in the less regular space ϕ + H 1 (R), we cannot use Proposition 2.15. Instead, we establish the following result using Strichartz estimate. Proposition 2.17. Consider the system (2.36)

. Let ϕ ∈ X 2 (R), ũ0 , ṽ0 ∈ L 2 (R).
There exists R > 0 such that if ∥ũ 0 ∥ L 2 + ∥ṽ 0 ∥ L 2 < R then there exist T > 0 and a unique solution (ũ, ṽ) of the system (2.36) verifying

ũ, ṽ ∈ C([-T, T ], L 2 (R)) ∩ L 4 ([-T, T ], L ∞ (R)).
Moreover, we have the following continuous dependence on initial data property: If

( u n 0 , v n 0 ) ∈ L 2 (R) × L 2 (R) is a sequence such that ∥ u n 0 -ũ0 ∥ 2 + ∥ v n 0 -ṽ0 ∥ 2 → 0 then for n large enough we have ∥ u n 0 ∥ 2 + ∥ v n 0 ∥ 2 < R
and the associated solutions (ũ n , ṽn ) satisfy:

∥ũ n -ũ∥ L ∞ L 2 ∩L 4 L ∞ + ∥ṽ n -ṽ∥ L ∞ L 2 ∩L 4 L ∞ → 0,
where we have used the following notation:

L ∞ L 2 = L ∞ ([-T, T ], L 2 (R)), L 4 L ∞ = L 4 ([-T, T ], L ∞ (R))
and the norm on L ∞ L 2 ∩L 4 L ∞ is dened, as usual for the intersection of two Banach spaces, as the sum of the norms on each space.

Proof. Let Q 1 , Q 2 be dened as in system (2.36). By direct computations, we have

Q 1 (ũ, ṽ, ϕ) = -i(ũ + ϕ) 2 ṽ - 1 2 |ϕ| 2 ϕ(ũ 2 + 2ũϕ) + 1 2 (|ũ + ϕ| 4 -|ϕ| 4 )ũ -∂ 2 ϕ, (2.44) Q 2 (ũ, ṽ, ϕ) = iũ ṽ + i 2 |ϕ| 2 ϕ 2 + iϕ ṽ + i 2 |ϕ| 2 ϕ 2 - i 2 |ϕ| 2 ϕ 2 + 3 2 |ũ + ϕ| 4 ṽ + 3 4 i|ϕ| 2 ϕ(|ũ + ϕ| 4 -|ϕ| 4 ) + ṽ(ũ + ϕ)|ũ + ϕ| 2 - i 2 |ϕ| 2 ϕ((ũ + ϕ) 2 |ũ + ϕ| 2 -|ϕ| 2 ϕ 2 ) - i 2 ∂ 2 (|ϕ| 2 ϕ). (2.45)
Thus,

|Q 1 (ũ, ṽ, ϕ)| ≲ |ṽ|(|ũ| 2 + |ϕ| 2 ) + |ϕ| 3 |ũ| 2 + |ϕ| 4 |ũ| + (|ũ| 5 + |ϕ| 4 |ũ|) + |ϕ|(|ũ| 4 + |ũ||ϕ| 3 ) + |∂ 2 ϕ| ≲ |ṽ||ũ| 2 + |ṽ||ϕ| 2 + |ũ| 5 + |ũ||ϕ| 4 + |∂ 2 ϕ|, |Q 2 (ũ, ṽ, ϕ)| ≲ |ũ|(|ṽ| 2 + |ϕ| 6 ) + |ϕ|(|ṽ| 2 + |ṽ||ϕ| 3 ) + |ṽ|(|ũ| 4 + |ϕ| 4 ) + |ϕ| 3 (|ũ| 4 + |ũ||ϕ| 3 ) + |ṽ|(|ũ| 3 + |ϕ| 3 ) + |ϕ| 3 (|ũ| 4 + |ϕ| 3 |ũ|) + |∂ 2 (|ϕ| 2 ϕ)| ≲ |ũ||ṽ| 2 + |ũ||ϕ| 6 + |ϕ||ṽ| 2 + |ϕ| 4 |ṽ| + |ũ| 4 |ṽ| + |ϕ| 3 |ũ| 4 + |ũ| 3 |ṽ| + |ϕ| 3 |ṽ| + |∂ 2 (|ϕ| 2 ϕ)|.
Consider the following problem

(ũ, ṽ) = S(t)(ũ 0 , ṽ0 ) -i t 0 S(t -s)Q(ũ, ṽ, ϕ) ds (2.46) where Q = (Q 1 , Q 2 ). Let Φ(ũ, ṽ) = S(t)(ũ 0 , ṽ0 ) -i t 0 S(t -s)Q ds.
Assume that ∥ũ 0 ∥ L 2 (R) + ∥ṽ 0 ∥ L 2 (R) ⩽ R 4 for R > 0 small enough. For T > 0 we dene the space X T,R by

X T,R = (ũ, ṽ) ∈ (C([-T, T ], L 2 (R)) ∩ L 4 ([-T, T ], L ∞ (R))) 2 : ∥(ũ, ṽ)∥ (L ∞ L 2 ∩L 4 L ∞ ) 2 ⩽ R .
We are going to prove that for R, T small enough the map Φ is a contraction from X T,R to itself.

We rst prove that for R, T small enough, Φ maps X T,R into X T,R . Let (ũ, ṽ) ∈ X T,R . By Strichartz estimates we have

∥Φ(ũ, ṽ)∥ (L ∞ L 2 ∩L 4 L ∞ ) 2 ≲ ∥(ũ 0 , ṽ0 )∥ L 2 ×L 2 + ∥Q∥ L 1 L 2 ×L 1 L 2 , ≲ R 4 + (∥Q 1 ∥ L 1 L 2 + ∥Q 2 ∥ L 1 L 2 ).
We have

∥Q 1 ∥ L 1 L 2 ≲ ∥|ũ| 2 ṽ∥ L 1 L 2 + ∥|ṽ||ϕ| 2 ∥ L 1 L 2 + ∥|ũ| 5 ∥ L 1 L 2 + ∥∂ 2 ϕ∥ L 1 L 2 ≲ ∥ṽ∥ L 2 L 2 ∥ũ∥ 2 L 4 L ∞ + ∥ṽ∥ L 2 L 2 ∥|ϕ|∥ 2 L 4 L ∞ + ∥ũ∥ 4 L 4 L ∞ ∥ũ∥ L ∞ L 2 + ∥∂ 2 ϕ∥ L 1 L 2 ≲ (2T ) 1 2 ∥ṽ∥ L ∞ L 2 ∥ũ∥ 2 L 4 L ∞ + (2T ) 1 2 ∥ṽ∥ L ∞ L 2 ∥ϕ∥ L ∞ (2T ) 1 4 + ∥ũ∥ 4 L 4 L ∞ ∥ũ∥ L ∞ L 2 + ∥∂ 2 ϕ∥ L 2 (2T ) ≲ (2T ) 1 2 R 3 + (2T ) 3 4 ∥ϕ∥ L ∞ R + R 5 + (2T )∥ϕ∥ X 2 < R 4 .
for T, R small enough. Similarly, we also have

∥ Q2 ∥ L 1 L 2 < R 4
for T, R small enough. Therefore, for T, R small enough, we have

∥Φ(ũ, ṽ)∥ (L ∞ L 2 ∩L 4 L ∞ ) 2 < 3R 4 < R.
Hence, Φ maps from X T,R into itself. We now show that for T, R small enough, the map Φ is a contraction from X T,R to itself. Indeed, let (u 1 , v 1 ), (u 2 , v 2 ) ∈ X T,R . By Strichartz estimates we have

∥Φ(u 1 , v 1 ) -Φ(u 2 , v 2 )∥ (L ∞ L 2 ∩L 4 L ∞ ) 2 = ∥ t 0 S(t -s) (Q(u 1 , v 1 ) -Q(u 2 , v 2 )) ds∥ (L ∞ L 2 ∩L 4 L ∞ ) 2 , ≲ ∥Q 1 (u 1 , v 1 ) -Q 1 (u 2 , v 2 )∥ L 1 L 2 + ∥Q 2 (u 1 , v 1 ) -Q 2 (u 2 , v 2 )∥ L 1 L 2 .
Using the same kind of arguments as before we obtain that Φ is a contraction on X T,R . Therefore, using the Banach xed-point theorem, there exist T > 0 and a unique solution (ũ, ṽ) (2.36). Thus, we proved the existence of a solution of (2.36). The uniqueness of solution of (2.36) is obtained by the uniqueness of solution of (2.46).

∈ C([-T, T ], L 2 (R)) ∩ L 4 ([-T, T ], L ∞ (R)) of the problem (2.46). As above, we see that if h, k ∈ C([-T, T ], L 2 (R)) ∩ L 4 ([-T, T ], L ∞ (R)) then Q 1 (h, k, ϕ), Q 2 (h, k, ϕ) ∈ L 1 ([-T, T ], L 2 (R)). By [17, Proposition 4.1.9], (ũ, ṽ) ∈ C([-T, T ], L 2 (R)) ∩ L 4 ([-T, T ], L ∞ (R)) solves (2.46) if only if (ũ, ṽ) solves
It is remains to prove the continuous dependence on initial data. Assume that

(u n 0 , v n 0 ) ∈ L 2 (R) × L 2 (R) is such that ∥u n 0 -ũ0 ∥ L 2 (R) + ∥v n 0 -ṽ0 ∥ L 2 (R) → 0, as n → ∞.
In particular, for n large enough, we have

∥u n 0 ∥ L 2 (R) + ∥v n 0 ∥ L 2 (R) < R.
There exists a unique maximal solution (u n , v n ) of system (2.36), and we may assume that for n large enough,

(u n , v n ) is dened on [-T, T ]. Assume that T small enough such that ∥ũ∥ L ∞ L 2 ∩L 4 L ∞ + ∥ṽ∥ L ∞ L 2 ∩L 4 L ∞ + sup n (∥u n ∥ L ∞ L 2 ∩L 4 L ∞ + ∥v n ∥ L ∞ L 2 ∩L 4 L ∞ ) ⩽ 2R.
(2.47) We have (ũ, ṽ) is a solution of the following system

(ũ, ṽ) = S(t)(ũ 0 , ṽ0 ) -i t 0 S(t -s)(Q 1 (ũ, ṽ, ϕ), Q 2 (ũ, ṽ, ϕ)).
Similarly, (u n , v n ) are solutions of the following system

(u n , v n ) = S(t)(u n 0 , v n 0 ) -i t 0 S(t -s)(Q 1 (u n , v n , ϕ), Q 2 (u n , v n , ϕ)).
Hence,

(u n -u, v n -v) = S(t)(u n 0 -ũ0 , v n 0 -ṽ0 ) -i t 0 S(t -s)(Q 1 (ũ, ṽ, ϕ) -Q 1 (u n , v n , ϕ), Q 2 (ũ, ṽ, ϕ) -Q 2 (u n , v n , ϕ)).
Using Strichartz estimates and (2.47), for all t ∈ [-T, T ] and R, T small enough, we have

∥u n -ũ∥ L ∞ L 2 ∩L 4 L ∞ + ∥v n -ṽ∥ L ∞ L 2 ∩L 4 L ∞ ≲ ∥u n 0 -ũ0 ∥ L 2 + ∥v n 0 -ṽ0 ∥ L 2 + ∥Q 1 (ũ, ṽ, ϕ) -Q 1 (u n , v n , ϕ)∥ L 1 L 2 + ∥Q 2 (ũ, ṽ, ϕ) -Q 2 (u n , v n , ϕ))∥ L 1 L 2 ≲ ∥u n 0 -ũ0 ∥ L 2 + ∥v n 0 -ṽ0 ∥ L 2 + R(∥u n -ũ∥ L ∞ L 2 ∩L 4 L ∞ + ∥v n -ṽ∥ L ∞ L 2 ∩L 4 L ∞ ).
For R < 1 2 small enough, we have

1 2 (∥u n -ũ∥ L ∞ L 2 ∩L 4 L ∞ + ∥v n -ṽ∥ L ∞ L 2 ∩L 4 L ∞ ) ⩽ ∥ũ 0 -u n 0 ∥ L 2 (R) + ∥ṽ 0 -v n 0 ∥ L 2 (R) .
Letting n → +∞ we obtain the desired result.

From the system to the equation Now, we nish the proof of Theorem 2.3.

Proof of Theorem 2.3. Let ϕ ∈ X 4 (R) be such that ∥∂ϕ∥ L 2 is small enough. Let

u 0 ∈ ϕ + H 1 (R) be such that ∥u 0 -ϕ∥ H 1 is small enough. Set v 0 = ∂u 0 + i 2 |u 0 | 2 u 0 , ũ0 = u 0 -ϕ and ṽ0 = v 0 -i 2 |ϕ| 2 ϕ. We have ṽ0 = ∂ ũ0 + i 2 (|ũ 0 + ϕ| 2 (ũ 0 + ϕ) -|ϕ| 2 ϕ) + ∂ϕ. Furthermore, ũ0 ∈ H 1 (R), ṽ0 ∈ L 2 (R) satisfy: ∥ũ 0 ∥ L 2 (R) + ∥ṽ 0 ∥ L 2 (R) ≲ ∥ũ 0 ∥ H 1 (R) + ∥∂ϕ∥ L 2 ,
which is small enough by the assumption. By Proposition 2.17, there exist T > 0 and a unique solution (ũ, ṽ)

∈ C([-T, T ], L 2 (R)) ∩ L 4 ([-T, T ], L ∞ (R)) of the system (2.36). Let u n 0 ∈ H 3 (R) satisfy ∥u n 0 -ũ0 ∥ H 1 (R) → 0 as n → +∞. Set v n 0 = ∂u n 0 + i 2 (|u n 0 + ϕ| 2 (u n 0 + ϕ) -|ϕ| 2 ) + ∂ϕ.
Let (u n , v n ) be the H 2 (R) solution of the system (2.36) obtained by Proposition 2.15 with data (u n 0 , v n 0 ). By Proposition 2.16 we have

v n = ∂u n + i 2 (|u n + ϕ| 2 (u n + ϕ) -|ϕ| 2 ϕ) + ∂ϕ. (2.48) Furthermore, ∥u n 0 -ũ0 ∥ L 2 (R) + ∥v n 0 -ṽ0 ∥ L 2 (R) → 0.
From the continuous dependence on the initial data obtained in Proposition 2.17, (u n , v n ), (ũ, ṽ) are solutions of the system (2.36) on [-T, T ] for n large enough, and

∥u n -ũ∥ L ∞ L 2 ∩L 4 L ∞ + ∥v n -ṽ∥ L ∞ L 2 ∩L 4 L ∞ → 0 as n → ∞. Letting n → ∞ on the two sides of (2.48), we obtain for all t ∈ [-T, T ] ṽ = ∂ ũ + i 2 (|ũ + ϕ| 2 (ũ + ϕ) -|ϕ| 2 ϕ) + ∂ϕ, (2.49)
which makes sense in

H -1 (R). From (2.49) we see that ∂ ũ ∈ C([-T, T ], L 2 (R)) and (2.49) makes sense in L 2 (R). Then ũ ∈ C([-T, T ], H 1 (R)) ∩ L 4 ([-T, T ], L ∞ (R)).
By the Sobolev embedding of

H 1 (R) in L ∞ (R) we obtain that ∥|ũ + ϕ| 2 (ũ + ϕ) -|ϕ| 2 ϕ∥ L 4 L ∞ ≲ ∥|ũ| 3 ∥ L 4 L ∞ + ∥|ũ||ϕ| 2 ∥ L 4 L ∞ < ∥ũ∥ L 4 L ∞ ∥ũ∥ L ∞ L ∞ + ∥ũ∥ L 4 L ∞ ∥ϕ∥ 2 L ∞ L ∞ < ∞. Hence, |ũ+ϕ| 2 (ũ+ϕ)-|ϕ| 2 ϕ ∈ L 4 L ∞ . From (2.49) we obtain that ∂ ũ ∈ L 4 L ∞ which implies ũ ∈ L 4 ([-T, T ], W 1,∞ (R)). Set u = ũ + ϕ, v = ṽ + i 2 |ϕ| 2 ϕ, then u -ϕ ∈ C([-T, T ], H 1 (R)) ∩ L 4 ([-T, T ], W 1,∞ (R)) and v -i 2 |ϕ| 2 ϕ ∈ C([-T, T ], L 2 (R)) ∩ L 4 ([-T, T ], L ∞ (R)). Moreover, v = ∂u + i 2 |u| 2 u.
Since (u, v) solves (2.35), we have

Lu = -iu 2 v + 1 2 |u| 4 u = -iu 2 ∂u.
The existence of a solution of the equation (2.1) follows. To prove the uniqueness property, assume that

U ∈ C([-T, T ], ϕ + H 1 (R)) ∩ L 4 ([-T, T ], ϕ + W 1,∞ (R)) is another solution of the equation (2.1). Set V = ∂U + i 2 |U | 2 U and Ũ = U -ϕ, Ṽ = V -i 2 |ϕ| 2 ϕ. Hence Ũ ∈ C([-T, T ], H 1 (R)) ∩ L 4 ([-T, T ], W 1,∞ (R)) and Ṽ ∈ C([-T, T ], L 2 (R))∩L 4 ([-T, T ], L ∞ (R)). Moreover, ( Ũ , Ṽ
) is a solution of the system (2.36). By the uniqueness of solutions of (2.36), we obtain that Ũ = ũ. Hence, u = U , which completes the proof.

2.4

Conservation of the mass, the energy and the momentum In this section, we prove Theorem 2.4. Let q 0 ∈ R and u ∈ q 0 + H 2 (R) be a solution of (2.1). Let χ and χ R be the functions dened as in (2.5) and (2.7). We have

∥χ ′ R ∥ L 2 (R) = R 1 R 2 χ ′ x R 2 dx 1 2 = 1 R 1 2 ∥χ ′ ∥ L 2 (R) → 0 as R → ∞. (2.50)
Similarly, for each a ∈ R, we have

∥χ ′ a,R ∥ L 2 (R) → 0 as R → ∞. (2.51)
By the continuous dependence on initial data property of solution, we can assume that u 0 ∈ q 0 + H 3 (R), so that

u ∈ C((T min , T max ), q 0 + H 3 (R)).
It is enough to prove conservation of generalized mass, conservation of energy (2.8) and conservation of momentum (2.9) for any closed interval

[T 0 , T 1 ] ∈ (T min , T max ). Let T 0 < 0, T 1 > 0 be such that [T 0 , T 1 ] ⊂ (T min , T max ). Let M > 0 be dened by M = sup t∈[T 0 ,T 1 ]
∥u -q 0 ∥ H 3 (R) .

Conservation of mass

Multiplying both sides of (2.1) by u and taking imaginary part to obtain

Re(u t u) + Im(∂ 2 uu) + Re(|u| 2 u∂u) = 0.
This implies that

0 = 1 2 ∂ t (|u| 2 ) + ∂(Im(∂uu)) + 1 4 ∂(|u| 4 ) = 1 2 ∂ t (|u| 2 -q 2 0 ) + ∂(Im(∂uu)) + 1 4 ∂(|u| 4 -q 4 0 ).
By multiplying both sides by χ R , integrating on space, and integrating by part we have

0 = ∂ t R 1 2 (|u| 2 -q 2 0 )χ R dx - R Im(∂uu)χ ′ R - R (|u| 4 -q 4 0 ) 4 χ ′ R dx = ∂ t R 1 2 (|u| 2 -q 2 0 )χ R dx - R Im(∂uu) + 1 4 (|u| 4 -q 4 0 ) χ ′ R dx. (2.52)
Denote the second term of (2.52) by K, using (2.50), we have

|K| ⩽ ∥Im(∂uu) + 1 4 (|u| 4 -q 4 0 )∥ L 2 ∥χ ′ R ∥ L 2 ≲ C(M ) 1 R 1 2 → 0 as R → ∞.
Thus, by integrating from 0 to t and taking R to innity we obtain

lim R→∞ R 1 2 (|u| 2 -q 2 0 )χ R dx - R 1 2 (|u 0 | 2 -q 2 0 )χ R dx = 0. (2.53)
Similarly, for each a ∈ R, we have

lim R→∞ R 1 2 (|u| 2 -q 2 0 )χ a,R dx - R 1 2 (|u 0 | 2 -q 2 0 )χ a,R dx = 0. (2.54) as R → ∞. This implies that m + (u(t)) and m -(u(t)) are conserved in time. In particular, if m + (u 0 ) = m -(u 0 ) = m(u 0 ) then m + (u(t)) = m -(u(t)) = m(u(t)) = m(u 0 )
. This completes the proof of conservation of mass.

Conservation of energy

Now, we prove the conservation of the energy. Since u solves (2.1), after an elementary calculation, we have

∂ t (|∂u| 2 ) = ∂ 2Re(∂u∂ t u) + Re(u 2 (∂u) 2 ) -|∂u|
∂ t (|∂u| 2 ) = ∂ 2Re(∂u∂ t u) + Re(u 2 (∂u) 2 ) -|u| 2 |∂u| 2 -|u| 4 Im(∂uu) - 1 2 Im(|u| 2 u∂ t u) + 1 2 ∂ t Im(|u| 2 u∂u) - 1 8 ∂(|u| 8 ) - 1 6 ∂ t (|u| 6 ).
Hence,

∂ t |∂u| 2 - 1 2 Im((|u| 2 u -q 3 0 )∂u) + 1 6 (|u| 6 -q 6 0 ) = ∂ 2Re(∂u∂ t u) + Re(u 2 (∂u) 2 ) -|u| 2 |∂u| 2 -|u| 4 Im(∂uu) - 1 2 Im(|u| 2 u∂ t u) - 1 8 (|u| 8 -q 8 0 ) + 1 2 q 3 0 Im∂ t ∂(u -q 0 ).
Multiplying both sides by χ R , integrating in space and integrating by part we obtain

∂ t R |∂u| 2 - 1 2 Im((|u| 2 u -q 3 0 )∂u) + 1 6 (|u| 6 -q 6 0 ) χ R dx = - R χ ′ R 2Re(∂u∂ t u) + Re(u 2 (∂u) 2 ) -|u| 2 |∂u| 2 -|u| 4 Im(∂uu) - 1 2 Im(|u| 2 u∂ t u) - 1 8 (|u| 8 -q 8 0 ) dx - q 3 0 2 Im∂ t R (u -q 0 )χ ′ R dx.
Integrating from 0 to t we obtain

R |∂u| 2 - 1 2 Im((|u| 2 u -q 3 0 )∂u) + 1 6 (|u| 6 -q 6 0 ) χ R dx (2.58) - R |∂u 0 | 2 - 1 2 Im((|u 0 | 2 u 0 -q 3 0 )∂u 0 ) + 1 6 (|u 0 | 6 -q 6 0 ) χ R dx (2.59) = t 0 R χ ′ R 2Re(∂u∂ t u) + Re(u 2 (∂u) 2 ) -|u| 2 |∂u| 2 -|u| 4 Im(∂uu) - 1 2 Im(|u| 2 u∂ t u) - 1 8 (|u| 8 -q 8 0 ) dx ds (2.60) - q 3 0 2 Im R (u -q 0 )∂χ R dx -Im R (u 0 -q 0 )χ ′ R dx . (2.61)
Denoting the term (2.60) by A R , using (2.50), we have

|A R | ⩽ ∥χ ′ R ∥ L 2 ∥2Re(∂u∂ t u) + Re(u 2 (∂u) 2 ) -|u| 2 |∂u| 2 -|u| 4 Im(∂uu) (2.62) - 1 2 Im(|u| 2 u∂ t u) - 1 8 (|u| 8 -q 8 0 )∥ L 2 (2.63) ≲ C(M )∥χ ′ R ∥ L 2 → 0 as R → ∞. (2.64)
Moreover, using (2.50) again, we have

Im R (u -q 0 )χ ′ R dx ⩽ ∥u -q 0 ∥ L 2 ∥χ ′ R ∥ L 2 ≲ C(M )∥χ ′ R ∥ L 2 → 0 as R → ∞. (2.65) Im R (u 0 -q 0 )χ ′ R dx ⩽ ∥u 0 -q 0 ∥ L 2 ∥χ ′ R ∥ L 2 ≲ C(M )∥χ ′ R ∥ L 2 → 0 as R → ∞. (2.66)
To deal with the term (2.58), we need to divide it into two terms. First, using

u ∈ q 0 + H 3 (R), as R → ∞, we have R |∂u| 2 - 1 2 Im((|u| 2 u -q 3 0 )∂u) χ R dx → R |∂u| 2 - 1 2 Im((|u| 2 u -q 3 0 )∂u) dx. (2.67)
Second, by easy calculations, we have

1 6 R (|u| 6 -q 6 0 )χ R - 1 6 R (|u 0 | 6 -q 6 0 )χ R dx (2.68) = 1 6 R (|u| 2 -q 2 0 )(|u| 4 + q 2 0 |u| 2 -2q 4 0 ) + 3q 4 0 (|u| 2 -q 2 0 ) χ R dx (2.69) - 1 6 R (|u 0 | 2 -q 2 0 )(|u 0 | 4 + q 2 0 |u 0 | 2 -2q 4 0 ) + 3q 4 0 (|u 0 | 2 -q 2 0 ) χ R dx = 1 6 R (|u| 2 -q 2 0 ) 2 (|u| 2 + 2q 2 0 )χ R dx - 1 6 R (|u 0 | 2 -q 2 0 ) 2 (|u 0 | 2 + 2q 2 0 )χ R dx (2.70) + q 4 0 2 R (|u| 2 -q 2 0 )χ R dx - q 4 0 2 R (|u 0 | 2 -q 2 0 )χ R dx. (2.71)
Denote the term (2.70) by B R , we have

B R → 1 6 R (|u| 2 -q 2 0 )(|u| 4 + q 2 0 |u| 2 -2q 4 0 ) dx - 1 6 R (|u 0 | 2 -q 2 0 )(|u 0 | 4 + q 2 0 |u 0 | 2 -2q 4 0 ) dx (2.72)
as R → +∞. The term (2.71) converges to 0 as R → ∞ by (2.53). Finally, we have

lim R→∞ 1 6 R (|u| 6 -q 6 0 )χ R dx - 1 6 R (|u 0 | 6 -q 6 0 )χ R dx = 1 6 R (|u| 2 -q 2 0 ) 2 (|u| 2 + 2q 2 0 ) dx - 1 6 R (|u 0 | 2 -q 2 0 ) 2 (|u 0 | 2 + 2q 2 0 ) dx. ( 2
) = R |∂u| 2 - 1 2 Im(|u| 2 u -q 3 0 )∂u) dx + 1 6 R (|u| 2 -q 2 0 ) 2 (|u| 2 + 2q 2 0 ) dx - R |∂u 0 | 2 - 1 2 Im(|u 0 | 2 u -q 3 0 )∂u 0 ) dx - 1 6 R (|u 0 | 2 -q 2 0 ) 2 (|u 0 | 2 + 2q
R |∂u| 2 - 1 2 Im(|u| 2 u -q 3 0 )∂u) dx + 1 6 R (|u| 2 -q 2 0 ) 2 (|u| 2 + 2q 2 0 ) dx = R |∂u 0 | 2 - 1 2 Im(|u 0 | 2 u 0 -q 3 0 )∂u 0 ) dx + 1 6 R (|u 0 | 2 -q 2 0 ) 2 (|u 0 | 2 + 2q 2 0 ) dx.
This implies (2.8).

Conservation of momentum

Now, we prove (2.9). Multiplying both sides of (2.1) by -∂u and taking real part we obtain

0 = -Re(iu t ∂u + ∂ 2 u∂u + iu 2 (∂u) 2 ) = Im(u t ∂u) + Im(u 2 (∂u) 2 ) - 1 2 ∂(|∂u| 2 ). (2.75)
Moreover, by an elementary calculation, we have

∂ t Im(u∂u) = 2Im(u t ∂u) + ∂Im(u∂ t u). Replacing Im(u t ∂u) = 1 2 (∂ t Im(u∂u) -∂Im(u∂ t u)) in (2.75), we obtain that 0 = 1 2 ∂ t Im(u∂u) - 1 2 ∂Im(u∂ t u) + 2Re(u∂u)Im(u∂u) - 1 2 ∂(|∂u| 2 ) = ∂ t 1 2 Im(u∂u) - 1 4 (|u| 4 -q 4 0 ) + ∂ Im(|u| 2 u∂u) - 1 2 |∂u| 2 - 1 6 (|u| 6 -q 6 0 ) .
Multiply both sides by χ R , integrating on space and integrating by part, we have

0 = ∂ t R 1 2 Im(u∂u) - 1 4 (|u| 4 -q 4 0 ) χ R dx - R Im(|u| 2 u∂u) - 1 2 |∂u| 2 - 1 6 (|u| 6 -q 6 0 ) χ ′ R dx = ∂ t R 1 2 Im(u∂u) - 1 4 (|u| 2 -q 2 0 ) 2 - 1 2 q 2 0 (|u| 2 -q 2 0 ) χ R dx - R Im(|u| 2 u∂u) - 1 2 |∂u| 2 - 1 6 (|u| 6 -q 6 0 ) χ ′ R dx. (2.76)
Denoting the second term of (2.76) by D R , we have

|D R | ⩽ ∥Im(|u| 2 u∂u) - 1 2 |∂u| 2 - 1 6 (|u| 6 -q 6 0 )∥ L 2 ∥χ ′ R ∥ L 2 ≲ C(M )∥χ ′ R ∥ L 2 → 0 (2.77)
as R → ∞. Integrating from 0 to t the two sides of (2.76) and taking R to innity, using (2.77) and (2.53), we have

R 1 2 Im(u∂u) - 1 4 (|u| 2 -q 2 0 ) 2 dx = R 1 2 Im(u 0 ∂u 0 ) - 1 4 (|u 0 | 2 -q 2 0 ) 2 dx. (2.78)
We thus obtain the conservation of momentum, which completes the proof of Theorem 2.4.

Stationary solutions

In this section, we give the proof of Theorem 2.7. To convenience for readers, we rst introduce a fundamental lemma which is a classical version of the Cauchy-Lipschitz theorem:

Lemma 2.18. Let C 1 , C 2 ∈ R and f : R → R be a C 1 function. There exists a unique real valued C 2 local solution of following equation

     u xx = f (u), u(0) = C 1 , u x (0) = C 2 .
(2.79)

Remark 2.19. Let C 1 , C 2 ∈ C and f be considered as C 1 function from R 2 to R 2 .
By using Picard's uniqueness and existence theorem for system equations, we obtain the existence and uniqueness of complex valued solution for (2.79). However, the Lemma 2.18 is sucient for our analysis in this paper. Now, we give the proof of Theorem 2.7. We use the similar of variable changing as in [ where R > 0 and R, θ ∈ C 2 (R) are real-valued functions. We have

ϕ x = e iθ (R x + iθ x R), ϕ xx = e iθ (R xx + 2iR x θ x + iRθ xx -Rθ 2 x ).
Hence, since ϕ satises (2.10) we obtain

0 = (R xx -Rθ 2 x + R 3 θ x ) + i(2R x θ x + Rθ xx + R 2 R x ).
This is equivalent to

0 = R xx -Rθ 2 x + R 3 θ x , (2.80
)

0 = 2R x θ x + Rθ xx + R 2 R x . (2.81)
The equation (2.81) is equivalent to

0 = ∂ x R 2 θ x + 1 4 R 4 .
Hence there exists B ∈ R such that

B = R 2 θ x + 1 4 R 4 . (2.82)
This implies

θ x = B R 2 - R 2 4 . (2.83)
Substituting the above equality in (2.80) we obtain

0 = R xx -R B R 2 - R 2 4 2 + R 3 B R 2 - R 2 4 = R xx - B 2 R 3 - 5R 5 16 + 3BR 2 .
(2.84)

We prove that the set V = {x ∈ R : R x (x) ̸ = 0} is dense in R. Indeed, assume there exists x ∈ R \ V . Thus, there exists

ε such that B(x, ε) ∈ R \ V . It implies that for all y ∈ B(x, ε), we have R x (y) = 0 so R ≡ C 0 on B(x, ε) for some constant C 0 . Let x 0 ∈ B(x, ε) then R(x 0 ) = C 0 and R x (x 0 ) = 0. By Lemma 2.18, R ≡ C 0 . By (2.83), θ x is constant. Thus, ϕ(x) is of form Ce iαx , for some constants C, α ∈ R. If α = 0, ϕ is a constant and if α ̸ = 0 ϕ is not in X 1 (R)
, which contradicts the assumption of ϕ. From (2.84), we have

0 = R x R xx - B 2 R 3 - 5R 5 16 + 3BR 2 = d dx 1 2 R 2 x + B 2 2R 2 - 5 96 R 6 + 3B 4 R 2 .
Hence there exists a ∈ R such that

a = 1 2 R 2 x + B 2 2R 2 - 5 96 R 6 + 3B 4 R 2 .
This is equivalent to

0 = R 2 x R 2 + B 2 - 5 48 R 8 + 3B 2 R 4 -2aR 2 = 1 4 [(R 2 ) x ] 2 + B 2 - 5 48 R 8 + 3B 2 R 4 -2aR 2 .
Set k = R 2 . We have

0 = 1 4 k 2 x + B 2 - 5 48 k 4 + 3B 2 k 2 -2ak. (2.85)
Dierentiating the two sides of (2.85) we have

0 = k x k xx 2 - 5 12 k 3 + 3Bk -2a
On the other hand, since k x = 2R x R ̸ = 0 for a.e x in R, we obtain the following equation for a.e x in R, hence, by continuity of k, it is true for all x in R:

0 = k xx 2 - 5 12 k 3 + 3Bk -2a. ( 2
a = 4B √ B 3 . Set h = k -2 √ B. Then from (2.86) h ∈ H 3 (R) solves 0 = h xx -5 6 h 3 -5 √ Bh 2 -4Bh, h > -2 √ B, (2.87)
Since h ∈ H 3 (R), there exists

x 0 ∈ R such that h x (x 0 ) = 0. Indeed, if h x does not change sign on R then |h(-∞)| > 0 or |h(∞)| > 0. This contradicts to h ∈ H 3 (R).
Multiplying both sides of (2.87) by h x we obtain

0 = 1 2 ∂ x (h 2 x ) - 5 24 ∂ x (h 4 ) - 5 √ B 3 ∂ x (h 3 ) -2B∂ x (h 2 ). Since h ∈ H 3 (R) we have h(∞) = h x (∞) = 0 and hence, 1 2 (h x ) 2 = 5 24 h 4 + 5 √ B 3 h 3 + 2Bh 2 . (2.88) Using h x (x 0 ) = 0, since (2.88), we have h(x 0 ) = 0 or h(x 0 ) = 4 5 (-5 ± √ 10) √ B. If h(x 0 ) = 0 then by using Lemma 2.18, we have h ≡ 0, this is a contradiction. Since h > -2 √ B, we obtain h(x 0 ) = 4 5 (-5 + √ 10) √ B. Dene v(x) = h(x + x 0 ). We have      0 = v xx -5 6 v 3 -5 √ Bv 2 -4Bv, v(0) = 4 5 (-5 + √ 10) √ B, v x (0) = 0.
(2.89) Using Lemma 2.18, there exists a unique solution v of (2.89). Moreover, we can check that the following function is a solution of (2.89):

v(x) = -1 5 72B cosh(2 √ Bx) + 5 12 √ B .
Hence,

h(x) = -1 5 72B cosh(2 √ B(x -x 0 )) + 5 12 √ B
This implies

k = 2 √ B + h = 2 √ B + -1 5 72B cosh(2 √ B(x -x 0 )) + 5 12 √ B . Furthermore, using θ x = B k -k 4 , there exists θ 0 ∈ R such that θ(x) = θ 0 - ∞ x B k - k 4 dy.
Now, assume that ϕ is a solution of (2.10) such that ϕ(∞) = 0. We prove ϕ ≡ 0 on R. Multiplying both sides of (2.10) by ϕ then taking the imaginary part we obtain

∂ x Im(ϕ x ϕ) + 1 4 ∂ x (|ϕ| 4 ) = 0
On the other hand, ϕ(∞) = ϕ x (∞) = 0 then on R we have

Im(ϕ x ϕ) + 1 4 |ϕ| 4 = 0. (2.90)
If there exists y 0 such that ϕ x (y 0 ) = 0 then from (2.90) we have ϕ(y 0 ) = 0. By the uniqueness of Cauchy problem we obtain ϕ ≡ 0 on R. Otherwise, ϕ x does not vanish on R. From now on, we will consider this case. Multiplying both sides of (2.10) by ϕ x then taking the real part, we have

0 = Re(ϕ xx ϕ x ) -Im(ϕ 2 ϕ x 2 ) = 1 2 d dx |ϕ x | 2 -2Re(ϕϕ x )Im(ϕϕ x ) = 1 2 d dx |ϕ x | 2 -∂ x (|ϕ| 2 ) 1 4 |ϕ| 4 = d dx 1 2 |ϕ x | 2 - 1 12 |ϕ| 6 .
This implies that

|ϕ x | 2 - 1 6 |ϕ| 6 = 0.
Hence, since ϕ x is non vanishing, ϕ is also non vanishing on R. We can write ϕ = ρe iθ for ρ > 0, ρ, θ ∈ C 2 (R). Similar to (2.80) we have

0 = -ρθ 2 x + ρ xx + ρ 3 θ x . (2.91)
Replacing ϕ = ρe iθ in (2.90) we have

0 = ρ 2 θ x + 1 4 ρ 4 .
Then θ x = -1 4 ρ 2 , replacing this equality in (2.91) we obtain

0 = ρ xx - 5 16 ρ 5 .
Multiplying both sides of the above equality by ρ x we obtain

0 = ρ xx ρ x - 5 16 ρ 5 ρ x = d dx 1 2 ρ 2 x - 5 96 ρ 6 .
Hence,

0 = ρ 2 x - 5 48 ρ 6 .
Moreover, ϕ is non vanishing on R then ρ > 0 and then ρ x is not change sign on R.

If ρ x > 0 then since ρ(∞) = 0 we have ρ < 0 on R, a contradiction. Hence, ρ x < 0 and ρ x = -5 48 ρ 3 . From this we easily check that

ρ 2 (x) = 1 ρ(0) 2 + 5/12 x ,
which implies the contradiction, for the right hand side is not a continuous function on R. This completes the proof.

Lemma 2.20. Let B > 0 be the constant given as the above. The following is true:

k -2 √ B ∈ L 2 (R), k ∈ X 3 (R).
Proof. Using ϕ ∈ L ∞ (R) we obtain k ∈ L ∞ (R). On the other hand, since ϕ ∈ X 3 (R),

we have ϕ x ∈ L 2 (R), ϕ xx ∈ L 2 (R)
and it easy to see that

|ϕ x | 2 = k 2 x 4k + kθ 2 x ∈ L 1 (R), |ϕ xx | 2 = k x θ x √ k + θ xx √ k 2 + k xx 2 √ k - √ kθ 2 x - k 2 x 4k √ k 2 ∈ L 1 (R).
This implies

k x 2 √ k ∈ L 2 (R) and √ kθ x ∈ L 2 (R) k x θ x √ k + θ xx √ k ∈ L 2 (R) and k xx 2 √ k - √ kθ 2 x - k 2 x 4k √ k ∈ L 2 (R). Using √ m < k < ∥k∥ L ∞ , θ x = 4B-k 2 4k ∈ L ∞ (R), k x = 2RR x ∈ L ∞ ( indeed |ϕ x | 2 = |R x | 2 + |Rθ x | 2 ∈ L ∞ (R)) we have k x ∈ L 2 and θ x ∈ L 2 , θ xx ∈ L 2 and k xx ∈ L 2 .

By using

θ x = 4B-k 2 4k ∈ L 2 (R), we have 4B -k 2 ∈ L 2 (R). Thus, B ⩾ 0 and 2 √ B -k ∈ L 2 (R). If B = 0 then k ∈ L 2 (R), hence, R ∈ L 2 (R).
Which contradicts to the assumption m > 0. Thus, B > 0. It remains to prove that k xxx ∈ L 2 (R). Indeed, from ϕ xxx ∈ L 2 (R) we have

|ϕ xxx | 2 = |θ xxx √ k + M| 2 + k xxx 2 √ k + N 2 ∈ L 1 (R) (2.92)
where M, N are functions of θ, θ x , θ xx , k, k x , k xx . We can easily check that M, N ∈ L 2 (R). Hence, from (2.92) and the facts that

θ x ∈ H 1 (R), k ∈ X 2 (R), k bounded from below we obtain θ xxx , k xxx ∈ L 2 (R)
. This implies the desired results.

From now on, we will denote ϕ B is the stationary solution of (2.10) given by Theorem 2.7 with θ 0 = 0. We have

ϕ B = e iθ √ k , (2.93) k(x) = 2 √ B + -1 5 72B cosh(2 √ Bx) + 5 12 √ B , (2.94)
θ(x) = - ∞ x B k(y) - k(y) 4
dy.

(2.95)

We have the following asymptotic properties for ϕ B at ∞.

Proposition 2.21. Let B > 0 and ϕ B be kink solution of (2.1). Then for x > 0,

we have

|ϕ B -2 √ B| ≲ e - √ Bx .
As consequence ϕ B converges to 2 √ B as x tends to ∞ and there exists limit of ϕ B as x tends to -∞.

Proof. Since (2.94) we have

|k -2 √ B| ≲ e -2 √ Bx .
Hence, for all x ∈ R we have

|ϕ B (x) -2 √ B| ≲ |e iθ(x) k(x) -k(x)| + | k(x) -2 √ B| (2.96) ≲ ∥k∥ 1 2 L ∞ |e iθ(x) -1| + e - √ Bx
(2.97)

Moreover, for x > 0, we have

|e iθ(x) -1| ⩽ |θ(x)| ⩽ ∞ x B k - k 4 dx ⩽ ∞ x B k - √ B 2 + √ B 2 - k 4 dx ≲ ∞ x k -2 √ B dx ≲ ∞ x e -2 √ Bx dx ≲ e -2 √ Bx .
Combining with (2.97) we obtain

|ϕ B (x) -2 √ B| ≲ e - √ Bx .
As consequence ϕ B converges to 2 √ B as x tends to ∞. Since (2.94), we have

|k -2 √ B| ∈ L 1 (R) and k > 2 - 1 5 12 + √ 5 72 √ B. Thus, B k -k 4 = 4B-k 2 4k ∈ L 1 (R).
Hence since (2.95) we have

lim x→-∞ θ(x) = - ∞ -∞ B k(y) - k(y) 4 dy.
Hence,

lim x→-∞ ϕ B (x) = exp -i ∞ -∞ B k(y) - k(y) 4 dy 2 √ B.
This completes the proof. Introduction

In this chapter, we consider the derivative nonlinear Schrödinger equation on [0, +∞) with Robin boundary condition at 0:

     iv t + v xx = i 2 |v| 2 v x -i 2 v 2 v x -3 16 |v| 4 v for x ∈ R + , v(0) = φ, ∂ x v(t, 0) = αv(t, 0) ∀t ∈ R, (3.1)
where α ∈ R is a given constant.

The linear parts of (3.1) can be rewritten in the following forms:

iv t + H α v = 0 for x ∈ R + , v(0) = φ, (3.2)
where H α are self-adjoint operators dened by

H α : D( H α ) ⊂ L 2 (R + ) → L 2 (R + ), H α u = u xx , D( H α ) = u ∈ H 2 (R + ) : u x (0 + ) = αu(0 + )) .
We call e i Hαt : R → L(L 2 (R + )) is group dening the solution of (3.2). The derivative nonlinear Schrödinger equation was originally introduced in Plasma Physics as a simplied model for Alfvén wave propagation. Since then, it has attracted a lot of attention from the mathematical community (see e.g [START_REF] Colliander | Global wellposedness for Schrödinger equations with derivative[END_REF][START_REF] Colliander | A rened global well-posedness result for Schrödinger equations with derivative[END_REF][START_REF] Hayashi | The initial value problem for the derivative nonlinear Schrödinger equation in the energy space[END_REF][START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF][START_REF] Herr | On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition[END_REF][START_REF] Kaup | An exact solution for a derivative nonlinear Schrödinger equation[END_REF][START_REF] Takaoka | Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces[END_REF][START_REF] Takaoka | A priori estimates and weak solutions for the derivative nonlinear Schrödinger equation on torus below H 1/2[END_REF]).

Consider the equation (3.1), and set

u(t, x) = exp 3i 4 x ∞ |v(t, y)| 2 dy v(t, x).
Using the Gauge transformation, we see that u solves

iu t + u xx = i∂ x (|u| 2 u), t ∈ R, x ∈ (0, ∞), (3.3)
under a boundary condition ∂ x u(t, 0) = αu(t, 0) + 3i 4 |u(t, 0)| 2 u(t, 0). In all line case, there are many papers to deal with Cauchy problem of (3.3) (see e.g [START_REF] Hayashi | Finite energy solutions of nonlinear Schrödinger equations of derivative type[END_REF][START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem[END_REF][START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation[END_REF]). In [START_REF] Hayashi | Finite energy solutions of nonlinear Schrödinger equations of derivative type[END_REF], the authors establish the local well posedness in H 1 (R) by using a Gauge transform. Indeed, since u solves (3.3) on R, by setting

h(t, x) = exp -i x -∞ |u(t, y)| 2 dy u(t, x), k = h x + i 2 |h| 2 h, (3.4)
we have h, k solve

ih t + h xx = -ih 2 k, ik t + k xx = ik 2 h. (3.5)
By classical arguments, we can prove that there exists a unique solution

h, k ∈ C([0, T ], L 2 (R)) ∩ L 4 ([0, T ], L ∞ (R)) given h 0 , k 0 ∈ L 2 (R)
:= {u 0 ∈ H 2 (R + ), xu 0 ∈ L 2 (R + )}.
In this paper, we give a proof of existence of blow up solution of (3.1) under Robin boundary condition.

To study equation (3.1), we start by the denition of solution on H 1 (R + ). Since (3.1) contains a Robin boundary condition, the notion of solution in H 1 (R + ) is not completely clear. We use the following denition. Let I be an open interval of R. We say that v is a

H 1 (R + ) solution of the problem (3.1) on I if v ∈ C(I, H 1 (R + )) satises the following equation v(t) = e i Hαt φ -i t 0 e i Hα(t-s) g(v(s)) ds, (3.6)
where g is the function dened by

g(v) = i 2 |v| 2 v x - i 2 v 2 v x - 3 16 |v| 4 v.
Let v ∈ C(I, D( H α )) be classical solution of (3.1). At least formally, we have

1 2 ∂ t (|v| 2 ) = -∂ x Im(v x v).
Therefore, using the Robin boundary condition we have

∂ t 1 2 ∞ 0 |v| 2 dx = -Im(v x v)(∞) + Im(v x v)(0) = Im(v x v)(0) = αIm(|v(0)| 2 ) = 0.
This implies the conservation of the mass. By an elementary calculation, we have

∂ t |v x | 2 - 1 16 |v| 6 = ∂ x 2Re(v x v t ) - 1 2 |v| 2 |v x | 2 + 1 2 v 2 v 2 x .
Hence, integrating the two sides in space, we obtain

∂ t R + |v x | 2 dx - 1 16 |v| 6 dx = -2Re(v x (0)v t (0)) + 1 2 |v(0)| 2 |v x (0)| 2 - 1 2 v(0) 2 v x (0) 2
Using the Robin boundary condition for v, we obtain

∂ t R + |v x | 2 dx - 1 16 |v| 6 dx = -2αRe(v(0)v t (0)) = -α∂ t (|v(0)| 2 ).
This implies the conservation of the energy.

In this paper, we will need the following assumption.

Assumption A. We assume that for all φ ∈ H 1 (R + ) there exist a solution v ∈ C(I, H 1 (R + )) of (3.1) for some interval I ⊂ R. Moreover, v satises the following conservation law:

M (v) := 1 2 ∥v∥ 2 H 1 (R + ) = M (φ), E(v) := 1 2 ∥v x ∥ 2 L 2 (R + ) - 1 32 ∥v∥ L 6 (R + ) + α 2 |v(0)| 2 .
The existence of blowing up solutions for classical nonlinear Schrödinger equations was considered by Glassey [START_REF] Glassey | On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations[END_REF] in 1977. He introduced a concavity argument based on the second derivative in time of ∥xu(t)∥ 2 L 2 to show the existence of blowing up solutions. In this paper, we are also interested in studying the existence of blowing-up solutions of (3.1). In the limit case α = +∞, which is formally equivalent to Dirichlet boundary condition if we write v(0) = 1 α v ′ (0) = 0. In [START_REF] Wu | Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space[END_REF], Wu proved the blow up in nite time of solutions of (3.3) with Dirichlet boundary condition and some conditions on the initial data. Using the method of Wu [START_REF] Wu | Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space[END_REF] we obtain the existence of blowing up solutions in the case α ⩾ 0, under a weighted space condition for the initial data and negativity of the energy. Our rst main result is the following. Theorem 3.1. We assume that Assumption A holds. Let α ⩾ 0 and φ ∈ Σ where

Σ = u ∈ D( H α ), xu ∈ L 2 (R + ))
such that E(φ) < 0. Then the solution v of (3.1) blows-up in nite time i.e T min > -∞ and T max < +∞. Remark 3.2. In (3.1), if we consider nonlinear term i|v| 2 v x instead of i

2 |v| 2 v x -i 2 v 2 v x - 3
16 |v| 4 v then there is no conservation of energy of solution. Indeed, set

u(t, x) = v(t, x) exp - i 4 x ∞ |v(t, y)| 2 dy . If v solves iv t + v xx = i|v| 2 v x , ∂ x v(t, 0) = αv(t, 0) then u solves iu t + u xx = i 2 |u| 2 u x -i 2 u 2 u x -3 16 |u| 4 u, ∂ x u(t, 0) = αu(t, 0) -i 4 |u(t, 0)| 2 u(t, 0). (3.7)
By an elementary calculation, since u solves (3.7), we have

∂ t |u x | 2 - 1 16 |u| 6 = ∂ x 2Re(u x u t ) - 1 2 |u| 2 |u x | 2 + 1 2 u 2 u x 2 .
Integrating the two sides in space, we obtain

∂ t R + |u x | 2 - 1 16 |u| 6 dx = -2Re(u x (0)u t (0)) + 1 2 |u(0)| 2 |u x (0)| 2 - 1 2 u(0) 2 u x (0) 2 .
Using the boundary condition of u, we obtain

∂ t R + |u x | 2 - 1 16 |u| 6 dx = -2αRe(u(0)u t (0)) - 1 2 Im(u(0)|u(0)| 2 u t (0)) + 1 2 |u(0)| 4 α 2 + 1 16 |u(0)| 4 -α + i 4 |u(0)| 2 2 = -α∂ t (|u(0)| 2 ) + A, where A = -1 2 Im(u(0)|u(0)| 2 u t (0)) + 1 2 |u(0)| 4 α 2 + 1 16 |u(0)| 4 -α + i 4 |u(0)| 2 2
. Moreover, we can not write A in form ∂ t B(u(0)), for some function B : C → C. Then, there is no conservation of energy of u and hence, there is no conservation of energy of v.

The stability of standing waves for classical nonlinear Schrödinger equations was originally studied by Cazenave and Lions [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF] with variational and compactness arguments. A second approach, based on spectral arguments, was introduced by Weinstein [START_REF] Weinstein | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF][START_REF] Weinstein | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF] and then considerably generalized by Grillakis, Shatah and Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] (see also [START_REF] De Bièvre | Orbital stability: analysis meets geometry[END_REF], [START_REF] De Bièvre | Orbital stability via the energy-momentum method: the case of higher dimensional symmetry groups[END_REF]). In our work, we use the variational techniques to study the stability of standing waves. First, we dene

S ω (v) := 1 2 ∥v x ∥ 2 L 2 (R + ) + ω∥v∥ 2 L 2 (R + ) + α|v(0)| 2 - 1 32 ∥v∥ 6 L 6 (R + ) , K ω (v) := ∥v x ∥ 2 L 2 (R + ) + ω∥v∥ 2 L 2 (R + ) + α|v(0)| 2 - 3 16 ∥v∥ 6 L 6 (R + )
. We are interested in the following variational problem:

d(ω) := inf S ω (v) | K ω (v) = 0, v ∈ H 1 (R + ) \ {0} . (3.8)
We have the following result.

Proposition 3.3. Let ω, α ∈ R such that ω > α 2 . All minimizers of (3.8) are of form e iθ φ, where θ ∈ R and φ is given by

φ = 2 4 √ ω sech 1 2 2 √ ωx + tanh -1 -α √ ω .
We give the denition of stability and instability by blow up in H 1 (R + ). Let w(t, x) = e iωt φ(x) be a standing wave solution of (3.1).

(1) The standing wave w is called orbitally stable in H 1 (R + ) if for all ε > 0, there

exists δ > 0 such that if v 0 ∈ H 1 (R + ) satises ∥v 0 -φ∥ H 1 (R + ) ⩽ δ,
then the associated solution v of (3.1) satises

sup t∈R inf θ∈R ∥v(t) -e iθ φ∥ H 1 (R + ) < ε.
Otherwise, w said to be unstable.

(2) The standing wave w is called unstable by blow up if there exists a sequence

(φ n ) such that lim n→∞ ∥φ n -φ∥ H 1 (R + ) =
0 and the associated solution v n of (3.1) blows up in nite time for all n.

Our second main result is the following. Theorem 3.4. Let α, ω ∈ R be such that ω > α 2 . The standing wave e iωt φ, where φ is the prole as in Proposition 3.3, solution of (3.1), satises the following properties.

(1) If α < 0 then the standing wave is orbitally stable in H 1 (R + ).

(2) If α > 0 then the standing wave is strongly unstable. Remark 3.5. To our knowledge, the conservation law play an important role to study the stability of standing waves. However, the existence of conservation of energy is not always true (see remark 3.2). Our work can only extend for the models with nonlinear terms provide the conservation law of solution.

This paper is organized as follows. First, under the assumption of local well posedness in H 1 (R + ), we prove the existence of blowing up solutions using a virial argument Theorem 3.1. In section 3.2.1, we give the proof of Theorem 3.1. Second, in the case α < 0, using similar arguments as in [START_REF] Colin | Stability of solitary waves for derivative nonlinear Schrödinger equation[END_REF], we prove the orbital stability of standing waves of (3.1). In the case α > 0, using similar arguments as in [START_REF] Coz | Standing waves in nonlinear Schrödinger equations[END_REF], we prove the instability by blow up of standing waves. The proof of Theorem 3.4 is obtained in Section 3.2.2.

Proof of the main results

We consider the equation (3.1) and assume that Assumption A holds.

The existence of a blow-up solutions

In this section, we give the proof of Theorem 3.1 using a virial argument (see e.g [START_REF] Glassey | On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations[END_REF] or [START_REF] Wu | Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space[END_REF] for similar arguments). Let α ⩾ 0. Let v be a solution of (3.1). To prove the existence of blowing up solutions we use similar arguments as in [START_REF] Wu | Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space[END_REF]. Set

I(t) = ∞ 0 x 2 |v(t)| 2 dx. Let u(t, x) = v(t, x) exp - i 4 +∞ x |v| 2 dy (3.9)
be a Gauge transform in H 1 (R + ). Then the problem (3.1) is equivalent with

iu t + u xx = i|u| 2 u x , u x (0) = αu(0) + i 4 |u(0)| 2 u(0).
(3.10)

The equation (3.10) has a simpler nonlinear form, but we pay this simplication with a nonlinear boundary condition. Observe that

I(t) = ∞ 0 x 2 |u(t)| 2 dx = ∞ 0 x 2 |v(t)| 2 dx.
By a direct calculation, we get

∂ t I(t) = 2Re ∞ 0 x 2 u(t, x)∂ t u(t, x) dx = 2Re ∞ 0 x 2 u(iu xx + |u| 2 u x ) dx (3.11) = 2Im ∞ 0 2xuu x dx - 1 2 ∞ 0 2x|u| 4 dx (3.12) = 4Im ∞ 0 xu x u dx - ∞ 0 x|u| 4 dx. (3.13)
Dene

J(t) = Im ∞ 0 xu x u dx.
We have

∂ t J(t) = ∞ 0 xu x u t dx + ∞ 0 xuu xt dx = -Im ∞ 0 xu t u x dx -Im ∞ 0 (xu) x u t dx = -2Im ∞ 0 xu t u x dx -Im ∞ 0 u t u dx = -2Im ∞ 0 xu x (iu xx + |u| 2 u x ) dx -Im ∞ 0 u(iu xx + |u| 2 u x ) dx = -2Re ∞ 0 xu x u xx dx -Re ∞ 0 uu xx dx -Im ∞ 0 |u| 2 u x u dx = - ∞ 0 x∂ x |u x | 2 dx -Re(uu x )(+∞) + Re(uu x )(0) + Re ∞ 0 u x u x dx -Im ∞ 0 |u| 2 u x u dx = ∞ 0 |u x | 2 dx + Re(u(0)u x (0)) + ∞ 0 |u x | 2 dx -Im ∞ 0 |u| 2 u x u dx = 2 ∞ 0 |u x | 2 dx -Im ∞ 0 |u| 2 u x u dx + Re(u(0)u x (0)).
Using the Robin boundary condition we have

∂ t J(t) = 2 ∞ 0 |u x | 2 dx -Im ∞ 0 |u| 2 u x u dx + α|u(0)| 2 .
Moreover using the expression of v in term of u given in (3.9), we get

∂ t J(t) = 2 ∞ 0 |v x | 2 dx - 1 8 ∞ 0 |v| 6 dx + α|v(0)| 2 = 4E(v) -α|v(0)| 2 ⩽ 4E(v) = 4E(φ).
By integrating the two sides of the above inequality in time we have

J(t) ⩽ J(0) + 4E(φ)t. (3.14)
Integrating the two sides of (3.11) in time we have

I(t) = I(0) + 4 t 0 J(s) ds - t 0 ∞ 0 x|u(s, x)| 4 dx ds ⩽ I(0) + 4 t 0 J(s) ds.
Using (3.14) we have

I(t) ⩽ I(0) + 4 t 0 (J(0) + 4E(φ)s) ds ⩽ I(0) + 4J(0)t + 8E(φ)t 2 .
From the assumption E(φ) < 0, there exists a nite time T * > 0 such that I(T * ) = 0,

I(t) > 0 for 0 < t < T * . Note that ∞ 0 |φ(x)| 2 dx = ∞ 0 |v(t, x)| 2 dx = -2Re ∞ 0 xv(t, x)v x (t, x) dx ⩽ 2∥xv∥ L 2 x (R + ) ∥v x ∥ L 2 x (R + ) = 2 I(t)∥v x ∥ L 2 x (R + ) .
Then there exists a constant C = C(φ) > 0 such that

∥v x ∥ L 2 x (R + ) ⩾ C 2 I(t) → +∞ as t → T * .
Then the solution v blows up in nite time in H 1 (R + ). This complete the proof of Theorem 3.1.

Stability and instability of standing waves

In this section, we give the proof of Theorem 3.4 and Proposition 3.3. First, we nd the form of the standing waves of (3.1).

Standing waves

Let v = e iωt φ be a solution of (3.1). Then φ solves

     0 = φ xx -ωφ + 1 2 Im(φ x φ)φ + 3 16 |φ| 4 φ, for x > 0 φ x (0) = αφ(0), φ ∈ H 2 (R + ). (3.15) Set A := ω - 1 2 Im(φ x φ) - 3 16 |φ| 4
By writing φ = f + ig for f and g real valued functions, for x > 0, we have

f xx = Af, g xx = Ag. Thus, ∂ x (f x g -g x f ) = f xx g -g xx f = 0 when x ̸ = 0.
Hence, by using f, g ∈ H 2 (R + ), we have

f x (x)g(x) -g x (x)f (x) = 0 when x ̸ = 0.
Then, for all x ̸ = 0, we have

Im(φ x (x)φ(x)) = g x (x)f (x) -f x (x)g(x) = 0, hence, (3.15) is equivalent to      0 = φ xx -ωφ + 3 16 |φ| 4 φ, for x > 0 φ x (0) = αφ(0), φ ∈ H 2 (R + ). (3.16)
We have the following description of the prole φ. Proposition 3.6. Let ω > α 2 . There exists a unique (up to phase shift) solution φ of (3.16), which is of the form

φ = 2 4 √ ω sech 1 2 2 √ ωx + tanh -1 -α √ ω , (3.17)
for all x > 0.

Proof. Let w be the even function dened by

w(x) = φ(x) if x ⩾ 0, φ(-x) if x ⩽ 0. Then w solves      0 = -w xx + ωw -3 16 |w| 4 w, for x ̸ = 0, w x (0 + ) -w x (0 -) = 2αw(0), w ∈ H 2 (R) \ {0} ∩ H 1 (R). (3.18)
Using the results of Fukuizumi and Jeanjean [START_REF] Fukuizumi | Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential[END_REF], we obtain that

w(x) = 2 4 √ ω sech 1 2 2 √ ω|x| + tanh -1 -α √ ω
up to phase shift provided ω > α 2 . Hence, for x > 0 we have

φ(x) = 2 4 √ ω sech 1 2 2 √ ω|x| + tanh -1 -α √ ω
up to phase shift. This implies the desired result.

The variational problems

In this section, we give the proof of Proposition 3.3. First, we introduce another variational problem:

d(ω) := inf S ω (v) | v even, K ω (v) = 0, v ∈ H 1 (R) \ {0} , (3.19)
where S ω , K ω are dened for all v ∈ H 1 (R) by

S ω (v) := 1 2 ∥v x ∥ 2 L 2 (R) + ω∥v∥ 2 L 2 (R) + 2α|v(0)| 2 - 1 32 ∥v∥ 6 L 6 (R) , K ω (v) := ∥v x ∥ 2 L 2 (R) + ω∥v∥ 2 L 2 (R) + 2α|v(0)| 2 - 3 16 ∥v∥ 6 L 6 (R) .
The functional K ω is called Nehari functional. The following result has proved in [START_REF] Fukuizumi | Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential[END_REF][START_REF] Fukuizumi | Nonlinear Schrödinger equation with a point defect[END_REF].

Proposition 3.7. Let ω > α 2 and φ satises

-φ xx + 2αδφ + ωφ -3 16 |φ| 4 φ = 0, φ ∈ H 1 (R) \ {0} . (3.20)
Then, there exists a unique positive solution φ of (3.20). This solution is the unique positive minimizer of (3.19). Furthermore, we have an explicit formula for φ

φ(x) = 2 4 √ ω sech 1 2 2 √ ω|x| + tanh -1 -α √ ω .
We have the following relation between the variational problems.

Proposition 3.8. Let ω > α 2 . We have

d(ω) = 1 2 d(ω).
Proof. Assume v is a minimizer of (3.8), dene the H 1 (R) function w by

w(x) = v(x) if x > 0, v(-x) if x < 0. The function w ∈ H 1 (R) \ {0} veries S ω (w) = 2S ω (v) = 2d(ω), K ω (w) = 2K ω (v) = 0. This implies that d(ω) ⩽ S ω (w) = 2d(ω). (3.21)
Now, assume v is a minimizer of (3.19). Let w be the restriction of v on R + , then,

K ω (w) = 1 2 K ω (v) = 0.
Hence, we obtain d(ω) = S ω (v) = 2S ω (w) ⩾ 2d(ω). This implies the desired result.

Proof of Proposition 3.3. Let v be a minimizer of (3.8). Dene w(x) ∈ H 1 (R) by

w(x) = v(x) if x > 0, v(-x) if x < 0.
Then, w is an even function. Moreover, w satises

K ω (w) = 2K ω (v) = 0, S ω (w) = 2S ω (v) = 2d(ω) = d(ω).
Hence, w is a minimizer of (3.19). From Propositions 3.7, 3.8, w is of the form e iθ φ, where θ ∈ R is a constant and φ is of the form

2 4 √ ω sech 1 2 2 √ ω|x| + tanh -1 -α √ ω .
Hence, v = w| R + satises v(x) = e iθ φ(x), for x > 0. This completes the proof of Proposition 3.3.

Stability and instability of standing waves

In this section, we give the proof of Theorem 3.4. We use the notations S ω and K ω as in Section 3.2.2. First, we dene

N (v) := 3 16 ∥v∥ 6 L 6 (R + ) , (3.23)
L(v) := ∥v x ∥ 2 L 2 (R + ) + ω∥v∥ 2 L 2 (R + ) + α|v(0)| 2 . (3.24)
We can rewrite S ω , K ω as follows

S ω = 1 2 L - 1 6 N, K ω = L -N.
We have the following classical properties of the above functions.

Lemma 3.9. Let (ω, α) ∈ R 2 such that ω > α 2 . The following assertions hold.

(1) There exists a constant C > 0 such that

L(v) ⩾ C∥v∥ 2 H 1 (R + ) ∀v ∈ H 1 (R + ).
(2) We have d(ω) > 0.

(

) If v ∈ H 1 (R + ) satises K ω (v) < 0 then L(v) > 3d(ω). 3
Proof. We have

|v(0)| 2 = - ∞ 0 ∂ x (|v(x)| 2 ) dx = -2Re ∞ 0 v(x)v x (x) dx ⩽ 2∥v∥ L 2 (R + ) ∥v x ∥ L 2 (R + ) .
Hence,

L(v) = ∥v x ∥ 2 L 2 (R + ) + ω∥v∥ 2 L 2 (R + ) + α|v(0)| 2 ⩾ ∥v x ∥ 2 L 2 (R + ) + ω∥v∥ 2 L 2 (R + ) -2|α|∥v∥ L 2 (R + ) ∥v x ∥ L 2 (R + ) ⩾ C∥v∥ 2 H 1 (R + ) + (1 -C)∥v x ∥ 2 L 2 (R + ) + (ω -C)∥v∥ 2 L 2 (R + ) -2|α|∥v∥ L 2 (R + ) ∥v x ∥ L 2 (R + ) ⩾ C∥v∥ 2 H 1 (R + ) + (2 (1 -C)(ω -C) -2|α|)∥v∥ L 2 (R + ) ∥v x ∥ L 2 (R + ) .
From the assumption ω > α 2 , we can choose C ∈ (0, 1) such that

2 (1 -C)(ω -C) -2|α| > 0.
This implies [START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF]. Now, we prove (2). Let v be an element of H 1 (R + ) satisfying K ω (v) = 0. We have

C∥v∥ 2 H 1 (R + ) ⩽ L(v) = N (v) ⩽ C 1 ∥v∥ 6 H 1 (R + ) .
Then,

∥v∥ 2 H 1 (R + ) ⩾ 4 C C 1 .
From the fact that, for v satisfying K ω (v) = 0, we have

S ω (v) = S ω (v) -1 6 K ω (v) = 1 3 L(v), this implies that d(ω) = 1 3 inf L(v) : v ∈ H 1 (R + ), K ω (v) = 0 ⩾ C 3 4 C C 1 > 0.
Finally, we prove (3). Let v ∈ H 1 (R + ) satisfying K ω (v) < 0. Then, there exists

λ 1 ∈ (0, 1) such that K ω (λ 1 v) = λ 2 1 L(v) -λ 6 1 N (v) = 0. Since v ̸ = 0, we have 3d(ω) ⩽ L(λ 1 v) = λ 2 1 L(v) < L(v). Dene Ñ (v) := 3 16 ∥v∥ 6 L 6 , (3.25)
L(v) := ∥v x ∥ 2 L 2 + ω∥v∥ 2 L 2 + 2α|v(0)| 2 . (3.26)
We can rewrite S ω , K ω as follows

Sω = 1 2 L - 1 6 Ñ , Kω = L -Ñ .
As consequence of the previous lemma, we have the following result.

Lemma 3.10. Let (ω, α) ∈ R 2 such that ω > α 2 . The following assertions hold.

(1) There exists a constant C > 0 such that

L(v) ⩾ C∥v∥ 2 H 1 ∀v ∈ H 1 (R).
(2) We have d(ω) > 0.

(

) If v ∈ H 1 satises Kω (v) < 0 then L(v) > 3 d(ω). 3
We introduce the following properties.

Lemma 3.11 (Brezis-Lieb [START_REF] Brézis | A relation between pointwise convergence of functions and convergence of functionals[END_REF]). Let 2 ⩽ p < ∞ and (f n ) be a bounded sequence in

L p (R). Assume that f n → f a.e in R.
Then we have

∥f n ∥ p L p -∥f n -f ∥ p L p -∥f ∥ p L p → 0.
Lemma 3.12. The following minimization problem is equivalent to the problem (3.19) i.e same minimum and the minimizers:

d := inf 1 16 ∥u∥ 6 L 6 : u even , u ∈ H 1 (R) \ {0}, Kω (u) ⩽ 0 . (3.27)
Proof. We see that the minimizer problem (3.19) is equivalent to following problem:

inf 1 16 ∥u∥ 6 L 6 : u even u ∈ H 1 (R) \ {0}, Kω (u) = 0 . (3.28)
Let v be a minimizer of (3.19) then Kω (v) ⩽ 0, hence, d(ω) = 1 16 ∥v∥ 6 L 6 ⩾ d. Now, let v be a minimizer of (3.27). We prove that Kω (v) = 0. Indeed, assuming Kω (v) < 0, we have

Kω (λv) = λ 2 ∥v x ∥ 2 L 2 + ω∥v∥ 2 L 2 + 2α|v(0)| 2 - 3λ 4 16 ∥v∥ 6 L 6 ⩽ 0,
as 0 < λ is small enough. Thus, by continuity, there exists a λ 0 ∈ (0, 1) such that Kω (λ 0 v) = 0. We have

d < d(ω) ⩽ 1 16 ∥λ 0 v∥ 6 L 6 < 1 16 ∥v∥ 6 L 6 = d.
Which is a contradiction. It implies that Kω (v) = 0 and v is a minimizer of (3.28), hence v is a minimizer of (3.19). This completes the proof. Now, using the similar arguments in [39, Proof of Proposition 2], we have the following result. Proposition 3.13. Let (ω, α) ∈ R 2 be such that α < 0, ω > α 2 and (w n ) ⊂ H 1 (R) be a even sequence satisfying the following properties

S ω (w n ) → d(ω), K ω (w n ) → 0.
as n → ∞. Then, there exists a minimizer w of (3.19) such that w n → w strongly in H 1 (R) up to subsequence.

Proof.

In what follows, we shall often extract subsequence without mentioning this fact explicitly. We divide the proof into two steps.

Step 1. Weakly convergence to a nonvanishing function of minimizer sequence We have

1 3 L(w n ) = Sω (w n ) - 1 6 Kω (w n ) → d(ω),
as n → ∞. Then, (w n ) is bounded in H 1 (R) and there exists w ∈ H 1 (R) even such that w n ⇀ w in H 1 (R) up to subsequence. We prove w ̸ = 0. Assume that w ≡ 0. Dene, for u ∈ H 1 (R),

S 0 ω (u) = 1 2 ∥u x ∥ 2 L 2 + ω 2 ∥u∥ 2 L 2 - 1 32 ∥u∥ 6 L 6 , K 0 ω (u) = ∥u x ∥ 2 L 2 + ω∥u∥ 2 L 2 - 3 16 ∥u∥ 6 L 6 .
Let ψ ω be minimizer of following problem

d 0 (ω) = inf S 0 ω (u) : u even , u ∈ H 1 (R) \ {0}, K 0 ω (u) = 0 = inf 1 16 ∥u∥ 6 L 6 : u even , u ∈ H 1 (R) \ {0}, K 0 ω (u) ⩽ 0 . We have K 0 ω (w n ) = Kω (w n ) -2α|w n (0)| 2 → 0, as n → ∞. Since, α < 0.
we have Kω (ψ ω ) < 0 and hence we obtain

d(ω) < 1 16 ∥ψ ω ∥ 6 L 6 = d 0 (ω) (3.29)
We set

λ n = ∥∂ x w n ∥ 2 L 2 + ω∥w n ∥ 2 L 2 3 16 ∥w n ∥ 6 L 6 1 4
.

We here remark that 0 < d(ω) = lim n→∞ 1 16 ∥w n ∥ 6 L 6 . It follows that

λ 4 n -1 = K 0 ω (w n ) 3 16 ∥w n ∥ 6 L 6 → 0,
as n → ∞. We see that K 0 ω (λ n w n ) = 0 and λ n w n ̸ = 0. By the denition of d 0 (ω), we have

d 0 (ω) ⩽ 1 16 ∥λ n w n ∥ 6 L 6 → d(ω) as n → ∞.
This contradicts to (3.29). Thus, w ̸ = 0.

Step 2. Conclude the proof Using Lemma 3.11 we have

Kω (w n ) -Kω (w n -w) -Kω (w) → 0, (3.30)
L(w n ) -L(w n -w) -L(w) → 0. (3.31)
Now, we prove Kω (w) ⩽ 0 by contradiction. Suppose that Kω (w) > 0. By the assumption Kω (w n ) → 0 and (3.30), we have Kω (w n -w) → -Kω (w) < 0.

Thus, Kω (w n -w) < 0 for n large enough. By Lemma 3.10 (3), we have L(w n -w) ⩾ 3 d(ω). Since L(w n ) → 3 d(ω), by (3.31), we have

L(w) = lim n→∞ ( L(w n ) -L(w n -w)) ⩽ 0.
Moreover, w ̸ = 0 and by Lemma 3.10 (1), we have L(w) > 0. This is a contradiction. Hence, Kω (w) < 0. By Lemma 3.10 (2), (3) and weakly lower semicontinuity of L, we have

3 d(ω) ⩽ L(w) ⩽ lim n→∞ inf L(w n ) = 3 d(ω).
Thus, L(w) = 3 d(ω). Combining with (3.31), we have L(w n -w) → 0, as n → ∞. By Lemma 3.10 (1), we have w n → w strongly in H 1 (R). Hence, w is a minimizer of (3.19). This completes the proof.

To prove the stability statement (1) for α < 0 in Theorem 3.4, we will use similar arguments as in the work of Colin and Ohta [START_REF] Colin | Stability of solitary waves for derivative nonlinear Schrödinger equation[END_REF]. We need the following property. Lemma 3.14

. Let α < 0, ω > α 2 . If a sequence (v n ) ⊂ H 1 (R + ) satises S ω (v n ) → d(ω), (3.32)
K ω (v n ) → 0, (3.33)
then there exist a constant θ 0 ∈ R such that v n → e iθ 0 φ, up to subsequence, where φ is dened as in Proposition 3.3.

Proof. Dene the sequence (w n ) ⊂ H 1 (R) as follows,

w n (x) = v n (x) for x > 0, v n (-x) for x < 0.
We can check that

S ω (w n ) = 2S ω (v n ) → 2d(ω) = d(ω), K ω (w n ) = 2K ω (v n ) → 0,
as n → ∞. Using Proposition 3.13, there exists a minimizer w 0 of (3.19) such that w n → w 0 strongly in H 1 (R), up to subsequence. For convenience, we assume that w n → w 0 strongly in H 1 (R). By Proposition 3.7, there exists a constant θ 0 ∈ R such that w 0 = e iθ 0 φ, where φ is dened by φ(x) = φ(x) for x > 0, φ(-x) for x < 0.

(3.34)

Hence, the sequence (v n ) is the restriction of the sequence (w n ) on R + , and satises v n → e iθ 0 φ, strongly in H 1 (R + ), up to subsequence. This completes the proof.

Dene

A + ω = v ∈ H 1 (R + ) \ {0} : S ω (v) < d(ω), K ω (v) > 0 , A - ω = v ∈ H 1 (R + ) \ {0} : S ω (v) < d(ω), K ω (v) < 0 , B + ω = v ∈ H 1 (R + ) \ {0} : S ω (v) < d(ω), N (v) < 3d(ω) , B - ω = v ∈ H 1 (R + ) \ {0} : S ω (v) < d(ω), N (v) > 3d(ω) .
We have the following result.

Lemma 3.15. Let ω, α ∈ R 2 such that α < 0 and ω > α 2 .

(1) The sets A + ω and A - ω are invariant under the ow of (3.1). ( 2)

A + ω = B + ω and A - ω = B - ω .
Proof. (1) Let u 0 ∈ A + ω and u(t) the associated solution for (3.1) on (T min , T max ). By u 0 ̸ = 0 and the conservation laws, we see that S ω (u(t)) = S ω (u 0 ) < d(ω) for t ∈ (T min , T max ). Moreover, by denition of d(ω) we have K ω (u(t)) ̸ = 0 on (T min , T max ). Since the function t → K ω (u(t)) is continuous, we have K ω (u(t)) > 0 on (T min , T max ). Hence, A + ω is invariant under ow of (3.1). By the same way, A - ω is invariant under ow of (3.1).

(2) If v ∈ A + ω then by (3.26), (3.25) we have

N (v) = 3S ω (v) -2K ω (v) < 3d(ω), which shows v ∈ B + ω , hence A + ω ⊂ B + ω . Now, let v ∈ B + ω .
We show K ω (v) > 0 by contradiction. Suppose that K ω (v) ⩽ 0. Then, by Lemma 3.10 (3), L(v) ⩾ 3d(ω). Thus, by (3.26) and (3.25), we have

S ω (v) = 1 2 L(v) - 1 6 N (v) ⩾ d(ω), which contradicts S ω (v) < d(ω). Therefore, we have K ω (v) > 0, which shows v ∈ A + ω and B + ω ⊂ A + ω . Next, if v ∈ A - ω ,
(v) = L(v) -K ω (v) > 3d(ω), which shows v ∈ B - ω . Thus, A - ω ⊂ B - ω . Finally, if v ∈ B - ω ,
(v) = 3S ω (v) -N (v) < 3d(ω) -3d(ω) = 0, which shows v ∈ A - ω , hence, B - ω ⊂ A - ω .
This completes the proof. From Proposition 3.3, we have

d(ω) = S ω (φ).
Since α < 0, we see that

d ′′ (ω) = ∂ ω ∥φ∥ 2 L 2 (R + ) = 1 2 ∂ ω ∥ φ∥ 2 L 2 (R) > 0,
where φ is dened as (3.34) and we know from [START_REF] Fukuizumi | Nonlinear Schrödinger equation with a point defect[END_REF], [START_REF] Fukuizumi | Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential[END_REF] that

∂ ω ∥ φ∥ 2 L 2 (R) > 0,
for α < 0. We dene the function h : (-ε 0 , ε 0 ) → R by

h(τ ) = d(ω ± τ ),
for ε 0 > 0 suciently small such that h ′′ (τ ) > 0 and the sign + oris selected such that h ′ (τ ) > 0 for τ ∈ (-ε 0 , ε 0 ). Without loss of generality, we can assume

h(τ ) = d(ω + τ ).
Lemma 3.16. Let (ω, α) ∈ R 2 such that ω > α 2 and let h be dened as above.

Then, for any ε ∈ (0, ε 0 ), there exists δ > 0 such that if v 0 ∈ H 1 (R + ) satises ∥v 0 -φ∥ H 1 (R + ) < δ, then the solution v of (3.1) with v(0) = v 0 satises 3h(-ε) < N (v(t)) < 3h(ε) for all t ∈ (T min , T max ).

Proof. The proof of the above lemma is similar to the one of [START_REF] Colin | Stability of solitary waves for derivative nonlinear Schrödinger equation[END_REF] or [START_REF] Shatah | Stable standing waves of nonlinear Klein-Gordon equations[END_REF]. Let if u 0 ∈ H 1 (R + ) satises ∥u 0 -φ∥ H 1 (R + ) < δ then we have 3h(0) = N (u 0 ) + O(δ) and 3h(-ε) < N (u 0 ) < 3h(ε) for suciently small δ > 0. Since h(±ε) = d(ω ± ε) and the set B ± ω are invariant under the ow of (3.1) by Lemma 3.15, to conclude the proof, we only have to show that there exists δ > 0 such that if

u 0 ∈ H 1 (R + ) satises ∥u 0 -φ∥ H 1 (R + ) < δ then S ω±ε (u 0 ) < h(±ε). Assume that u 0 ∈ H 1 (R + ) satises ∥u 0 -φ∥ H 1 (R + ) < δ. We have S ω±ε (u 0 ) = S ω±ε (φ) + O(δ) = S ω (φ) ± εM (φ) + O(δ) = h(0) ± εh ′ (0) + O(δ).
On the other hand, by the Taylor expansion, there exists

τ 1 = τ 1 (ε) ∈ (-ε 0 , ε 0 ) such that h(±ε) = h(0) ± εh ′ (0) + ε 2 2 h ′′ (τ 1 ).
Since h ′′ (τ 1 ) > 0 by denition of h, we see that there exists δ > 0 such that if

u 0 ∈ H 1 (R + ) satises ∥u 0 -φ∥ H 1 (R + ) < δ then S ω±ε (u 0 ) < h(±ε)
. This completes the proof.

Proof of Theorem 3.4 [START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF]. Assume that e iωt φ is not stable for (3.1). Then, there exists a constant ε 1 > 0, a sequence of solutions (v n ) to (3.1), and a sequence

{t n } ∈ (0, ∞) such that v n (0) → φ in H 1 (R + ), inf θ∈R ∥v n (t n ) -e iθ φ∥ H 1 (R + ) ⩾ ε 1 . (3.35)
By using the conservation laws of solutions of (3.1), we have

S ω (v n (t n )) = S ω (v n (0)) → S ω (φ) = d(ω). (3.36)
Using Lemma 3.16, we have

N (v n (t n )) → 3d(ω). (3.37)
Combined (3.36) and (3.37), we have

K ω (v n (t n )) = 2S ω (v n (t n )) - 2 3 N (v n (t n )) → 0.
Therefore, using Lemma 3.14, there exists θ 0 ∈ R such that (v n (t n , .)) has a subsequence (we denote it by the same letter) that converges to e iθ 0 φ in H 1 (R + ), where φ is dened as in Proposition 3.3. Hence, we have

inf θ∈R ∥v n (t n ) -e iθ φ∥ H 1 (R + ) → 0, (3.38)
as n → ∞, this contradicts (3.35). Hence, we obtain the desired result.

Next, we give the proof of Theorem 3.4 (2). We divide the proof in two cases.

First, let α = 0. In this case, we use similar arguments as in Weinstein [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF]. We have

E ω (v) = 1 2 ∥v x ∥ 2 L 2 (R + ) - 1 32 ∥v∥ 6 L 6 (R + ) , P (v) = ∥v x ∥ 2 L 2 (R + ) - 1 16 ∥v∥ 6 L 6 (R + ) .
Thus, E(φ ω ) = P (φ ω ) = 0. Let ε > 0 and φ ω,ε = (1 + ε)φ ω . We have

E(φ ω,ε ) = (1+ε) 2 1 2 ∥φ ω ∥ 2 L 2 (R + ) -(1+ε) 6 1 32 ∥φ ω ∥ 6 L 6 (R + ) = ((1+ε) 2 -(1+ε) 6 ) 1 2 ∥φ ω ∥ 2 L 2 (R + ) < 0.
In the addition, |x|φ ω,ε (x) ∈ L 2 (R + ) by exponential decay of φ ω . Using Theorem 3.1, the solution associated to φ ω,ε blows up in nite time. As φ ω,ε → φ ω in H 1 (R + ), we obtain the instability by blow-up of standing waves. Now, let α > 0 and e iωt φ be the standing wave solution of (3.1). We use similar arguments as in [START_REF] Coz | Standing waves in nonlinear Schrödinger equations[END_REF]. Introduce the scaling

v λ (x) = λ 1 2 v(λx).
Let S ω , K ω be dened as in Proposition 3.3, for convenience, we will remove the index ω. Dene

P (v) := ∂ ∂λ S(v λ )| λ=1 = ∥v x ∥ 2 L 2 (R + ) - 1 16 ∥v∥ 6 L 6 (R + ) + α 2 |v(0)| 2 .
In the following lemma, we investigate the behaviour of the above functional under scaling.

Lemma 3.17. Let v ∈ H 1 (R + ) \ {0} be such that v(0) ̸ = 0, P (v) ⩽ 0. Then there

exists λ 0 ∈ (0, 1] such that (i) P (v λ 0 ) = 0, (ii) λ 0 = 1 if only if P (v) = 0, (iii) ∂ ∂λ S(v λ ) = 1 λ P (v λ ), (iv) ∂ ∂λ S(v λ ) > 0 on (0, λ 0 ) and ∂ ∂λ S(v λ ) < 0 on (λ 0 , ∞), (v) The function λ → S(v λ ) is concave on (λ 0 , ∞).

Proof. A simple calculation leads to

P (v λ ) = λ 2 ∥v x ∥ 2 L 2 (R + ) - λ 2 16 ∥v∥ 6 L 6 (R + ) + λα 2 |v(0)| 2 .
Then, for λ > 0 small enough, we have

P (v λ ) > 0.
By continuity of P , there exists λ 0 ∈ (0, 1] such that P (v λ 0 ) = 0. Hence (i) is proved. If λ 0 = 1 then P (v) = 1. Conversely, if P (v) = 0 then

0 = P (v λ 0 ) = λ 2 0 P (v) + λ 0 -λ 2 0 2 α|v(0)| 2 = λ 0 -λ 2 0 2 α|v(0)| 2 .
By the assumption v(0) ̸ = 0, we have λ 0 = 1, hence (ii) is proved. Item (iii) is obtained by a simple calculation. To obtain (iv), we use (iii). We have

P (v λ ) = λ 2 λ -2 0 P (v λ 0 ) + λα 2 - λ 2 λ -1 0 α 2 |v(0)| 2 = λα(λ 0 -λ) 2λ 0 |v(0)| 2 .
Hence, P (v λ ) > 0 if λ < λ 0 and P (v λ ) < 0 if λ > λ 0 . This proves (iv). Finally, we have

∂ 2 ∂λ 2 S(v λ ) = P (v) - α 2 |v(0)| 2 < 0.
This proves (v).

In the case of functions such that v(0) = 0, we have the following lemma.

Lemma 3.18. Let v ∈ H 1 (R + ) \ {0}, v(0) = 0 and P (v) = 0 then we have v) for all λ > 0.

S(v λ ) = S(
Proof. The proof is simple, using the fact that

∂ ∂λ S(v λ ) = 1 λ P (v λ ) = λP (v) = 0.
Hence, we obtain the desired result. Now, consider the minimization problems

d M := inf {S(v) : v ∈ M} , (3.39)
m := inf S(v), v ∈ H 1 (R + ) \ 0, S ′ (v) = 0 , (3.40)
where

M = v ∈ H 1 (R + ) \ 0, P (v) = 0, K(v) ⩽ 0 .
By classical arguments, we can prove the following property.

Proposition 3.19. Let m be dened as above. Then, we have

m = inf S(v) : v ∈ H 1 (R + ) \ 0, K(v) = 0 .
We have the following relation between the minimization problems m and d M .

Lemma 3.20. Let m and d M be dened as above. We have

m = d M .
Proof. Let G be the set of all minimizers of (3.40). If φ ∈ G then S ′ (φ) = 0. By the denition of S, P , K we have P (φ) = 0 and K(φ) = 0. Hence, φ ∈ M, this implies

S(φ) ⩾ d M . Thus, m ⩾ d M . Conversely, let v ∈ M. If K(v) = 0 then S(v) ⩾ m, using Proposition 3.19. Otherwise, K(v) < 0. Using the scaling v λ (x) = λ 1 2 v(λx), we have K(v λ ) = λ 2 ∥v x ∥ 2 L 2 (R + ) - 3λ 2 16 ∥v∥ 6 L 6 (R + ) + ω∥v∥ 2 L 2 (R + ) + αλ 2 |v(0)| 2 → ω∥v∥ 2 L 2 (R + ) > 0,
as λ → 0. Hence, K(v λ ) > 0 as λ > 0 is small enough. Thus, there exists λ 1 ∈ (0, 1) such that K(v λ 1 ) = 0. Using Proposition 3.19, S(v λ 1 ) ⩾ m. We consider two cases. First, if v(0) = 0 then using Lemma 3.18, we have S(v) = S(v λ 1 ) ⩾ m. Second, if v(0) ̸ = 0 then using Lemma 3.17, we have S(v) ⩾ S(v λ 1 ) ⩾ m. In any case, S(v) ⩾ m. This implies d M ⩾ m, and completes the proof.

Dene

V := v ∈ H 1 (R + ) \ {0} : K(v) < 0, P (v) < 0, S(v) < m .
We have the following important lemma.

Lemma 3.21. If v 0 ∈ V then the solution v of (3.1) associated with v 0 satises v(t) ∈ V for all t in the time of existence.

Proof. Since S(v 0 ) < 0, by conservation of the energy and the mass we have

S(v) = E(v) + ωM (v) = E(v 0 ) + ωM (v 0 ) = S(v 0 ) < m. (3.41)
If there exists t 0 > 0 such that K(v(t 0 )) ⩾ 0 then by continuity of K and v, there exists t 1 ∈ (0, t 0 ] such that K(v(t 1 )) = 0. This implies S(v(t 1 )) ⩾ m, using Proposition 3.19. This contradicts (3.41). Hence, K(v(t)) < 0 for all t in the time of existence of v. Now, we prove P (v(t)) < 0 for all t in the time of existence of v.

Assume that there exists t 2 > 0 such that P (v(t 2 )) ⩾ 0, then, there exists t 3 ∈ (0, t 2 ] such that P (v(t 3 )) = 0. Using the previous lemma, S(v(t 3 )) ⩾ m, which contradicts (3.41). This completes the proof.

Using the above lemma, we have the following property of solutions of (3.1) when the initial data lies on V. Lemma 3.22. Let v 0 ∈ V, v be the corresponding solution of (3.1) in (T min , T max ).

There exists δ > 0 independent of t such that P (v(t)) < -δ, for all t ∈ (T min , T max ).

Proof. Let t ∈ (T min , T max ), u = v(t) and u λ (x) = λ 1 2 u(λx). Using Lemma 3.17, there exists λ 0 ∈ (0, 1) such that P (u λ 0 ) = 0. If K(u λ 0 ) ⩽ 0 then we keep λ 0 . Otherwise, K(u λ 0 ) > 0, then, there exists λ 0 ∈ (λ 0 , 1) such that K(u λ 0 ) = 0. We replace λ 0 by λ 0 . In any case, we have

S(u λ 0 ) ⩾ m. (3.42)
By (v) of Lemma 3.17 we have

S(u) -S(u λ 0 ) ⩾ (1 -λ 0 ) ∂ ∂λ S(u λ )| λ=1 = (1 -λ 0 )P (u).
In addition P (u) < 0, we obtain S(u) -S(u λ 0 ) ⩾ (1 -λ 0 )P (u) > P (u).

S(v 0 ) -m = S(v(t)) -m = S(u) -m ⩾ S(u) -S(u λ 0 ) > P (u) = P (v(t)). Setting -δ := S(v 0 ) -m,
we obtain the desired result.

Using the previous lemma, if the initial data lies on V and satises a weight condition then the associated solution blows up in nite time on H 1 (R + ). More precisely, we have the following result. Proposition 3.23. Let φ ∈ V such that |x|φ ∈ L 2 (R + ). Then the corresponding solution v of (3.1) blows up in nite time on H 1 (R + ).

Proof. By Lemma 3.22, there exists δ > 0 such that P (v(t)) < -δ for t ∈ (T min , T max ).

Remember that

∂ ∂t ∥xv(t)∥ 2 L 2 (R + ) = J(t) - R + x|v| 4 dx, (3.44)
where J(t) satises

∂ t J(t) = 4 2∥v x ∥ 2 L 2 (R + ) - 1 8 ∥v∥ 6 L 6 (R + ) + α|v(0)| 2 = 8(P (v(t))) < -8δ.
This implies that

J(t) = J(0) + 8 t 0 P (v(s)) ds < J(0) -8δt.
Hence, from (3.44), we have

∥xv(t)∥ 2 L 2 (R + ) = ∥xv(0)∥ 2 L 2 (R + ) + t 0 J(s) ds - t 0 R + x|v| 4 dx ds ⩽ ∥xv(0)∥ 2 L 2 (R + ) + t 0 (J(0) -8δs) ds ⩽ ∥xv(0)∥ 2 L 2 (R + ) + J(0)t -4δt 2 .
Thus, for t suciently large, there is a contradiction with ∥xv∥ L 2 (R + ) ⩾ 0. Hence, T max < ∞ and T min > -∞. By the blow up alternative, we have

lim t→Tmax ∥v x ∥ L 2 (R + ) = lim t→T min ∥v x ∥ L 2 (R + ) = ∞.
This completes the proof.

Proof of Theorem 3.4 (2). Using Proposition 3.23, we need to construct a sequence

(φ n ) ⊂ V such that φ n converges to φ in H 1 (R + ). Dene φ λ (x) = λ 1 2 φ(λx).
We have S(φ) = m, P (φ) = K(φ) = 0, φ(0) ̸ = 0.

By (iv) of Lemma 3.17, S(φ λ ) < m for all λ > 0.

In the addition, P (φ λ ) < 0 for all λ > 1.

Moreover, In this chapter, we consider the derivative nonlinear Schrödinger equation:

∂ ∂λ K(φ λ ) = 2λ ∥φ x ∥ 2 L 2 (R + ) - 3 16 ∥φ∥ 6 L 6 (R + ) + α|φ(0)| 2 = 2λ(K(φ) -ω∥φ∥ 2 L 2 (R + ) -α|φ(0)| 2 ) + α|φ(0)| 2 = -2ωλ∥φ∥ 2 L 2 (R + ) -α(2λ -1)|φ(0)| 2 < 0, when λ > 1. Thus, K(φ λ ) < K(φ) = 0 when λ > 1. This implies φ λ ∈ V when λ > 1. Let λ n > 1 such that λ n → 1 as n → ∞. Dene, for n ∈ N * φ n = φ λn ,
iu t + u xx + iα|u| 2 u x + iµu 2 u x + f (u) = 0, u(0) = u 0 . (4.1)
where α, µ ∈ R, f : C → C is a given function and u is a complex valueed function of (t, x) ∈ R × R.

In [START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem[END_REF][START_REF] Tsutsumi | On solutions of the derivative nonlinear Schrödinger equation[END_REF], Tsutsumi and Fukuda used an approximation argument to prove the existence of solutions of (4.1) in the case α = -2, µ = -1. In this case with f = 0, Biagioni and Linares [START_REF] Biagioni | Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations[END_REF] proved that the solution map from H s (R) to C([-T, T ], H s (R)) is not locally uniformly continuous, for T > 0 and s < 1 2 . The H 1 2 solution in this case is global if ∥u 0 ∥ 2 L 2 < 2π by the work of Miao-Wu-Xu [START_REF] Miao | Global well-posedness for Schrödinger equation with derivative in H 1 2 (R)[END_REF]. Later, Guo and Wu [START_REF] Guo | Global well-posedness for the derivative nonlinear Schrödinger equation in H 1 2 (R)[END_REF] improved this result; that is, H

1 2 solution is global if ∥u 0 ∥ 2 L 2 < 4π.
The Cauchy problem of (4.1) was also studied as in [START_REF] Takaoka | Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity[END_REF], where gauge transformation and Fourier restriction method are used to obtain local well-posedness in H s , s ⩾ 1/2. In [START_REF] Ozawa | On the nonlinear Schrödinger equations of derivative type[END_REF], Ozawa studied the Cauchy problem and gave a sucient condition of global well-posedness for (4.1). The proof was used gauge transformations which reduce the original equations to systems of equations without derivative nonlinearities. In [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF][START_REF] Hayashi | Finite energy solutions of nonlinear Schrödinger equations of derivative type[END_REF], in the case α = 2µ, Hayashi-Ozawa proved the unique global existence of solutions to (4.1) in Sobolev spaces and in the weighted spaces with smallness on the initial data ∥u 0 ∥ 2 L 2 < 4π |α| . In the case α = -2, µ = -1, f = 0, Wu [START_REF] Wu | Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space[END_REF] improved the global results in [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF][START_REF] Hayashi | Finite energy solutions of nonlinear Schrödinger equations of derivative type[END_REF]. More precisely, the author proved that the solutions exist globally in time under smallness on the initial data ∥u 0 ∥ L 2 < √ 2π + ε * , where ε * is a small positive constant. Later, Wu [START_REF] Wu | Global well-posedness on the derivative nonlinear Schrödinger equation[END_REF] improved this results for larger bounded on the initial data ∥u 0 ∥ L 2 < √ 4π.

The proof combines a gauge transformation and conservation laws with a sharp Gagliardo-Nirenberg inequality. In [START_REF] Fukaya | A sucient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation[END_REF], by using variational argument, Fukaya-Hayashi-Inui gave results covering the result of Wu [START_REF] Wu | Global well-posedness on the derivative nonlinear Schrödinger equation[END_REF]. The authors showed that in the case f = 0, α = 1, µ = 0, the H 1 solutions of (4.1) exist globally in time for the initial satises ∥u 0 ∥ 2 L 2 < 4π or ∥u 0 ∥ 2 L 2 = 4π and P (u 0 ) < 0, where P is the momentum functional which is conserved under the ow of (4.1). In [START_REF] Colliander | A rened global well-posedness result for Schrödinger equations with derivative[END_REF],

Colliander-Keel-Stalani-Takaoka-Tao proved by the so-called I-method the global well posedness in H s (R), s > 1 2 of (4.1) if ∥u 0 ∥ 2 L 2 < 2π (see also [START_REF] Colliander | Global wellposedness for Schrödinger equations with derivative[END_REF]). In the case f = 0 and µ = 0, (4.1) is a completely integrable equation. The complete integrability structure of equation was used to prove global existence of solutions in H 2,2 (R) by [START_REF] Jenkins | Global existence for the derivative nonlinear Schrödinger equation with arbitrary spectral singularities[END_REF] and in H s (R), s > 1 2 by [START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF]. In the case µ = 0 and f (u) = b|u| 4 u, there were a lot of works on studying stability and instability of solitons of (4.1). The family of solitons of (4.1) has two parameters (ω, c). In the case b = 0, Guo and Wu [START_REF] Guo | Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation[END_REF] proved that the solitons are orbitally stable when ω > c 2 4 and c < 0 by using the abstract theory of Grillakis-Shatah-Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF]. After that, Colin and Ohta [START_REF] Colin | Stability of solitary waves for derivative nonlinear Schrödinger equation[END_REF] improved this result for all ω > c 2 4 using variational techniques. In [START_REF] Ohta | Instability of solitary waves for nonlinear Schrödinger equations of derivative type[END_REF], Ohta proved that for each b > 0 there exists a unique s * = s * (b) > 0 ∈ (0, 1) such that the soliton u ω,c is orbitally stable if

-2 √ ω < c < 2s * √ ω and orbitally unstable if 2s * √ ω < c < 2 √ ω.
In the case b < 0, the stability result is obtained in [START_REF] Hayashi | Stability of algebraic solitons for nonlinear schrödinger equations of derivative type: variational approach[END_REF]. In the case b = 0, Kwon-Wu [START_REF] Kwon | Orbital stability of solitary waves for derivative nonlinear Schrödinger equation[END_REF] proved a stability result of solitons in the zero mass case. Removing the eect of scaling in the stability result of this work is an open question.

Multi-solitons

First, we focus on studying the following special form of (4.1):

iu t + u xx + i|u| 2 u x + b|u| 4 u = 0. (4.2)
Our rst goal in this paper is to study the long time behaviour of solutions of (4.2). More precisely, we study the multi-solitons theory of (4.2). The existence of multi-solitons is a step towards the proof of the soliton resolution conjecture, which states that all global solutions of a dispersive equation behave at large times as a sum of a radiative term and solitons. The theory of multi-soliton has attracted a lot of interest. In [START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF], Le Coz-Li-Tsai proved existence and uniqueness of nite and innite soliton and kink-soliton trains of classical nonlinear Schrödinger equations, using xed point arguments around of the desired prole. Another method was introduced in [START_REF] Martel | Multi solitary waves for nonlinear Schrödinger equations[END_REF] for the simple power nonlinear Schrödinger equation with L 2subcritical nonlinearities. The proof was established by two ingredients: uniform estimates and a compactness property. The arguments were later modied to obtain the results for L 2 -supercritical equations [START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF] and for proles made with excited states [START_REF] Côte | High-speed excited multi-solitons in nonlinear Schrödinger equations[END_REF]. One can also cite the works on the logarithmic Schrödinger equation (logNLS) in the focusing regime in [START_REF] Ferriere | Existence of multi-solitons for the focusing logarithmic non-linear Schrödinger equation[END_REF]. In [START_REF] Zakharov | Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear media[END_REF], the inverse scattering transform method (IST) was used to construct multi-solitons of the one dimensional cubic focusing NLS. We would like also to mention the works on the non-linear Klein-Gordon equation [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF] and [START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF], and on the stability of multi-solitons for generalized Korteweg-de Vries equations and L 2 -subcritical nonlinear Schrödinger equations from Martel, Merle and Tsai [START_REF] Martel | Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations[END_REF], [START_REF] Martel | Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations[END_REF]. In [START_REF] Coz | Stability of multisolitons for the derivative nonlinear Schrödinger equation[END_REF], Le Coz-Wu proved a stability result of multi-solitons of (4.2) in the case b = 0. Our motivation is to prove the existence of a multi-solitons in a similar sense as in [START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF][START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF]. The method used in [START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF][START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF] cannot apply directly in our case. The reason is the appearance of the derivative nonlinearities. To overcome this diculty, we use a Gauge transformation to obtain a system of Schrödinger equations without derivative nonlinearities. We may use Strichartz estimates and xed point argument to construct a suitable solution of this system. This solution satises a relation which is proved by using the Grönwall inequality and the condition on the parameters and we obtain a solution of (4.2). This solution satises the desired property.

Consider equation (4.2). The soliton of equation ( 4.2) is a solution of the form R ω,c (t, x) = e iωt ϕ ω,c (x -ct), where ϕ ω,c ∈ H 1 (R) solves

-ϕ xx + ωϕ + icϕ x -i|ϕ| 2 ϕ x -b|ϕ| 4 ϕ = 0, x ∈ R. (4.3)
Applying the following gauge transform to ϕ ω,c

ϕ ω,c (x) = Φ ω,c (x) exp i c 2 x - i 4 x -∞ |Φ ω,c (y)| 2 dy ,
it is easy to verify that Φ ω,c (see e.g [23, Proof of Lemma 2]) satises the following equation.

-

Φ xx + ω - c 2 4 Φ + c 2 |Φ| 2 Φ - 3 16 γ|Φ| 4 Φ = 0, γ := 1 + 16 3 b. (4.4)
The positive even solution of (4.4) is explicitly obtained by: if γ > 0 (b > -3 16 ),

Φ 2 ω,c (x) = 2(4ω-c 2 ) √ c 2 +γ(4ω-c 2 ) cosh( √ 4ω-c 2 x)-c if -2 √ ω < c < 2 √ ω, 4c (cx) 2 +γ if c = 2 √ ω, (4.5
)

and if γ ⩽ 0 (b ⩽ -3 16 ), Φ 2 ω,c (x) = 2(4ω -c 2 ) c 2 + γ(4ω -c 2 ) cosh( √ 4ω -c 2 x) -c if -2 √ ω < c < -2s * √ ω,
where

s * = s * (γ) = -γ
1-γ . We note that the following condition on the parameters γ and (ω, c) is a necessary and sucient condition for the existence of non-trivial solutions of (4.2) vanishing at innity (see [START_REF] Berestycki | Nonlinear scalar eld equations. I. Existence of a ground state[END_REF]):

if γ > 0(⇔ b > -3 16 ) then -2 √ ω < c ⩽ 2 √ ω, if γ ⩽ 0(⇔ b ⩽ -3 16 ) then -2 √ ω < c < -2s * √ ω.
Let (c j , ω j ) satisfying for each 1 ⩽ j ⩽ K the condition of existence of soliton. For each j ∈ {1, 2, .., K}, we set

R j (t, x) = e iθ j R ω j ,c j (t, x).
The prole of a multi-soliton is a sum of the form:

R = K j=1 R j . (4.6)
A solution of (4.2) is called a multi-soliton if

∥u(t) -R(t)∥ H 1 ⩽ Ce -λt ,
for some C, λ > 0 and t large enough. For convenience, we set h j = 4ω j -c 2 j . We rewrite

Φ ω j ,c j (x) = √ 2h j c 2 j + γh 2 j cosh(h j x) -c j -1 2 . (4.7)
As each soliton is in H ∞ (R), we have R ∈ H ∞ (R). Our rst main result is the following.

Theorem 4.1. Let K ∈ N * and for each 1 ⩽ j ⩽ K, let (θ j , c j , ω j ) be a set of

parameters such that θ j ∈ R, c j ̸ = c k , for j ̸ = k and c j such that -2 √ ω j < c j < 2 √ ω j if γ > 0 and -2 √ ω j < c j < -2s * √ ω j if γ ⩽ 0.
The multi-soliton prole R is given as in (4.6). Then there exists a certain positive constant C * such that if the parameters (ω j , c j ) satisfy

C * (1 + ∥R x ∥ L ∞ t L ∞ x )(1 + ∥R∥ L ∞ t L ∞ x ) + ∥R∥ 4 L ∞ t L ∞ x ⩽ v * := inf j̸ =k h j |c j -c k |, (4.8)
then there exist T 0 > 0 depending on ω 1 , ..., ω K , c 1 , ..., c K and a solution u of (4.2) on [T 0 , ∞) such that

∥u -R∥ H 1 ⩽ Ce -λt , ∀t ⩾ T 0 , (4.9)
where λ = v * 16 and C is a positive constant depending on the parameters ω 1 , ..., ω K , c 1 , ..., c K .

We observe that the formula for solitons in the case γ > 0 and in the case γ ⩽ 0 is similar. Thus, in the proof of Theorem 4.1, we only consider the case γ > 0. The case γ ⩽ 0 is treated by similar arguments. Remark 4.2. We give an example of parameters satisfying (4.8). Let d j < 0,

h j ∈ R for all j ∈ {1, 2, ..., K} such that d j ̸ = d k for all j ̸ = k. Let (c j , ω j ) = M d j , 1 4 (h 2 j + M 2 d 2 j
) . We prove that for M large enough, the condition (4.8) is satised. By this choosing, we have h j ≪ |c j | and c j < 0 for all j. We have

∥Φ ω j ,c j ∥ 2 L ∞ ⩽ 2h 2 j c 2 j + γh 2 j -c j ≲ h 2 j |c j | .
Moreover,

∂Φ ω j ,c j = - √ 2 2 h 2 j c 2 j + γh 2 j sinh(h j x) c 2 j + γh 2 j cosh(h j x) -c j -3 2 .
Thus, for all j, we obtain

|∂Φ ω j ,c j | ≲ h 2 j c 2 j + γh 2 j | sinh(h j x)| c 2 j + γh 2 j cosh(h j x) -c j -3 2 ≲ h 2 j c 2 j + γh 2 j cosh(h j x) -c j -1 2 ≈ h j |Φ ω j ,c j | ≲ h 2 j |c j | .
In the addition, we have

∥∂R j ∥ L ∞ = ∥∂ϕ ω j ,c j ∥ L ∞ ≈ ∥∂Φ ω j ,c j ∥ L ∞ + c j 2 Φ ω j ,c j -Φ 3 ω j ,c j L ∞ ⩽ ∥∂Φ ω j ,c j ∥ L ∞ + |c j | 2 ∥Φ ω j ,c j ∥ L ∞ + ∥Φ ω j ,c j ∥ 3 L ∞ ≲ h 2 j |c j | + h j |c j | + h 3 j |c j | 3 .
Thus, the left hand side of (4.8) is bounded by

C * 1 + 1⩽j⩽K h 2 j |c j | + h j |c j | + h 3 j |c j | 3 1 + 1⩽j⩽K h j |c j | + 1⩽j⩽K h 4 j c 2 j .
(4.10) By our choosing, (4.10) is order M 1 2 and the right hand side of (4.8) is order M 1 . Thus, (4.8) is satised for M large enough.

Multi kink-solitons

Second, we consider another special case of (4.1) as follows

iu t + u xx + iu 2 u x + b|u| 4 u = 0. (4.11)
Our goal is to construct multi kink-solitons of (4.11). The motivation comes from [START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF][START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF], where the authors have constructed an innite multi kink-soliton train for classical nonlinear Schrödinger equations by using xed point arguments. However, in the case of (4.11), this method can not directly be used due to the appearing of a derivative term. To overcome this diculty, use a transformation and work on a system of two equations without derivative nonlinearites. Consider the equation (4.11). First, we would like to dene a kink solution of (4.11). Let R ω,c be a smooth solution of (4.11) of the form:

R ω,c (t, x) = e iωt ϕ ω,c (x -ct), (4.12)
where ϕ ω,c is smooth and solves

-ϕ xx + ωϕ + icϕ x -iϕ 2 ϕ x -b|ϕ| 4 ϕ = 0, x ∈ R. (4.13) If ϕ ω,c | R + ∈ H 1 (R +
) then the following Gauge transform is well dened:

Φ ω,c = exp -i c 2 x + i 4 x ∞ |ϕ ω,c (y)| 2 dy ϕ ω,c .
Since ϕ ω,c solves (4.13), Φ ω,c is smooth and solves

-Φ xx + ω - c 2 4 Φ- 3 2 Im(ΦΦ x )Φ- c 2 |Φ| 2 Φ+ 3 16 γ|Φ| 4 Φ = 0, γ := 5 3 - 16 3 b. (4.14) Since Φ ω,c | R + ∈ H 2 (R + ),
Im(Φ ω,c ∂ x Φ ω,c ) = 0.
Thus, Φ ω,c solves

-Φ xx + ω - c 2 4 Φ - c 2 |Φ| 2 Φ + 3 16 γ|Φ| 4 Φ = 0. (4.15)
Now, we give the denition of a half-kink of (4.2).

Denition 4.3. The function R ω,c is called a half-kink solution of (4.2) if R ω,c is of the form (4.12) and the associated Φ ω,c is a real valued function solving (4.15) and satisfying:

   lim x→±∞ Φ(x) ̸ = 0, lim x→∓∞ Φ(x) = 0, (4.16)
where ω = ω -c 2 4 , f : R → R. For more convenience, we dene

f (s) = c 2 s 3 - 3 16 γs 5 .
The following result about the existence of a half-kink prole is stated in [START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF] as follows.

Proposition 4.4. Let f : R → R be a C 1 function with f (0) = 0 and dene

F (s) := s 0 f (t) dt. For ω ∈ R, let ζ(ω) := inf ζ > 0, F (ζ) - 1 2 ωζ 2 = 0
and assume that there exists ω1 ∈ R such that

ζ(ω 1 ) > 0, f ′ (0) -ω1 < 0, f (ζ(ω 1 )) -ω1 ζ(ω 1 ) = 0. (4.17)
Then, for ω = ω1 , there exists a half-kink prole Φ ∈ C 2 (R) of (4.16) i.e Φ is unique (up to translation), positive and satises Φ ′ > 0 on R and the boundary conditions

lim x→-∞ Φ(x) = 0, lim x→∞ Φ(x) = ζ(ω 1 ) > 0. (4.18)
If in addition,

f ′ (ζ(ω 1 )) -ω1 < 0, (4.19)
then for any 0 < a < ω1 -max{f ′ (0), f ′ (ζ(ω 1 ))} there exists D a > 0 such that

|Φ ′ (x)| + |Φ(x)1 x<0 | + |(ζ(ω 1 ) -Φ(x))1 x>0 | ⩽ D a e -a|x| , ∀x ∈ R. (4.20)
We have the following remarks.

Remark 4.5.

(1) As in [72, Remark 1.15], using the symmetry x → -x and Proposition 4.4 implies the existence and uniqueness of half-kink prole Φ satisfying

lim x→-∞ Φ(x) = ζ(ω 1 ) > 0, lim x→∞ Φ(x) = 0.
(2) In our case, f (s) = c 2 s 3 -3 16 γs 5 . We may check that if γ > 0, c > 0 then there exist ω1 = c 2 4γ and ζ(ω 1 ) = 2c γ satisfying the conditions (4.17), (4.19) and the denition of the function ζ. Thus, using Proposition 4.4, if γ > 0, c > 0 then there exists a half-kink solution of (4.2) and the constant a in Proposition 4.4 satisfy 0 < a < c 2 4γ .

(3) Consider the half-kink prole Φ of Proposition 4.4. Since Φ solves (4.16) and satises (4.20), we have

|Φ ′′ (x)| + |Φ ′′′ (x)| ⩽ D a e -a|x| . Now, we assume γ > 0. Let K > 0, θ 0 , ω 0 , c 0 ∈ R be such that 2 √ ω 0 > c 0 > √ 2γ. For 1 ⩽ j ⩽ K, let (θ j , ω j , c j ) ∈ R be such that c j > c 0 , c j ̸ = c k for j ̸ = k, 2 √ ω j > c j > 2s * √ ω j for s * = γ 1+γ
. Set R j = e iθ j R ω j ,c j , where R ω j ,c j ∈ H 1 (R) is the soliton solution of (4.11) with the associated prole dened in (4.5). Let Φ 0 be the half-kink prole given in Remark 4.5 (1) associated with the parameters ω 0 , c 0 and R ω 0 ,c 0 be the associated half-kink solution of (4.11). Set R 0 = e iθ R ω 0 ,c 0 . The multi kink-soliton prole of (4.11) is dened as follows:

V = R 0 + K j=1 R j . (4.21)
Our second main result is the following.

Theorem 4.6. Considering (4.11), we assume that b < 5 16 (γ > 0). Let V be given as in (4.21). There exists a certain positive constant C * such that if the parameters (ω j , c j ) satisfy

C * 1 + ∥V x ∥ L ∞ t L ∞ x 1 + ∥V ∥ L ∞ t L ∞ x + ∥V ∥ 4 L ∞ t L ∞ x ⩽ v * := min inf j̸ =k h j |c j -c k |, inf j̸ =0 |c j -c 0 | , (4.22)
then there exist a solution u to (4.11) such that

∥u -V ∥ H 1 ⩽ Ce -λt . ∀t ⩾ T 0 , (4.23)
where λ = v * 16 and C, T 0 are positive constants depending on the parameters ω 0 , ..., ω K , c 0 , ..., c K .

We have some following discussions about the above theorem.

Remark 4.7.

(1) The condition c 2 0 > 2γ in Theorem 4.6 is a technical condition and we can remove this. The constant a in Proposition 4.4 satises

0 < a < c 2 0 4γ .
Thus, under the condition c 2 0 > 2γ, we can choose a = 1 2 . This fact makes the proof easier and we have

|Φ ′′′ 0 (x)| + |Φ ′′ 0 (x)| + |Φ ′ 0 (x)| + |Φ 0 (x)1 x>0 | + 2c 0 γ -Φ 0 (x) 1 x<0 ≲ e -1 2 |x| . (4.24)
(2) By similar arguments as above, we can prove that there exists a half-kink solution of (4.2) which satises the denition 4.3. To our knowledge, there are no result about stability or instability of this kind of solution.

(3) Let γ > 0. We give an example of parameters satisfying the condition (4.22) of Theorem 4.6. As in Remark 4.2, we have

Φ ω j ,c j = √ 2h j c 2 j -γh 2 j cosh(h j x) + c j - 1 2
, ∀j = 1, ..., K.

Hence, choosing h j ≪ c j , for all j, we have

∥Φ ω j ,c j ∥ 2 L ∞ ⩽ 2h 2 j c 2 j -γh 2 j + c j ≲ h 2 j c j .
By similar arguments as in Remark 4.2, for all 1 ⩽ j ⩽ K, we have

∥∂R j ∥ L ∞ ≲ h 2 j √ c j + h j √ c j + h 3 j c 3 j .
Now, we treat to the case j = 0. Let Φ 0 be the prole given as in Proposition 4.4 associated to the parameters c 0 , ω 0 and R 0 be the associated half-kink solution of (4.2). From (4.20), Remark 4.5 and Remark 4.7 we have

∥Φ 0 ∥ L ∞ ≲ √ c 0 , ∥∂Φ 0 ∥ L ∞ ≲ 1, Thus, ∥R 0 ∥ L ∞ L ∞ ≲ √ c 0 , ∥∂R 0 ∥ L ∞ L ∞ ≲ 1 + c 3 2 0 ≲ c 3 2
0 . This implies that for h j ≪ c j (j = 1, .., K) the left hand side of (4.22) is estimated by:

C *     1 + c 3 2 0 + K j=1   h 2 j √ c j + h j √ c j + h 3 j c 3 j     1 + √ c 0 + K j=1 h j √ c j   .
Choosing c 0 ≈ 1, the above expression is estimated by:

C *     1 + K j=1   h 2 j √ c j + h j √ c j + h 3 j c 3 j     1 + K j=1 h j √ c j   . (4.25) Let h j , d j ∈ R + , d j ̸ = d k for all j ̸ = k, 1 ⩽ j, k ⩽ K. Set c j = M d j , ω j = 1 4 (h 2 j + M 2 d 2 j ).
We have (4.25) is of order M 0 and the right hand side of (4.22) is of order M 1 . Thus, by these choices of parameters, when M is large enough, the condition (4.22) is satised.

(ii) Let I be an interval of R and t 0 ∈ I. Let (γ, ρ) be an admissible pair and

f ∈ L γ ′ (I, L ρ ′ (R N )).
Then, for all admissible pair (q, r), the function

t → Φ f (t) = t t 0 S(t -s)f (s) ds belong to L q (I, L r (R N )) ∩ C(I, L 2 (R N )).
Moreover, there exists a constant C independent of I such that

∥Φ f ∥ L q (I,L r ) ⩽ C∥f ∥ L γ ′ (I,L ρ ′ ) , for all f ∈ L γ ′ (I, L ρ ′ (R N )).
Notation.

(1) For t > 0, the Strichartz space S([t, ∞)) is dened via the norm

∥u∥ S([t,∞)) = sup (q,r) admissible ∥u∥ L q τ L r x ([t,∞)×R)
The dual space is denoted by Particularly, we denote a ≲ p b if there exists a constant C depending only on the parameters ω 1 , ..., ω K , c 1 , ..., c K such that a ⩽ Cb.

N ([t, ∞)) = S([t, ∞)) * . ( 2
(5) Let f ∈ C 1 (R). We use ∂f or f x to denote the derivative in space of the function f . (6) Let f (x, y, z, ..) be a function. We

denote |df | = |f x | + |f y | + |f z | + ....

4.2

Proof of Theorem 4.1

In this section, we give the proof of Theorem 4.1. We divide our proof into three steps.

Step 1. Preliminary analysis

Considering the following transform:

φ(t, x) = exp i 2 x -∞ |u(t, y)| 2 dy u(t, x), ψ = ∂φ -i 2 |φ| 2 φ. (4.26)
By similar arguments as in [START_REF] Hayashi | Finite energy solutions of nonlinear Schrödinger equations of derivative type[END_REF] and [START_REF] Ozawa | On the nonlinear Schrödinger equations of derivative type[END_REF], we see that if u solves (4.2) then (φ, ψ) solves the following system

           Lφ = iφ 2 ψ -b|φ| 4 φ, Lψ = -iψ 2 φ -3b|φ| 4 ψ -2b|φ| 2 φ 2 ψ, φ | t=0 = φ 0 = exp i 2 x -∞ |u 0 (y)| 2 dy u 0 , ψ | t=0 = ψ 0 = ∂φ 0 -i 2 |φ 0 | 2 φ 0 , (4.27)
where

L = i∂ t + ∂ xx . Dene P (φ, ψ) = iφ 2 ψ -b|φ| 4 φ, Q(φ, ψ) = -iψ 2 φ -3b|φ| 4 ψ -2b|φ| 2 φ 2 ψ.
Let R be the multi soliton prole given in (4.6). Since R j solves (4.2), for all j, by an elementary calculation, we have

iR t +R xx +i|R| 2 R x +b|R| 4 R = i |R| 2 R x - K j=1 |R j | 2 R jx +b |R| 4 R - K j=1 |R j | 4 R j .
(4.28) From Lemma 4.11, we have

|R| 2 R x - K j=1 |R j | 2 R jx H 2 + |R| 4 R - K j=1 |R j | 4 R j H 2 ⩽ e -λt , (4.29)
where λ = 1 16 v * . Thus, we rewrite (4.28) as follows

iR t + R xx + i|R| 2 R x + b|R| 4 R = e -λt v(t, x), (4.30)
where

v(t) ∈ H 2 (R) is such that ∥v(t)∥ H 2 is uniformly bounded in t. Dene h(t, x) = exp i 2 x -∞ |R| 2 dy R(t, x), (4.31)
k = h x - i 2 |h| 2 h. (4.32)
By an elementary calculation, we have

Lh = ih 2 k -b|h| 4 h + e -tλ m(t, x) = P (h, k) + e -tλ m(t, x), Lk = -ik 2 h -3b|h| 4 k -2b|h| 2 h 2 k + e -tλ n(t, x) = Q(h, k) + e -tλ n(t, x),
where m, n satisfy

m = v exp i 2 x -∞ |R| 2 dy -h x -∞
Im(vR) dy, (4.33)

n = m x -i|h| 2 m + i 2 h 2 m. (4.34)
From Lemma 4.12, we have ∥m(t)∥ H 1 + ∥n(t)∥ H 1 uniformly bounded in t. Set φ = φ -h and ψ = ψ -k. Then φ, ψ solve:

L φ = P (φ, ψ) -P (h, k) -e -tλ m(t, x), L ψ = Q(φ, ψ) -Q(h, k) -e -tλ n(t, x). (4.35)
Set η = ( φ, ψ), W = (h, k), H = -e -tλ (m, n) and f (φ, ψ) = (P (φ, ψ), Q(φ, ψ)). We express solutions of (4.35) in the following form:

η(t) = i ∞ t S(t -s)[f (W + η) -f (W ) + H](s) ds, (4.36)
where S(t) is the Schrödinger group. Moreover, by using ψ = ∂φ -i 2 |φ| 2 φ, we have

ψ = ∂ φ - i 2 (| φ + h| 2 ( φ + h) -|h| 2 h). (4.37)
Step 2. Existence a solution of (4.35) From Lemma 4.13, there exists T * ≫ 1 such that for T 0 ⩾ T * there exists a unique solution η dened on [T 0 , ∞) of (4.35) such that

e tλ (∥η∥ S([t,∞))×S([t,∞)) ) + e tλ (∥η x ∥ S([t,∞))×S([t,∞)) ) ⩽ 1, ∀t ⩾ T 0 , (4.38)
Thus, for all t ⩾ T 0 , we have

∥ φ∥ H 1 + ∥ ψ∥ H 1 ≲ e -λt , (4.39)
Step 3. Existence of multi-solitons

Let η be the solution of (4.35) found in step 1. We prove that the solution η = ( φ, ψ) of (4.35) satises the relation (4.37).

Set φ = φ + h, ψ = ψ + k and v = ∂φ - i 2 |φ| 2 φ.
Since h solves Lh = P (h, k) + e -tλ m(t, x) and φ solves L φ = P (φ, ψ) -P (h, k)e -tθ m(t, x), we have Lφ = P (φ, ψ). Similarly, Lψ = Q(φ, ψ). We have

Lφ = P (φ, ψ), Lψ = Q(φ, ψ).
Thus,

Lψ -Lv = Q(φ, ψ) -∂Lφ - i 2 L(|φ| 2 φ) = Q(φ, ψ) -∂Lφ - i 2 (L(φ 2 )φ + φ 2 L(φ) + 2∂(φ 2 )∂φ) = Q(φ, ψ) -∂Lφ - i 2 (2Lφ|φ| 2 + 2(∂φ) 2 φ -φ 2 Lφ + 2φ 2 ∂ xx φ) + 4φ|∂φ| 2 ) . (4.40)
Moreover,

Lφ = P (φ, ψ) = iφ 2 ψ -b|φ| 4 φ = iφ 2 (ψ -v) + iφ 2 v -b|φ| 4 φ. (4.41)
Combining (4.41) and (4.40) and by an elementary calculation, we obtain

Lψ -Lv = Q(φ, ψ) -∂(iφ 2 (ψ -v)) -|φ| 2 φ 2 (ψ -v) - 1 2 |φ| 4 (ψ -v) -Q(φ, v) = (Q(φ, ψ) -Q(φ, v)) -2iφ∂φ(ψ -v) -iφ 2 ∂(ψ -v) -|φ| 2 φ 2 (ψ -v) - 1 2 |φ| 4 (ψ -v) = -i(ψ 2 -v 2 )φ -3b|φ| 4 (ψ -v) -2b|φ| 2 φ 2 (ψ -v) -2iφ v + i 2 |φ| 2 φ (ψ -v) -iφ 2 ∂(ψ -v) -|φ| 2 φ 2 (ψ -v) - 1 2 |φ| 4 (ψ -v). (4.42)
Dene ṽ = v -k. Since ψ -ṽ = ψ -v and (4.42) we have

L ψ-Lṽ = ( ψ-ṽ)A( ψ, ṽ, φ, h, k)+( ψ -ṽ)B( ψ, ṽ, φ, h, k)-i( φ+h) 2 ∂( ψ -ṽ), (4.43)
where

A = -i( ψ + ṽ + 2k)( φ + h) -3b| φ + h| 4 - 1 2 | φ + h| 4 , B = -2b| φ + h| 2 ( φ + h) 2 -2i( φ + h) ṽ + k + i 2 | φ + h| 2 ( φ + h) -| φ + h| 2 ( φ + h) 2 .
We see that A, B are polynomials of degree at most 4 in ( ψ, ṽ, φ, h, k). Multiplying both sides of (4.43) by ψ -ṽ then taking imaginary part and integrating over space using integration by parts, we obtain

1 2 ∂ t ∥ ψ -ṽ∥ 2 L 2 = Im R ( ψ -ṽ) 2 A( ψ, ṽ, φ, h, k) + ( ψ -ṽ) 2 B( ψ, ṽ, φ, h, k) + i 2 ∂( φ + h) 2 ( ψ -ṽ) 2 dx.
Thus,

1 2 ∂ t ∥ ψ -ṽ∥ 2 L 2 ≲ ∥ ψ -ṽ∥ 2 L 2 (∥A∥ L ∞ + ∥B∥ L ∞ + ∥∂( φ + h) 2 ∥ L ∞ ).
By using Grönwall inequality, we obtain

∥ ψ(t) -ṽ(t)∥ 2 L 2 ≲ ∥ ψ(N ) -ṽ(N )∥ 2 L 2 exp N t (∥A∥ L ∞ + ∥B∥ L ∞ + ∥∂( φ + h) 2 ∥ L ∞ ds . (4.44)
Combining (4.38), (4.39), using

k = h x -i 2 |h| 2 h, ṽ = ∂ φ -i 2 (| φ + h| 2 ( φ + h) -|h| 2 h), |h| = |R| and the Sobolev embedding H 1 (R) → L ∞ , we have, for t ⩾ T 0 : ∥ φ + h∥ L ∞ ≲ 1 + ∥h∥ L ∞ , ∥ṽ∥ L ∞ = ∂ φ - i 2 (| φ + h| 2 ( φ + h) -|h| 2 h) L ∞ ≲ ∥∂ φ∥ L ∞ + ∥ φ∥ 3 L ∞ + ∥ φ∥ L ∞ ∥h∥ 2 L ∞ ≲ 1 + ∥∂ φ∥ L ∞ + ∥h∥ 2 L ∞ . Thus, N t (∥A∥ L ∞ + ∥B∥ L ∞ + ∥∂( φ + h) 2 ∥ L ∞ ) ds ≲ N t (∥ ψ∥ L ∞ + ∥ṽ∥ L ∞ + ∥k∥ L ∞ )∥ φ + h∥ L ∞ + ∥ φ + h∥ 4 L ∞ + ∥ φ + h∥ L ∞ ∥ṽ + k∥ L ∞ + (∥ φ∥ L ∞ + ∥h∥ L ∞ )(∥∂ φ∥ L ∞ + ∥h x ∥ L ∞ ) ds ≲ N t (1 + ∥ṽ∥ L ∞ + ∥k∥ L ∞ )(1 + ∥h∥ L ∞ ) + 1 + ∥h∥ 4 L ∞ + (1 + ∥h∥ L ∞ )(∥ṽ∥ L ∞ + ∥k∥ L ∞ ) + (1 + ∥h∥ L ∞ )(∥∂ φ∥ L ∞ + ∥h x ∥ L ∞ ) ds ≲ N t 1 + ∥h∥ 4 L ∞ + ∥k∥ L ∞ (1 + ∥h∥ L ∞ ) + ∥ṽ∥ L ∞ (1 + ∥h∥ L ∞ ) + (1 + ∥h∥ L ∞ )(∥∂ φ∥ L ∞ + ∥h x ∥ L ∞ ) ds ≲ N t 1 + ∥h∥ 4 L ∞ + ∥k∥ L ∞ (1 + ∥h∥ L ∞ ) + ∥∂ φ∥ L ∞ (1 + ∥h∥ L ∞ ) + (1 + ∥h∥ L ∞ )(∥∂ φ∥ L ∞ + ∥k∥ L ∞ + ∥h∥ 3 L ∞ ) ds ≲ N t 1 + ∥h∥ 4 L ∞ + ∥k∥ L ∞ (1 + ∥h∥ L ∞ ) + ∥∂ φ∥ L ∞ (1 + ∥h∥ L ∞ ) ds ≲ (N -t)(1 + ∥h∥ 4 L ∞ L ∞ + ∥k∥ L ∞ L ∞ (1 + ∥h∥ L ∞ L ∞ )) + ∥∂ φ∥ L 4 (t,N )L ∞ (∥1∥ L 4 3 (t,N ) + ∥h∥ L 4 3 (t,N )L ∞ ) ≲ (N -t)(1 + ∥R∥ 4 L ∞ L ∞ + (∥h x ∥ L ∞ L ∞ + ∥R∥ 3 L ∞ L ∞ )(1 + ∥R∥ L ∞ L ∞ )) + (N -t) 3 4 (1 + ∥R∥ 4 3 L ∞ L ∞ ) ≲ (N -t)(1 + ∥R∥ 4 L ∞ L ∞ + ∥R x ∥ L ∞ L ∞ (1 + ∥R∥ L ∞ L ∞ )) + (N -t) 3 4 (1 + ∥R∥ 4 3 L ∞ L ∞ ).
Thus, there exists a certain positive constant C 0 such that

N t (∥A∥ L ∞ + ∥B∥ L ∞ + ∥∂( φ + h) 2 ∥ L ∞ ) ds ⩽ C 0 (N -t)(1 + ∥R∥ 4 L ∞ L ∞ + ∥R x ∥ L ∞ L ∞ (1 + ∥R∥ L ∞ L ∞ )) + (N -t) 3 4 (1 + ∥R∥ 4 3 L ∞ L ∞ ) .
Let C * = 32C 0 . From the assumption (4.8), we have

C 0 (1 + ∥R x ∥ L ∞ L ∞ )(1 + ∥R∥ L ∞ L ∞ ) + ∥R∥ 4 L ∞ L ∞ ⩽ v * 32 = λ 2 .
Hence, x t and let N large enough, we have

N t (∥A∥ L ∞ + ∥B∥ L ∞ + ∥∂( φ + h) 2 ∥ L ∞ ) ds ⩽ (N -t)λ.
Combining with (4.39) and (4.44), we obtain, for N large enough:

∥ ψ(t) -ṽ(t)∥ 2 L 2 ≲ e -2λN e (N -t)λ = e -λN -tλ .
Let N → ∞, we obtain ∥ ψ(t) -ṽ(t)∥ 2 L 2 = 0. This implies that ψ = ṽ and we have

ψ = v = ∂φ - i 2 |φ| 2 φ. (4.45) Dene u = exp -i 2 x -∞ |φ(y)| 2 dy φ. Combining (4.45) with the fact that (φ, ψ) solves Lφ = P (φ, ψ), Lψ = Q(φ, ψ),
we obtain that u solves (4.2). Moreover,

∥u -R∥ H 1 = exp - i 2 x -∞ |φ(y)| 2 dy φ -exp - i 2 x -∞ |h(y)| 2 dy h H 1 ≲ ∥φ -h∥ H 1 = ∥ φ∥ H 1
Combining with (4.39), for t ⩾ T 0 , we have

∥u -R∥ H 1 ⩽ Ce -λt ,
for a constant C depending on the parameters ω 1 , ..., ω K , c 1 , ..., c K . This completes the proof of Theorem 4.1.

4.3

Proof of Theorem 4.6

In this section, we prove Theorem 4.6. We use the similar idea in the proof of Theorem 4.1. However, the argument used in this section cannot apply to (4.2) (see Remark 4.10). We divide our proof into three steps:

Step 1. Preliminary analysis

Set v := u x + i 2 |u| 2 u.
By an elementary calculation, we see that if u solves (4.2) then (u, v) solves the following system:

         Lu = -iu 2 v + 1 2 -b |u| 4 u, Lv = iv 2 u + 3 2 -3b |u| 4 v + (1 -2b)|u| 2 u 2 v, u | t=0 = u 0 , v | t=0 = v 0 = ∂u 0 + i 2 |u 0 | 2 u 0 . (4.46)
Dene

P (u, v) = -iu 2 v + 1 2 -b |u| 4 u, Q(u, v) = iv 2 u + 3 2 -3b |u| 4 v + (1 -2b)|u| 2 u 2 v.
Let V be the multi kink-soliton prole dened in (4.21). Since R j solves (4.2), for all j, by an elementary calculation, we have

iV t + V xx + iV 2 V x + b|V | 4 V = i V 2 V x - K j=0 R 2 j R jx + b |V | 4 V - K j=0 |R j | 4 R j .
(4.47) From Lemma 4.14, we have

V 2 V x - K j=0 R 2 j R jx H 2 + |V | 4 V - K j=0 |R j | 4 R j H 2 ⩽ e -λt , (4.48)
for λ = 1 16 v * . Thus, we rewrite (4.47) as follows

iV t + V xx + iV 2 V x + b|V | 4 V = e -λt m(t, x), (4.49)
where

m(t) ∈ H 2 (R) such that ∥m(t)∥ H 2 uniformly bounded in t. Dene h = V, k = h x + i 2 |h| 2 h.
By an elementary calculation, h, k satisfy the following system.

Lh = -ih 2 k + 1 2 -b |h| 4 h + e -tλ m = P (h, k) + e -tλ m, Lk = ik 2 h + 3 2 -3b |h| 4 k + (1 -2b)|h| 2 h 2 k + e -tλ n = Q(h, k) + e -tλ n.
where n = m x + i|h| 2 m -i 2 h 2 m satises ∥n(t)∥ H 1 uniformly bounded in t. Let ũ = u -h and ṽ = v -k. Then (ũ, ṽ) solves:

Lũ = P (u, v) -P (h, k) -e -tλ m, Lṽ = Q(u, v) -Q(h, k) -e -tλ n. (4.50) Dene η = (ũ, ṽ), W = (h, k), H = e -tλ (m, n) and f (u, v) = (P (u, v), Q(u, v)).
We nd a solution of (4.50) in the Duhamel form

η = -i ∞ t S(t -s)[f (W + η) -f (W ) + H](s) ds. (4.51) Moreover, from v = u x + i 2 |u| 2 u, we have ṽ = ũx + i 2 (|ũ + h| 2 (ũ + h) -|h| 2 h). (4.52)
Step 2. Existence a solution of (4.51)

From Lemma 4.13, there exists T * ≫ 1 such that for T 0 ≫ T * there exists a unique solution η dened on [T 0 , ∞) of (4.51) such that

e tλ ∥η∥ S([t,∞))×S([t,∞)) + e tλ ∥η x ∥ S([t,∞))×S([t,∞)) ⩽ 1, ∀t ⩾ T 0 , (4.53)
where λ = v * 16 . Thus, for all t ⩾ T 0 , we have

∥ũ∥ H 1 + ∥ṽ∥ H 1 ≲ e -tλ . (4.54)
Step 3. Existence of multi kink-solitons By using similar arguments as in the proof of Theorem 4.1 we can prove that the solution η = ( φ, ψ) of (4.51) satises the relation (4.52) provided assumption (4.22) is veried. This implies that

ṽ = ũx + i 2 (|ũ + h| 2 (ũ + h) -|h| 2 h). Set u = ũ + h, v = ṽ + k. We have v = u x + i 2 |u| 2 u. (4.55)
Since (ũ, ṽ) solves (4.50), we infer that u, v solve

Lu = P (u, v), Lv = Q(u, v).
Combining with (4.55), we have u solves (4.2). Moreover, for t ⩾ T 0 , we have

∥u -V ∥ H 1 = ∥ũ∥ H 1 ≲ e -λt .
This completes the proof of Theorem 4.6.

Remark 4.10. We do not have the proof for the construction of multi kink-solitons for (4.2). The reason is that if the prole R in the proof of Theorem 4.1 is not in H 1 (R) then the function h dened as in (4.31) is not in H 1 (R). Thus, the functions m, n dened as in (4.33) and (4.34) are not in H 1 (R) and we can not apply Lemma 4.13 to construct a solution of system (4.35).

4.4 Some technical lemmas

Properties of solitons

In this section, we prove some estimates on the multi-soliton prole used in the proof of Theorem 4.1.

Lemma 4.11. There exist T 0 > 0 and a constant λ > 0 such that the estimate (4.29) is uniformly true for t ⩾ T 0 .

Proof. First, we need some estimates on the soliton prole. We have

|R j (x, t)| = |Φ ω j ,c j (x -c j t)| = √ 2h j c 2 j + γh 2 j cosh(h j (x -c j t)) -c j -1 2 ≲ h j ,|c j | e -h j 2 |x-c j t| .
Moreover,

|∂R j (x, t)| = |∂ϕ ω j ,c j (x -c j t)| = - √ 2 2 h 2 j c 2 j + γh 2 j |sinh(h j (x -c j t)| c 2 j + γh 2 j cosh(h j (x -c j t)) -c j -3 2 ≲ h j ,|c j | e -h j 2 |x-c j t| .
By an elementary calculation, we have

|∂ 2 R j (x, t)| + |∂ 3 R j (x, t)| ≲ h j ,|c j | e -h j 2 |x-c j t| .
For convenience, we set

χ 1 = i|R| 2 R x -i K j=1 |R j | 2 R jx , (4.56)
χ 2 = |R| 4 R - K j=1 |R j | 4 R j . (4.57) Fix t > 0. For x ∈ R, choose m = m(x) ∈ {1, 2, ..., K} so that |x -c m t| = min j |x -c j t|.
For j ̸ = m, we have

|x -c j t| ⩾ 1 2 |c j t -c m t| = t 2 |c j -c m |.
Thus, we have

|(R -R m )(x, t)| + |(∂R -∂R m (x, t))| + |∂ 2 R -∂ 2 R m | + |∂ 3 R -∂ 3 R m | ⩽ j̸ =m (|R j (x, t)| + |∂R j (x, t)| + |∂ 2 R j (x, t)| + |∂ 3 R j (x, t)|) ≲ h 1 ,..,h K ,|c 1 |,..,|c K | δ m (x, t) := j̸ =m e -h j 2 |x-c j t| Recall that v * = inf j̸ =k h j |c j -c k |.
We have

|(R-R m )(x, t)|+|(∂R-∂R m (x, t))|+|∂ 2 R-∂ 2 R m |+|∂ 3 R-∂ 3 R m | ≲ δ m (x, t) ≲ e -1 4 v * t .
Let f 1 , g 1 , r 1 and f 2 , g 2 , r 2 be the polynomials of u, u x , u xx , u xxx and conjugates satisfying:

i|u| 2 u x = f 1 (u, u, u x ), |u| 4 u = f 2 (u, u), ∂(i|u| 2 u x ) = g 1 (u, u x , u xx , u, ..), ∂(|u| 4 u) = g 2 (u, u x , u, ..), ∂ 2 (i|u| 2 u x ) = r 1 (u, u x , u xx , u xxx , u, ..), ∂ 2 (|u| 4 u) = r 2 (u, u x , u xx , u, ..). Denote A = sup |u|+|ux|+|uxx|+|uxxx|⩽ K j=1 ∥R j ∥ H 4 (|df 1 | + |df 2 | + |dg 1 | + |dg 2 | + |dr 1 | + |dr 2 |),
We have

|χ 1 | + |χ 2 | + |∂χ 1 | + |∂χ 2 | + |∂ 2 χ 1 | + |∂ 2 χ 2 | ⩽ |f 1 (R, R x ) -f 1 (R m , R mx )| + |f 2 (R) -f 2 (R m )| + j̸ =m (|f 1 (R j , R jx )| + |f 2 (R j )|) + |g 1 (R, R x , R xx , ..) -g 1 (R m , R mx , R mxx , ..)| + |g 2 (R, R x , ..) -g 2 (R m , R mx , ..)| + j̸ =m (g 1 (R j , R jx , R jxx , ..) + g 2 (R j , R jx ), ..) + |r 1 (R, R x , R xx , R xxx , ..) -r 1 (R m , R mx , R mxx , R mxxx , ..)| + |r 2 (R, R x , R xx , ..) -r 2 (R m , R mx , R mxx , ..)| + j̸ =m (r 1 (R j , R jx , R jxx , R jxxx , ..) + r 2 (R j , R jx , R jxx , ..)) ⩽ A(|R -R m | + |R x -R mx | + |R xx -R mxx | + |R xxx -R mxxx |) + j̸ =m A(|R j | + |R jx | + |R jxx | + |R jxxx |) ⩽ 2A j̸ =m (|R j | + |R jx | + |R jxx | + |R jxxx |) ≲ p δ m (t, x). In particular, ∥χ 1 ∥ W 2,∞ + ∥χ 2 ∥ W 2,∞ ≲ p e -1 4 v * t . Moreover, we have ∥χ 1 ∥ W 2,1 + ∥χ 2 ∥ W 2,1 ≲ K j=1 (∥|R j | 2 R jx ∥ L 1 + ∥∂(|R j | 2 R jx )∥ L 1 + ∥∂ 2 (|R j | 2 R jx )∥ L 1 + ∥R 5 j ∥ L 1 + ∥∂(|R j | 4 R j )∥ L 1 + ∥∂ 2 (|R j | 4 R j )∥ L 1 ) ≲ K j=1 (∥R j ∥ 3 H 1 + ∥R j ∥ 3 H 2 + ∥R j ∥ 3 H 3 + ∥R j ∥ 5 H 1 + ∥R j ∥ 5 H 1 + ∥R j ∥ 5 H 2 ) < C < ∞
By Holder inequality, for 1 < r < ∞, we have

∥χ 1 ∥ W 2,r + ∥χ 2 ∥ W 2,r ≲ p e -(1-1 r ) 1 4 v * t , ∀r ∈ (1, ∞).
Choosing r = 2 we obtain:

∥χ 1 ∥ H 2 + ∥χ 2 ∥ H 2 ≲ p e -v *
8 t , Thus, for t ⩾ T 0 , where T 0 large enough depend on the parameters ω 1 , ..., ω K , c 1 , ..., c K , we have

∥χ 1 ∥ H 2 + ∥χ 2 ∥ H 2 ⩽ e -v * 16 t , ∀t ⩾ T 0 . Let λ = v *
16 , we obtain the desired result.

Prove the boundedness of v, m, n

Let v, m and n be given as in (4.28), (4.33) and (4.34) respectively. In this section, we prove the uniform in time boundedness in H 2 (R) of v and in H 1 (R) of m, n. We have the following result.

Lemma 4.12. There exist C > 0 and T 0 > 0 such that for all t > T 0 the functions v, m, n satisfy

∥v(t)∥ H 2 + ∥m(t)∥ H 1 + ∥n(t)∥ H 1 ⩽ C,
Proof. Let χ 1 and χ 2 be dened as in (4.56) and (4.57) respectively. We have

e -λt v = χ 1 + bχ 2 .
By Lemma 4.11, we have ∥v(t)∥ H 2 ⩽ D, for some constant D > 0. From (4.33), we have

∥m∥ H 2 ≲ ∥v∥ H 2 + ∥h∥ H 2 ∥v∥ H 2 ∥R∥ H 2 ⩽ C 1 ,
for some constant C 1 > 0. From, (4.34), we have

∥n∥ L 2 ≲ ∥m x ∥ L 2 + ∥h∥ 2 H 1 ∥m∥ H 1 ⩽ ∥m∥ H 1 (1 + ∥h∥ 2 H 1 ) ⩽ C 2 ,
for some constant C 2 > 0. Moreover, we have

∥n x ∥ L 2 ≲ ∥m xx ∥ L 2 + ∥h∥ 2 H 1 ∥m∥ H 1 ⩽ ∥m∥ H 2 (1 + ∥h∥ 2 H 1 ) ⩽ C 3 ,
for some constant C 3 > 0. Choosing C = D + C 1 + C 2 + C 3 , we obtain the desired result.

Existence solution of system equation

In this section, we prove the existence of solutions of (4.36). For convenience, we recall the equation:

η(t) = i ∞ t S(t -s)[f (W + η) -f (W ) + H](s) ds, (4.58)
where

η = (ũ, ṽ) is unknown function, W = (h, k), H = -e -tλ (m, n) and f (u, v) = (P (u, v), Q(u, v))
, where P, Q are dened by

P (u, v) = -iu 2 v + 1 2 -b |u| 4 u, Q(u, v) = iv 2 u + 3 2 -3b |u| 4 v + (1 -2b)|u| 2 u 2 v.
The existence of solutions of (4.58) is established in the following lemma.

Lemma 4. [START_REF] Biagioni | Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations[END_REF].

Let H = H(t, x) : [0, ∞) × R → C 2 , W = W (t, x) : [0, ∞) × R → C 2
be given vector functions which satisfy for some C 1 > 0, C 2 > 0, λ > 0, T 0 ⩾ 0:

∥W (t)∥ L ∞ ×L ∞ + e λt ∥H(t)∥ L 2 ×L 2 ⩽ C 1 ∀t ⩾ T 0 , (4.59)
∥∂W (t)∥ L 2 ×L 2 + ∥∂W (t)∥ L ∞ ×L ∞ + e λt ∥∂H(t)∥ L 2 ×L 2 ⩽ C 2 , ∀t ⩾ T 0 . (4.60)
Consider equation (4.58). There exists a constant λ * such that if λ ⩾ λ * then there exists a unique solution η to (4.58) on [T 0 , ∞) × R satisfying

e λt ∥η∥ S([t,∞))×S([t,∞)) + e λt ∥∂η∥ S([t,∞))×S([t,∞)) ⩽ 1, ∀t ⩾ T 0 .
Proof. We use similar arguments as in [START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF]. We rewrite (4.58) into η = Φη. We shall show that, for λ suciently large, Φ is a contraction map in the ball

B = η : ∥η∥ X := e λt ∥η∥ S([t,∞))×S([t,∞)) + e λt ∥∂η∥ S([t,∞))×S([t,∞)) ⩽ 1 .
Step

1. Proof that Φ maps B into B Let t ⩾ T 0 , η = (η 1 , η 2 ) ∈ B, W = (w 1 , w 2 ) and H = (h 1 , h 2 ). By Strichartz estimates, we have ∥Φη∥ S([t,∞))×S([t,∞)) ≲ ∥f (W + η) -f (W )∥ N ([t,∞))×N ([t,∞)) (4.61)
+ ∥H∥ L 1 τ L 2 x ([t,∞))×L 1 τ L 2 x ([t,∞)) . (4.62)
For (4.62), using (4.59), we have

∥H∥ L 1 τ L 2 x ([t,∞))×L 1 τ L 2 x ([t,∞)) = ∥h 1 ∥ L 1 τ L 2 x ([t,∞)) + ∥h 2 ∥ L 1 τ L 2 x ([t,∞)) ≲ ∞ t e -λτ dτ ⩽ 1 λ e -λt .
For (4.61), we have

|P (W + η) -P (W )| = |P (w 1 + η 1 , w 2 + η 2 ) -P (w 1 , w 2 )| ≲ |(w 1 + η 1 ) 2 (w 2 + η 2 ) -w 2 1 w 2 | + ||η 1 + w 1 | 4 (η 1 + w 1 ) -|w 1 | 4 w 1 | ≲ |η 1 | + |η 2 | + |η 1 | 5 Thus, ∥P (W + η) -P (W )∥ N ([t,∞)) ≲ ∥η 1 ∥ N ([t,∞)) + ∥η 2 ∥ N ([t,∞)) + ∥η 5 1 ∥ N ([t,∞)) ≲ ∥η 1 ∥ L 1 τ L 2 x (t,∞) + ∥η 2 ∥ L 1 τ L 2 x (t,∞) + ∥η 5 1 ∥ L 1 τ L 2 x (t,∞) ≲ ∞ t e -λτ dτ + ∞ t ∥η 1 (τ )∥ 5 L 10 dτ ≲ 1 λ e -λt + ∞ t ∥η 1 (τ )∥ 7 2 L 2 ∥∂η 1 (τ )∥ 3 2 L 2 ≲ 1 λ e -λt + ∞ t e -(7/2λ+3/2λ)τ dτ ≲ 1 λ e -λt + 1 7/2λ + 3/2λ e -(7/2λ+3/2λ)t ≲ 1 λ e -λt .
By similar arguments as above, we have

∥Q(W + η) -Q(W )∥ N ([t,∞)) ≲ 1 λ e -λt .
Thus, for λ large enough, we have

∥Φη∥ S([t,∞)×S([t,∞))) ⩽ 1 10 e -λt .

It remains to estimate ∥∂Φη∥ S([t,∞)×S([t,∞)))

. By Strichartz estimate we have

∥∂Φη∥ S([t,∞)×S([t,∞))) ≲ ∥∂(f (W + η) -f (W ))∥ N ([t,∞))×N ([t,∞)) (4.63)
+ ∥∂H∥ N ([t,∞))×N ([t,∞)) . (4.64)
For (4.64), using (4.60), we have

∥∂H∥ N ([t,∞))×N ([t,∞)) ⩽ ∥∂h 1 ∥ L 1 τ L 2 x ([t,∞)) + ∥∂h 2 ∥ L 1 τ L 2 x ([t,∞)) ≲ ∞ t e -λτ dτ = 1 λ e -λt . (4.65)
For (4.63), we have

∥∂(f (W + η) -f (W ))∥ N ([t,∞))×N ([t,∞)) = ∥∂(P (W + η) -P (W ))∥ N ([t,∞)) + ∥∂(Q(W + η) -Q(W ))∥ N ([t,∞))
Furthermore,

|∂(P (W + η) -P (W ))| ≲ |∂((w 1 + η 1 ) 2 (w 2 + η 2 ) -w 2 1 w 2 )| + |∂(|w 1 + η 1 | 4 (w 1 + η 1 ) -|w 1 | 4 w 1 )| ≲ |∂η|(|η| 2 + |W | 2 ) + |∂W |(|η| 2 + |W ||η|) + |∂η|(|η| 4 + |W | 4 ) + |∂W |(|η| 4 + |η||W | 3 ).
Thus, we have

∥∂(P (W + η) -P (W ))∥ N ([t,∞)) ≲ ∥|∂η|(|η| 2 + |W | 2 )∥ N ([t,∞)) + ∥|∂W |(|η| 2 + |W ||η|)∥ N ([t,∞)) (4.66)
+ ∥|∂η|(|η| 4 + |W | 4 )∥ N ([t,∞)) + ∥|∂W |(|η| 4 + |η||W | 3 )∥ N ([t,∞)) . (4.67)
For (4.66), using (4.59) and (4.60) and the assumption η ∈ B we have

∥|∂η|(|η| 2 + |W | 2 )∥ N ([t,∞)) + ∥|∂W |(|η| 2 + |W ||η|)∥ N ([t,∞)) ≲ ∥|∂η||η| 2 ∥ L 1 τ L 2 x ([t,∞)) + ∥|∂η||W | 2 ∥ L 1 τ L 2 x ([t,∞)) + ∥|∂W ||η| 2 ∥ L 1 τ L 2 x ([t,∞)) + ∥|∂W ||W ||η|∥ L 1 τ L 2 x ([t,∞)) ≲ ∥|∂η|∥ L 2 τ L 2 x ([t,∞)) ∥|η|∥ 2 L 4 τ L ∞ + ∥|∂η|∥ L 1 τ L 2 x ([t,∞)) ∥|W |∥ 2 L ∞ L ∞ + ∥|∂W |∥ L ∞ L ∞ ∥|η|∥ L 4 τ L ∞ x ([t,∞)) ∥|η|∥ L 4/3 τ L 2 x ([t,∞)) + ∥|W |∥ L ∞ L ∞ ∥|∂W |∥ L ∞ L ∞ ∥|η|∥ L 1 τ L 2 x ([t,∞)) ≲ 1 λ e -λt .
For (4.67), using (4.59) and (4.60) and the assumption η ∈ B we have

∥|∂η|(|η| 4 + |W | 4 )∥ N ([t,∞)) + ∥|∂W |(|η| 4 + |η||W | 3 )∥ N ([t,∞)) ≲ ∥|∂η|(|η| 4 + |W | 4 )∥ L 1 τ L 2 x ([t,∞)) + ∥|∂W |(|η| 4 + |η||W | 3 )∥ L 1 τ L 2 x ([t,∞)) ≲ ∥∂η∥ L ∞ τ L 2 x ([t,∞)) ∥η∥ 4 L 4 τ L ∞ x ([t,∞)) + ∥W ∥ 4 L ∞ L ∞ ∥∂η∥ L 1 τ L 2 x ([t,∞)) + ∥∂W ∥ L ∞ L 2 ∥η∥ 4 L 4 τ L ∞ x ([t,∞)) + ∥∂W ∥ L ∞ L ∞ ∥|W |∥ 3 L ∞ L ∞ ∥η∥ L 1 τ L 2 x ([t,∞)) ≲ 1 λ e -λt .
Hence,

∥∂(P (W + η) -P (W ))∥ N ([t,∞)) ≲ 1 λ e -λt . (4.68)
By similar arguments, we have

∥∂(Q(W + η) -Q(W ))∥ N ([t,∞)) ≲ 1 λ e -λt . (4.69)
Combining (4.68) and (4.69), we obtain

∥∂(f (W + η) -f (W ))∥ N ([t,∞))×N ([t,∞)) ≲ 1 λ e -λt . (4.70)
Combining (4.65) and (4.70), we obtain

∥∂Φη∥ S([t,∞))×S([t,∞)) ≲ 1 λ e -λt ⩽ 1 10 e -λt ,
if λ > 0 is large enough. Thus, for λ > 0 large enough

∥Φη∥ X ⩽ 1. (4.71)
This implies that Φ map B onto B.

Step 2. Φ is contraction map on B By using (4.59) and (4.60) and similar estimates as for the proof of (4.71), we can show that, for any η ∈ B, κ ∈ B,

∥Φη -Φκ∥ X ⩽ 1 2 ∥η -κ∥ X .
By Banach xed point theorem there exists a unique solution on B of (4.58).

Properties of multi kink-solitons prole

In this section, we prove some estimates on the multi kink-solitons prole used in the proof of Theorem 4.6.

Lemma 4.14. There exist T 0 > 0 and a constant λ > 0 such that the estimate (4.48) is uniformly true for t ⩾ T 0 .

Proof. For convenience, set

R = K j=1 R j .
By similar arguments in the proof of Lemma 4.11, we have

|R j (x, t)| + |∂R j (x, t)| + |∂ 2 R j (x, t)| + |∂ 3 R j (x, t)| ≲ h j ,|c j | e -h j 2 |x-c j t| ,
for all 1 ⩽ j ⩽ K. Dene

χ 1 = iV 2 V x -i K j=0 R 2 j R jx , χ 2 = |V | 4 V - K j=0 |R j | 4 R j . Fix t > 0. For x ∈ R, we choose m = m(x) ∈ N such that |x -c m t| = min j∈N |x -c j t|.
If m ⩾ 1 then by the assumption c 0 < c j for j > 0 we have x > c 0 t. Thus, by the asymptotic behaviour of Φ 0 as in Remark 4.7, we can see R 0 as a soliton. More precise, we have

|R 0 (t, x)| + |R ′ 0 (t, x)| + |R ′′ 0 (t, x)| + |R ′′′ 0 (t, x)| ≲ e -1 2 |x-c 0 t| ≲ e -1 4 v * t .
Using similar argument as in the proof of Lemma 4.11, we have:

|(R -R m )(x, t)| + |(∂R -∂R m )(x, t)| + |(∂ 2 R -∂ 2 R m )(x, t)| + |∂ 3 R -∂ 3 R m | ≲ e -1 4 v * t .
Let f 1 , g 1 , r 1 and f 2 , g 2 , r 2 be the polynomials of u, u x , u xx , u xxx and their conjugates such that for all u ∈ H 3 (R):

iu 2 u x = f 1 (u, u, u x ), |u| 4 u = f 2 (u, u), ∂(iu 2 u x ) = g 1 (u, u x , u xx , u, ..), ∂(|u| 4 u) = g 2 (u, u x , u, ..), ∂ 2 (iu 2 u x ) = r 1 (u, u x , u xx , u xxx , u, ..), ∂ 2 (|u| 4 u) = r 2 (u, u x , u xx , u, ..). Denote A = sup |u|+|ux|+|uxx|+|uxxx|⩽∥R 0 ∥ W 4,∞ + K j=1 ∥R j ∥ H 4 (R) (|df 1 |+|df 2 |+|dg 1 |+|dg 2 |+|dr 1 |+|dr 2 |).
On one hand,

∥χ 1 (t)∥ W 2,1 ≲ K j=0 (∥R 2 j R jx ∥ L 1 + ∥∂(R 2 j R jx )∥ L 1 + ∥∂ 2 (R 2 j R jx )∥ L 1 ) ≲ K j=1 ∥R j ∥ 3 H 3 + ∥∂R 0 ∥ W 2,1 < C < ∞.
On the other hand,

∥χ 2 (t)∥ W 2,1 ≲ ∥|V | 4 V -|R 0 | 4 R 0 ∥ W 2,1 + K j=1 ∥|R j | 5 ∥ W 2,1 ≲ |R 0 | 4 K j=1 |R j | + K j=1 |R j | 5 W 2,1 + K j=1 ∥R j ∥ 5 W 2,1 ≲ K j=1 ∥|R 0 | 4 |R j |∥ W 2,1 + K j=1 ∥R j ∥ 5 W 2,1 ≲ K j=1 (∥R j ∥ W 2,1 ∥R 0 ∥ 4 W 2,∞ + ∥R j ∥ 5 H 3 ) < C < ∞. Thus, ∥χ 1 (t)∥ W 2,1 + ∥χ 1 (t)∥ W 2,1 < ∞. (4.73)
From (4.72) and (4.73), using Hölder inequality, we have

∥χ 1 (t)∥ H 2 + ∥χ 2 (t)∥ H 2 ≲ p e -1 8 v * t .
Let T 0 be large enough, we have

∥χ 1 (t)∥ H 2 + ∥χ 2 (t)∥ H 2 ⩽ e -1 16 v * t , ∀t ⩾ T 0 .
Setting λ = 1 16 v * , we obtain the desired result.

tary waves solutions given by

ψ ω,c (t, x) = φ ω,c (x -ct) exp i ωt + c 2 (x -ct) - 1 2σ + 2 x-ct -∞ φ 2σ ω,c (η) dη , (5.2
) where ω > c 2 4 and

φ 2σ ω,c (y) = (σ + 1)(4ω -c 2 ) 2 √ ω cosh(σ √ 4ω -c 2 y) -c 2 √ ω . (5.3)
The prole φ ω,c is a positive solution of

-∂ 2 y φ ω,c + ω - c 2 4 φ ω,c + c 2 |φ ω,c | 2σ φ ω,c - 2σ + 1 (2σ + 2) 2 |φ ω,c | 4σ φ ω,c = 0. (5.4) Dene ϕ ω,c (y) = φ ω,c (y)e iθω,c(y) , (5.5)
where

θ ω,c (y) = c 2 y - 1 2σ + 2 y -∞ φ 2σ ω,c (η) dη. (5.6)
Clearly, we have

ψ ω,c (x, t) = e iωt ϕ ω,c (x -ct). (5.7)
and ϕ ω,c solves

-∂ 2 y ϕ ω,c + ωϕ ω,c + ic∂ y ϕ ω,c -i|ϕ ω,c | 2σ ∂ y ϕ ω,c = 0, y ∈ R. (5.8)
Let K ∈ N. For each 1 ⩽ j ⩽ K, let (ω j , c j , x j , θ j ) ∈ R 4 be parameters such that

ω j > c 2 j 4 . Dene, for each j = 1, ..., K R j (t, x) = e iθ j ψ ω j ,c j (t, x -x j )
and dene the multi-soliton prole by

R = K j=1 R j . (5.9)
For convenience, dene h j = 4ω j -c 2 j , for each j = 1, ..., K. Our main result is the following. Theorem 5.1. Let σ ⩾ 5 2 , K ∈ N * and for each 1 ⩽ j ⩽ K, (θ j , ω j , c j , x j ) be a sequence of parameters such that x j ∈ R, θ j ∈ R, c j ̸ = c k , for j ̸ = k. The multisoliton prole R is given as in (5.9). There exists a certain positive constant C * such that if the parameters (ω j , c j ) satisfy

C * (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 )(1 + ∥∂ x R∥ L ∞ L ∞ + ∥R∥ 2σ+1 L ∞ L ∞ ) ⩽ v * = inf j̸ =k h j |c j -c k |, (5.10)
then there exists a solution u of (5.1) such that

∥u -R∥ H 1 ⩽ Ce -λt , ∀t ⩾ T 0 ,
for positive constants C, T 0 depending only on the parameters ω 1 , ..., ω K , c 1 , ..., c K and λ = 1 16 v * .

We have the following comment about the restriction σ ⩾ 5 2 .

Remark 5.2. By Lemma 5.7, the following inequality holds for σ ⩾ 2:

(a + b) 2(σ-2) -a 2(σ-2) ≲ b 2(σ-2) + ba 2(σ-2)-1
, for all a, b > 0.

(5.11)

The condition σ ⩾ 5 2 ensures that the order of b on the right hand side of (5.11) is larger than 1. This is used in the proof of Lemma 5.9.

The condition (5.10) is an implicit condition on the parameters. Below, we show that for large, negative and enough separated velocities, the condition (5.10) holds.

Remark 5.3. We prove that there exist parameters (ω j , c j , θ j , x j ) for 1 ⩽ j ⩽ K such at the condition (5.10) is satised. Let M > 0, h j > 0, d j < 0, for each 1 ⩽ j ⩽ K. We chose (c j , ω j ) = M d j , 1 4 (h 2 j + M 2 d 2 j ) . We verify that this choice satises the condition (5.10) for M large enough. Indeed, we see that c j < 0 and h j ≪ |c j | for M large enough. We have

φ 2σ ω j ,c j ≈ h 2 j 2
√ ω j cosh(σh j y) -

c j 2 √ ω j ∂ x φ ω j ,c j ≈ h 2 j 2 √ ω j 1 2σ
-sinh(σh j y) cosh(σh j y) -

c j 2 √ ω j 1+ 1 2σ . Using | sinh(x)| ⩽ | cosh(x)| for all x ∈ R we have |∂ x φ ω j ,c j | ⩽ h 2 j 2 √ ω j 1 2σ
1 (cosh(σh j y) -

c j 2 √ ω j ) 1 2σ ≲ |φ ω j ,c j |. Thus, ∥R j ∥ L ∞ L ∞ = ∥φ ω j ,c j ∥ L ∞ ≲ 2σ h 2 j |c j | ≪ 1 ∥∂ x R j ∥ L ∞ L ∞ = ∥∂ x ϕ ω j ,c j ∥ L ∞ L ∞ ≲ ∥∂ x φ ω j ,c j ∥ L ∞ + c j 2 φ ω j ,c j - 1 2σ + 2 φ 2σ+1 ω j ,c j L ∞ ≲ ∥φ ω j ,c j ∥ L ∞ + |c j |∥φ ω j ,c j ∥ L ∞ ≲ 2σ h 2 j |c j | + |c j | 2σ h 2 j |c j | . Hence, ∥R∥ L ∞ L ∞ ≲ j 2σ h 2 j |c j | ≲ 1 ∥∂ x R∥ L ∞ L ∞ ≲ j   2σ h 2 j |c j | + |c j | 2σ h 2 j |c j |   .
Furthermore,

∥R j ∥ 2 L ∞ H 1 = ∥R j ∥ 2 L ∞ L 2 + ∥∂ x R j ∥ 2 L ∞ L 2 = ∥φ ω j ,c j ∥ 2 L 2 + ∥∂ x φ ω j ,c j ∥ 2 L 2 ≲ ∥φ ω j ,c j ∥ 2 L 2 ≲ h 2 j 2 √ ω j 1 σ 1 cosh(σh j y) 1 2σ 2 L 2 ≲ h 2 j 2 √ ω j 1 σ ∥e -h j 2 |y| ∥ 2 L 2 ≈ h 2 j 2 √ ω j 1 σ 1 h j ≲ h 1 σ j h -1 j = h 1 σ -1 j
, where we use h j ⩽ 2 √ ω j . Thus,

∥R∥ 2 L ∞ H 1 ≲ j h 1 σ -1 j
.

The condition (5.10) satises if the following estimate holds:

1 + j h 1 σ -1 j   1 + j   2σ h 2 j |c j | + |c j | 2σ h 2 j |c j |     ≪ inf j̸ =k h j |c j -c k |.
(5.12)

We see that the left hand side of (5.12) is order M 1-1 2σ and the right hand side of (5.12) is order M 1 . Hence, the condition (5.10) satises if we choose M large enough.

Remark 5.4. We may replace the condition (5.10) by the following condition

(1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 )(1 + ∥∂ x R∥ L ∞ L ∞ + ∥R∥ 2σ+1 L ∞ L ∞ ) ⩽ Cv * = inf j̸ =k h j |c j -c k |, (5.13
) where C is a certain positive number. We do not know exactly what this constant is. The condition (5.10) says that we can choose the parameters such that the right hand side of (5.10) is arbitrary larger than the left hand side and hence the condition (5.13) satises.

Our strategy of the proof of Theorem 5.1 is as follows. First, we dene φ, ψ based on u in such a way that φ and ψ satisfy a system of nonlinear Schrödinger equations without derivatives (see (5.16)). Let R be a multi-soliton prole which satises the assumptions of Theorem 5.1. Then R solves (5.1) up to a small perturbation. Let (h, k) be dened in a similar way as (φ, ψ) but replace u by R. We see that (h, k) solves (5.16) up to small perturbations. Setting φ = φ -h and ψ = ψ -k, we see that if u solves (5.1) then ( φ, ψ) solves a system and a relation between φ and ψ holds and vice versa. By using the Banach xed point theorem, we prove that there exists a solution ( φ, ψ) of this system which exponential decays in time on H 1 (R) for t large. Combining with the assumption (5.10), we can prove a relation between φ and ψ. Thus, we easily obtain the solution u of (5.1) satisfying the desired property.

This chapter is organized as follows. In Section 5.2, we prove the existence of multi-soliton trains for the equation (5.1). In Section 5.3, we prove some technical results which are used in the proof of the main result Theorem 5.1. More precisely, we prove the exponential decay of perturbations in the equations of h, k (Lemma 5.6) and the existence of decaying solutions for the system of equations of φ, ψ (Lemma 5.9).

Before proving the main result, we introduce some notation used in this chapter.

Notation.

(1) We denote the Schrödinger operator as follows

L = i∂ t + ∂ 2 x .
(2) Given a time t ∈ R, the Strichartz space S([t, ∞)) is dened via the norm

∥u∥ S([t,∞)) = sup (q,r) admissible ∥u∥ L q t L r x ([t,∞)×R) .
We denote the dual space by N [t, ∞) = S([t, ∞)) * . Hence for any (q, r) admissible pair we have

∥u∥ N ([t,∞)) ⩽ ∥u∥ L q ′ t L r ′
x ([t,∞)×R) .

(

Proof of the main result

In this section we give the proof of Theorem 5.1. We use the Banach xed point theorem and Strichartz estimates. We divide our proof in three steps.

Step 1. Preliminary analysis. Let u ∈ C(I, H 1 (R)) be a H 1 (R) solution of (5.1) on I. Consider the following transform:

φ(t, x) = exp(iΛ)u(t, x), (5.14)
ψ = exp(iΛ)∂ x u = ∂ x φ - i 2 |φ| 2σ φ, (5.15)
where

Λ = 1 2 x -∞ |u(t, y)| 2σ dy.
As in [56, section 4], we have

∂ t Λ = -σIm(|u| 2(σ-1) u∂ x u) + σIm x -∞ ∂ x (|u| 2(σ-1) u)∂ x u dy - 1 4 |u| 4σ .
Thus, using |u| = |φ| and Im(u∂ x u) = Im(φψ), we have

∂ t Λ = -σ|φ| 2(σ-1) Im(φψ) + σ x -∞ ∂ x (|u| 2(σ-1) )Im(u∂ x u) dx - 1 4 |φ| 4σ = -σ|φ| 2(σ-1) Im(φψ) + σ x -∞ ∂ x (|φ| 2(σ-1) )Im(φψ) dx - 1 4 |φ| 4σ .
Since u solves (5.1), we have

Lφ = L(exp(iΛ))u + exp(iΛ)Lu + 2∂ x (exp(iΛ))∂ x u = L(exp(iΛ))u + exp(iΛ)(Lu + i|u| 2σ u) = L(exp(iΛ))u = (i∂ t + ∂ 2 x )(exp(iΛ))u, = -exp(iΛ)∂ t Λ + ∂ x (exp(iΛ) i 2 |u| 2σ ) u = -φ∂ t Λ + exp(iΛ) -1 4 |u| 2σ + i 2 exp(iΛ)∂ x (|u| 2σ ) u = -φ∂ t Λ + φ - 1 4 |φ| 4σ + i 2 ∂ x (|φ| 2σ ) = σ|φ| 2(σ-1) φIm(φψ) -σφ x -∞ ∂ x (|φ| 2(σ-1) )Im(φψ) dx + 1 4 |φ| 4σ φ - 1 4 φ|φ| 4σ + iσ|φ| 2(σ-1) φRe(φ∂ x φ) = σ|φ| 2(σ-1) φ(Im(φψ) + iRe(φ∂ x φ)) -σφ x -∞ |φ| 2(σ-2) (σ -1)∂ x (|φ| 2 )Im(φψ) dx = σ|φ| 2(σ-1) φ(Im(φψ) + iRe(φψ)) -σ(σ -1)φ x -∞ |φ| 2(σ-2) 2Re(φψ)Im(φψ) dx = iσ|φ| 2(σ-1) φ 2 ψ -σ(σ -1)φ x -∞ |φ| 2(σ-2) Im(ψ 2 φ 2 ) dy.
As in [56, section 4], we have

Lψ = L(exp(iΛ)∂ x u) = exp(iΛ) - i 2 ∂ x (|u| 2σ )∂ x u + σ|u| 2(σ-1) Im(u∂ x u)∂ x u -σ x -∞ Im(∂ x (|u| 2(σ-1) u)∂ x u) dy∂ x u = - i 2 ∂ x (|φ| 2σ )ψ + σ|φ| 2(σ-1) Im(φψ)ψ -σ x -∞ ∂ x (|u| 2(σ-1) )Im(u∂ x u) dyψ = - i 2 ∂ x (|φ| 2σ )ψ + σ|φ| 2(σ-1) ψIm(φψ) -σψ x -∞ ∂ x (|φ| 2(σ-1) )Im(φψ) dy = σ|φ| 2(σ-1) ψ(Im(φψ) -iRe(φ∂ x φ)) -σψ x -∞ (σ -1)|φ| 2(σ-1) 2Re(φ∂φ)Im(φψ) dy = σ|φ| 2(σ-1) ψ(Im(φψ) -iRe(φψ)) -σ(σ -1)ψ x -∞ |φ| 2(σ-2) 2Re(φψ)Im(φψ)Im(φψ) dy = -iσ|φ| 2(σ-1) ψ 2 φ -σ(σ -1)ψ x -∞ |φ| 2(σ-2) Im(ψ 2 φ 2 ) dy.
Thus, if u solves (5.1) then (φ, ψ) solves

Lφ = iσ|φ| 2(σ-1) φ 2 ψ -σ(σ -1)φ x -∞ |φ| 2(σ-2) Im(ψ 2 φ 2 ) dy, Lψ = -iσ|φ| 2(σ-1) ψ 2 φ -σ(σ -1)ψ x -∞ |φ| 2(σ-2) Im(ψ 2 φ 2 ) dy.
(5.16) For convenience, we dene

P (φ, ψ) = iσ|φ| 2(σ-1) φ 2 ψ -σ(σ -1)φ x -∞ |φ| 2(σ-2) Im(ψ 2 φ 2 ), (5.17)
Q(φ, ψ) = -iσ|φ| 2(σ-1) ψ 2 φ -σ(σ -1)ψ x -∞ |φ| 2(σ-2) Im(ψ 2 φ 2 ). (5.18)
Let R be the multi-soliton prole dened in (5.9). Dene h, k by

h(t, x) = exp i 2 x -∞ |R(t, x)| 2σ dy R(t, x), k = ∂ x h - i 2 |h| 2σ h.
Since R j solves (5.1) for each 1 ⩽ j ⩽ K, we have

LR + i|R| 2σ R x = - j i|R j | 2σ R jx + i|R| 2σ R x .
(5.19) By Lemma 5.6 for t ≫ T 0 large enough we have

- j i|R j | 2σ R jx + i|R| 2σ R x H 2 ⩽ e -λt .
(5.20) Thus, we rewrite (5.19) as follows:

LR + i|R| 2σ R x = e -λt v, (5.21)
where v = e λt (-

j i|R j | 2σ R jx + i|R| 2σ R x ). (5.22)
By an elementary calculation, we have

Lh = iσ|h| 2(σ-1) h 2 k -σ(σ -1)h x -∞ |h| 2(σ-2) Im(k 2 h 2 ) dy + e -λt m(t, x), Lk = -iσ|h| 2(σ-1) k 2 h -σ(σ -1)k x -∞ |h| 2(σ-2) Im(k 2 h 2 )
dy + e -λt n(t, x).

(5.23) where

m = exp i 2 x -∞ |R| 2σ dy v -σh x -∞
|R| 2(σ-1) Im(Rv) dy, (5.24)

n = exp i 2 x -∞ |R| 2σ dy e -λt (∂ x v -σ∂ x R x -∞
|R| 2(σ-1) Im(Rv) dy).

(5.25)

Since v is uniformly bounded in time in H 2 (R), we see that m, n are uniformly bounded in time in H 1 (R). Let φ = φ -h and ψ = ψ -k. Then ( φ, ψ) solves:

L φ = P (φ, ψ) -P (h, k) -e -λt m(t, x), L ψ = Q(φ, ψ) -Q(h, k
) -e -λt n(t, x).

(5.26)

Set η = ( φ, ψ), W = (h, k) and f (φ, ψ) = (P (φ, ψ), Q(φ, ψ) and H = e -λt (m, n).

We nd a solutions of (5.26) in Duhamel form:

η(t) = i ∞ t [f (W + η) -f (W ) + H](s) ds, (5.27)
where S(t) denote the Schrödinger group. Moreover, since

ψ = ∂ x φ -i 2 |φ| 2σ φ, we have ψ = ∂ x φ - i 2 (| φ + h| 2σ ( φ + h) -|h| 2σ h).
(5.28)

Step 2. Existence of a solution of the system From Lemma 5.9, there exists T * ≫ 1 such that for T 0 ⩾ T * there exists a unique solution η of (5.26) dened on [T 0 , T * ) such that

∥η∥ X := e λt ∥η∥ S([t,∞))×S([t,∞)) + e λt ∥∂ x η∥ S([t,∞))×S([t,∞)) ⩽ 1 ∀t ⩾ T 0 . (5.29)
Thus, for all t ⩾ T 0 , we have

∥ φ∥ H 1 + ∥ ψ∥ H 1 ≲ e -λt .
(5.30)

Step 3. Existence of a multi-soliton train

We prove that the solution η = ( φ, ψ) of (5.26) satises the relation (5.28). Set φ = φ + h, ψ = ψ + k and v = ∂ x φ -i 2 |φ| 2 φ and ṽ = v -k. Since ( φ, ψ) solves (5.26) and (h, k) solves (5.23), we have (φ, ψ) solves (5.16). Furthermore,

Lv = ∂ x Lφ - i 2 L(|φ| 2σ φ). (5.31)
Moreover,

L(|φ| 2σ φ) = (i∂ t + ∂ 2 x )(φ σ+1 φ σ ) = i∂ t (φ σ+1 φ σ ) + ∂ 2 x (φ σ+1 φ σ ) = i(σ + 1)|φ| 2σ ∂ t φ + iσ|φ| 2(σ-1) φ 2 ∂ t φ + ∂ x ((σ + 1)|φ| 2σ ∂ x φ + σ|φ| 2(σ-1) φ 2 ∂ x φ) = i(σ + 1)|φ| 2σ ∂ t φ + iσ|φ| 2(σ-1) φ 2 ∂ t φ + (σ + 1) ∂ 2 x φ|φ| 2σ + ∂ x φ∂ x (|φ| 2σ ) + σ ∂ 2 x φ|φ| 2(σ-1) φ 2 + (σ + 1)|∂ x φ| 2 |φ| 2(σ-1) φ + (σ -1)|φ| 2(σ-2) φ 3 (∂ x φ) 2 = (σ + 1)|φ| 2σ (i∂ t φ + ∂ 2 x φ) + σ|φ| 2(σ-1) φ 2 (i∂ t φ + ∂ 2 x φ) + (σ + 1)∂ x φ∂ x (|φ| 2σ ) + σ(σ + 1)|∂ x φ| 2 |φ| 2(σ-1) φ + σ(σ -1)(∂ x φ) 2 |φ| 2(σ-2) φ 3 = (σ + 1)|φ| 2σ Lφ + σ|φ| 2(σ-1) φ 2 (-Lφ + 2∂ 2 x φ) + (σ + 1)∂ x φ∂ x (|φ| 2σ ) + σ(σ + 1)|∂ x φ| 2 |φ| 2(σ-1) φ + σ(σ -1)(∂ x φ) 2 |φ| 2(σ-2) φ 3 .
Combining with (5.31) and using (5.16), we have

Lv = ∂ x Lφ - i 2 L(|φ| 2σ φ) = ∂ x Lφ - i 2 (σ + 1)|φ| 2σ Lφ + σ|φ| 2(σ-1) φ 2 (-Lφ + 2∂ 2 x φ) +(σ + 1)∂ x φ∂ x (|φ| 2σ ) + σ(σ + 1)|∂ x φ| 2 |φ| 2(σ-1) φ + σ(σ -1)(∂ x φ) 2 |φ| 2(σ-2) φ 3 = ∂ x (P (φ, ψ) -P (φ, v)) + ∂ x P (φ, v) - i 2 (σ + 1)|φ| 2σ (P (φ, ψ) -P (φ, v)) - i 2 (σ + 1)|φ| 2σ P (φ, v) + i 2 σ|φ| 2(σ-1) φ 2 (P (φ, ψ) -P (φ, v)) + i 2 σ|φ| 2(σ-1) φ 2 P (φ, v) -iσ|φ| 2(σ-1) φ 2 ∂ 2 x φ - i 2 (σ + 1)∂ x φ∂ x (|φ| 2σ ) + σ(σ + 1)|∂ x φ| 2 |φ| 2(σ-1) φ +σ(σ -1)(∂ x φ) 2 |φ| 2(σ-2) φ 3 = ∂ x (P (φ, ψ) -P (φ, v)) - i 2 (σ + 1)|φ| 2σ (P (φ, ψ) -P (φ, v)) + i 2 σ|φ| 2(σ-1) φ 2 (P (φ, ψ) -P (φ, v)) + G(φ, v),
where G(φ, v) contains the remaining ingredients and G(φ, v) only depends on φ and v:

G(φ, v) = ∂ x P (φ, v) - i 2 (σ + 1)|φ| 2σ P (φ, v) + i 2 σ|φ| 2(σ-1) φ 2 P (φ, v) -iσ|φ| 2(σ-1) φ 2 ∂ 2 x φ - i 2 (σ + 1)∂ x φ∂ x (|φ| 2σ ) + σ(σ + 1)|∂ x φ| 2 |φ| 2(σ-1) φ +σ(σ -1)(∂ x φ) 2 |φ| 2(σ-2) φ 3 .
(5.32)

As the calculations of Lψ in the step 1, noting that the role of v is similar to the role of ψ in the process of calculation, we have G(φ, v) = Q(φ, v) (see Lemma 5.8 for a detailed proof). Hence,

Lψ -Lv = Q(φ, ψ) -Q(φ, v) -∂ x (P (φ, ψ) -P (φ, v)) + i 2 (σ + 1)|φ| 2σ (P (φ, ψ) -P (φ, v)) - i 2 σ|φ| 2(σ-1) φ 2 (P (φ, ψ) -P (φ, v)).
Thus,

L ψ -Lṽ = Lψ -Lv = Q(φ, ψ + k) -Q(φ, ṽ + k) -∂ x (P (φ, ψ + k) -P (φ, ṽ + k) + i 2 (σ + 1)|φ| 2σ (P (φ, ψ + k) -P (φ, ṽ + k)) - i 2
σ|φ| 2(σ-1) φ 2 (P (φ, ψ + k) -P (φ, ṽ + k)).

(5.33)

Multiplying both side of (5.33) by ψ -ṽ, taking imaginary part and integrating over space with integration by parts we obtain

1 2 ∂ t ∥ ψ -ṽ∥ 2 L 2 = Im R (Q(φ, ψ + k) -Q(φ, ṽ + k))( ψ -ṽ) dx (5.34) -Im R ∂ x (P (φ, ψ + k) -P (φ, ṽ + k))( ψ -ṽ) dx (5.35) + (σ + 1)Im R i 2 |φ| 2σ (P (φ, ψ + k) -P (φ, ṽ + k))( ψ -ṽ) dx (5.36) -σIm R i 2
|φ| 2(σ-1) φ 2 (P (φ, ψ + k) -P (φ, ṽ + k))( ψ -ṽ) dx.

(5.37)

We denote by A, B, C, D the terms (5.34), (5.35), (5.36) and (5.37) respectively.

First, we try to estimate A, B, C, D in term of R. We have

|A| ≲ R (Q(φ, ψ + k) -Q(φ, ṽ + k))( ψ -ṽ) dx ≲ R |φ| 2(σ-1) φ(( ψ + k) 2 -(ṽ + k) 2 )( ψ -ṽ) dx + R ( ψ + k) x -∞ |φ| 2(σ-2) Im(( ψ + k) 2 φ 2 ) dy -(ṽ + k) x -∞ |φ| 2(σ-2) Im((ṽ + k) 2 φ 2 ) dy ( ψ -ṽ) dx ≲ R |φ| 2(σ-1) φ(( ψ + k) 2 -(ṽ + k) 2 )( ψ -ṽ) dx + R ( ψ -ṽ) x -∞ |φ| 2(σ-2) Im(( ψ + k) 2 φ 2 ) dy ( ψ -ṽ) dx + R (ṽ + k) x -∞ |φ| 2(σ-2) Im(φ 2 (( ψ + k) 2 -(ṽ + k) 2 )) dy ( ψ -ṽ) dx ≲ ∥ ψ -ṽ∥ 2 L 2 ∥φ∥ 2σ-1 L ∞ ∥ ψ + ṽ + 2k∥ L ∞ + ∥ ψ -ṽ∥ 2 L 2 x -∞ |φ| 2(σ-2) Im(( ψ + k) 2 φ 2 ) dy L ∞ x + ∥ ψ -ṽ∥ L 2 ∥ṽ + k∥ L 2 x -∞ |φ| 2(σ-2) Im(φ 2 (( ψ + k) 2 -(ṽ + k) 2 )) dy L ∞ x ≲ ∥ ψ -ṽ∥ 2 L 2 ∥φ∥ 2σ-1 L ∞ ∥ ψ + ṽ + 2k∥ L ∞ + ∥ ψ -ṽ∥ 2 L 2 ∥φ 2(σ-1) ( ψ + k) 2 ∥ L 1 x + ∥ ψ -ṽ∥ L 2 ∥ṽ + k∥ L 2 ∥φ 2(σ-1) (( ψ + k) 2 -(ṽ + k) 2 )∥ L 1 ≲ ∥ ψ -ṽ∥ 2 L 2 ∥φ∥ 2σ-1 L ∞ ∥ ψ + ṽ + 2k∥ L ∞ + ∥ ψ -ṽ∥ 2 L 2 ∥φ 2(σ-1) ( ψ + k) 2 ∥ L 1 + ∥ ψ -ṽ∥ 2 L 2 ∥ṽ + k∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2 ≲ ∥ ψ -ṽ∥ 2 L 2 K 1 , (5.38)
where,

K 1 := ∥φ∥ 2σ-1 L ∞ ∥ ψ + ṽ + 2k∥ L ∞ + ∥φ 2(σ-1) ( ψ + k) 2 ∥ L 1 + ∥ṽ + k∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2 . Furthermore, |B| ≲ R ∂ x (|φ| 2(σ-1) φ 2 ( ψ -ṽ))( ψ -ṽ) dx + R ∂ x φ x -∞ |φ| 2(σ-2) Im(φ 2 (( ψ + k) 2 -(ṽ + k) 2 )) dy ( ψ -ṽ) dx ≲ R ∂ x (|φ| 2(σ-1) φ 2 )( ψ -ṽ) 2 dx + |φ| 2(σ-1) φ 2 1 2 ∂ x (( ψ -ṽ) 2 ) dx (5.39) + R ∂ x φ x -∞
|φ| 2(σ-2) Im(φ 2 ( ψ -ṽ)( ψ + ṽ + 2k)) dy( ψ -ṽ) dx + R φ|φ| 2(σ-2) Im(φ 2 ( ψ -ṽ)( ψ + ṽ + 2k))( ψ -ṽ) dx .

By using integration by parts for the second term of (5.39) and using Hölder inequality we have

|B| ≲ ∥ ψ -ṽ∥ 2 L 2 ∥∂ x (|φ| 2(σ-1) φ 2 )∥ L ∞ + ∥ ψ -ṽ∥ 2 L 2 ∥∂ x (|φ| 2(σ-1) φ 2 )∥ L ∞ + ∥∂ x φ∥ L 2 ∥ x -∞ |φ| 2(σ-2) Im(φ 2 ( ψ -ṽ)( ψ + ṽ + 2k)) dy∥ L ∞ x ∥ ψ -ṽ∥ L 2 + ∥ ψ -ṽ∥ 2 L 2 ∥φ 2σ-1 ( ψ + ṽ + 2k)∥ L ∞ ≲ ∥ ψ -ṽ∥ 2 L 2 ∥∂ x (|φ| 2(σ-1) φ 2 )∥ L ∞ + ∥ ψ -ṽ∥ 2 L 2 ∥∂ x (|φ| 2(σ-1) φ 2 )∥ L ∞ + ∥∂ x φ∥ L 2 ∥ ψ -ṽ∥ L 2 ∥φ 2(σ-1) ( ψ -ṽ)( ψ + ṽ + 2k)∥ L 1 x (5.40) + ∥ ψ -ṽ∥ 2 L 2 ∥φ 2σ-1 ( ψ + ṽ + 2k)∥ L ∞ ≲ ∥ ψ -ṽ∥ 2 L 2 ∥∂ x (|φ| 2(σ-1) φ 2 )∥ L ∞ + ∥ ψ -ṽ∥ 2 L 2 ∥∂ x (|φ| 2(σ-1) φ 2 )∥ L ∞ + ∥∂ x φ∥ L 2 ∥ ψ -ṽ∥ 2 L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2 + ∥ ψ -ṽ∥ 2 L 2 ∥φ 2σ-1 ( ψ + ṽ + 2k)∥ L ∞ = ∥ ψ -ṽ∥ 2 L 2 K 2 , (5.41)
where

K 2 := ∥∂ x (|φ| 2(σ-1) φ 2 )∥ L ∞ +∥∂ x φ∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ +2k)∥ L 2 +∥φ 2σ-1 ( ψ + ṽ +2k)∥ L ∞ .
Using (5.17), we have

|C| ≲ R |φ| 2σ |φ| 2(σ-1) φ 2 ( ψ -ṽ) 2 dx + R |φ| 2σ φ x -∞ |φ| 2(σ-2) Im(φ 2 (( ψ + k) 2 -(ṽ + k) 2 )) dy( ψ -ṽ) dx ≲ ∥ ψ -ṽ∥ 2 L 2 ∥φ 4σ ∥ L ∞ + ∥ ψ -ṽ∥ L 2 ∥φ 2σ+1 ∥ L 2 ∥ x -∞ |φ| 2(σ-2) Im(φ 2 ( ψ -ṽ)( ψ + ṽ + 2k)) dy∥ L ∞ x ≲ ∥ ψ -ṽ∥ 2 L 2 ∥φ 4σ ∥ L ∞ + ∥ ψ -ṽ∥ L 2 ∥φ 2σ+1 ∥ L 2 ∥φ 2(σ-1) ( ψ -ṽ)( ψ + ṽ + 2k)∥ L 1 ≲ ∥ ψ -ṽ∥ 2 L 2 ∥φ 4σ ∥ L ∞ + ∥ ψ -ṽ∥ 2 L 2 ∥φ 2σ+1 ∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2 = ∥ ψ -ṽ∥ 2 L 2 K 3 , (5.42)
where

K 3 := ∥φ 4σ ∥ L ∞ + ∥φ 2σ+1 ∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2 .
Now, we give an estimate for D. We have

|D| ≲ R |φ| 2(σ-1) φ 2 |φ| 2(σ-1) φ 2 ( ψ -ṽ)( ψ -ṽ) dx + R |φ| 2(σ-1) φ 2 φ x -∞ |φ| 2(σ-2) Im(φ 2 (( ψ + k) 2 -(ṽ + k) 2 )) dy( ψ -ṽ) dx ≲ ∥ ψ -ṽ∥ 2 L 2 ∥φ 4σ ∥ L ∞ + ∥ ψ -ṽ∥ L 2 ∥φ 2σ+1 ∥ L 2 ∥ x -∞ |φ| 2(σ-2) Im(φ 2 (( ψ + k) 2 -(ṽ + k) 2 )) dy∥ L ∞ x ≲ ∥ ψ -ṽ∥ 2 L 2 ∥φ 4σ ∥ L ∞ + ∥ ψ -ṽ∥ L 2 ∥φ 2σ+1 ∥ L 2 ∥φ 2(σ-1) ( ψ -ṽ)( ψ + ṽ + 2k)∥ L 1 ≲ ∥ ψ -ṽ∥ 2 L 2 ∥φ 4σ ∥ L ∞ + ∥ ψ -ṽ∥ 2 L 2 ∥φ 2σ+1 ∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2 = ∥ ψ -ṽ∥ 2 L 2 K 4 , (5.43)
where

K 4 := ∥φ 4σ ∥ L ∞ + ∥φ 2σ+1 ∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2 .
Combining (5.38), (5.41), (5.42) and (5.43), we have

∂ t ∥ ψ -ṽ∥ 2 L 2 ≲ ∥ ψ -ṽ∥ 2 L 2 (K 1 + K 2 + K 3 + K 4 ).
Using the Grönwall inequality, we have

∥ ψ(t) -ṽ(t)∥ 2 L 2 ≲ ∥ ψ(N ) -ṽ(N )∥ 2 L 2 exp N t (K 1 + K 2 + K 3 + K 4 ) ds ⩽ e -2λN exp N t (K 1 + K 2 + K 3 + K 4 ) ds .
(5.44)

Now, we try to estimate K 1 + K 2 + K 3 + K 4 in term of R. When we have this kind of estimate, we will use the assumption (5.10) to obtain that ψ = ṽ. We have

N t (K 1 + K 2 + K 3 + K 4 ) ds = N t ∥φ∥ 2σ-1 L ∞ ∥ ψ + ṽ + 2k∥ L ∞ + ∥φ 2(σ-1) ( ψ + k) 2 ∥ L 1
+ ∥ṽ + k∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2 ds (5.45)

+ N t ∥∂ x (|φ| 2(σ-1) φ 2 )∥ L ∞ + ∥∂ x φ∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2
+ ∥φ 2σ-1 ( ψ + ṽ + 2k)∥ L ∞ ds (5.46)

+ N t ∥φ 4σ ∥ L ∞ + ∥φ 2σ+1 ∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2 ds
(5.47)

+ N t ∥φ 4σ ∥ L ∞ + ∥φ 2σ+1 ∥ L 2 ∥φ 2(σ-1) ( ψ + ṽ + 2k)∥ L 2 ds (5.48)
The above estimate is not enough explicit. As said above, we would like to estimate the right hand side of (5.52) in terms of R. Noting that |h| = |R| and |k| = |∂ x R|, we have

W 1 (h, k) = ∥∂ x R∥ L ∞ L ∞ (1 + ∥R∥ 2σ-1 L ∞ L ∞ ) + (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥∂ x R∥ 2 L ∞ L 2 ) + ∥∂ x R∥ L ∞ L ∞ (1 + ∥∂ x R∥ L ∞ L 2 )(1 + ∥R∥ L ∞ L 2 )(1 + ∥R∥ 2(σ-1) L ∞ L ∞ ) ≲ (1 + ∥R∥ 2(σ-1) L ∞ L ∞ ) [∥∂ x R∥ L ∞ L ∞ (1 + ∥R∥ L ∞ L ∞ ) + (1 + ∥∂ x R∥ L ∞ L 2 ) +∥∂ x R∥ L ∞ L ∞ (1 + ∥∂ x R∥ L ∞ L 2 )(1 + ∥R∥ L ∞ L 2 )] ≲ (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )× × ∥∂ x R∥ L ∞ L ∞ (1 + ∥R∥ L ∞ H 1 ) + (1 + ∥R∥ 2 L ∞ H 1 ) + ∥∂ x R∥ L ∞ L ∞ (1 + ∥R∥ 2 L ∞ H 1 ) ≲ (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 )(1 + ∥∂ x R∥ L ∞ L ∞ ).
Similarly, by noting that |∂ x h| ⩽ |k| + |h| 2σ+1 , we have

W 2 (h, k) ≲ (∥k∥ L ∞ L ∞ + ∥h∥ 2σ+1 L ∞ L ∞ )(1 + ∥h∥ 2(σ-1) L ∞ L ∞ )(1 + ∥h∥ L ∞ L ∞ ) + (1 + ∥h∥ 2(σ-1) )(1 + ∥k∥ L ∞ L 2 ) + ∥k∥ L ∞ L ∞ (1 + ∥h∥ 2(σ-1) L ∞ L ∞ )(1 + ∥h∥ L ∞ L ∞ ) ≲ (1 + ∥h∥ 2(σ-1) )× × (∥k∥ L ∞ L ∞ + ∥h∥ 2σ+1 L ∞ L ∞ )(1 + ∥h∥ L ∞ L ∞ ) +(1 + ∥k∥ L ∞ L 2 ) + ∥k∥ L ∞ L ∞ (1 + ∥h∥ L ∞ L ∞ )] ≲ (1 + ∥h∥ 2(σ-1) )× × (1 + ∥h∥ L ∞ L ∞ )(∥k∥ L ∞ L ∞ + ∥h∥ 2σ+1 L ∞ L ∞ ) + (1 + ∥k∥ L ∞ L 2 ) = (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )× × (1 + ∥R∥ L ∞ L ∞ )(∥∂ x R∥ L ∞ L ∞ + ∥R∥ 2σ+1 L ∞ L ∞ ) + (1 + ∥∂ x R∥ L ∞ L 2 ) ≲ (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ L ∞ H 1 )(1 + ∥∂ x R∥ L ∞ L ∞ + ∥R∥ 2σ+1 L ∞ L ∞ ) ≲ (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 )(1 + ∥∂ x R∥ L ∞ L ∞ + ∥R∥ 2σ+1 L ∞ L ∞ ),
and

W 3 (h, k) = (1 + ∥R∥ 4σ L ∞ L ∞ ) + (1 + ∥R∥ L ∞ L 2 )(1 + ∥R∥ 4σ-2 L ∞ L ∞ )(1 + ∥∂ x R∥ L ∞ L 2 ) ≲ (1 + ∥R∥ 4σ-2 L ∞ L ∞ ) (1 + ∥R∥ 2 L ∞ L ∞ ) + (1 + ∥R∥ L ∞ L 2 )(1 + ∥∂ x R∥ L ∞ L 2 ) ≲ (1 + ∥R∥ 4σ-2 L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 ).
Combining the above estimates, we have

W 1 (h, k) + W 2 (h, k) + W 3 (h, k) ≲ (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 )(1 + ∥∂ x R∥ L ∞ L ∞ + ∥R∥ 2σ+1 L ∞ L ∞ ) + (1 + ∥R∥ 4σ-2 L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 ) ≲ (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 )(1 + ∥∂ x R∥ L ∞ L ∞ + ∥R∥ 2σ+1 L ∞ L ∞ ) + (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ 2σ L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 ) ≲ (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 )(1 + ∥∂ x R∥ L ∞ L ∞ + ∥R∥ 2σ+1 L ∞ L ∞ ).
Thus, there exists a positive constant C 0 such that

W 1 (h, k) + W 2 (h, k) + W 3 (h, k) ⩽ C 0 (1 + ∥R∥ 2(σ-1) L ∞ L ∞ )(1 + ∥R∥ 2 L ∞ H 1 )(1 + ∥∂ x R∥ L ∞ L ∞ + ∥R∥ 2σ+1 L ∞ L ∞ ) .
Let C * = 16C 0 . Using the assumption (5.10), we have

W 1 (h, k) + W 2 (h, k) + W 3 (h, k) ⩽ v * 16 = λ,
for t large enough. Thus, by (5.52), we have

∥ ψ(t) -ṽ(t)∥ 2 L 2 ⩽ e -2λN +(N -t)λ ,
for t large enough. Letting N → ∞ in the above estimate, we obtain

∥ ψ(t) -ṽ∥ 2 L 2 = 0,
for all t large enough. This implies that

ψ = ∂ x φ - i 2 |φ| 2 φ -k, (5.53)
and then

ψ = ∂ x φ - i 2 |φ| 2 φ.
Moreover, since ( ψ, φ) solves (5.26) we have (ψ, φ) solves (5.16). Combining with (5.53), if we set

u = exp - i 2 x -∞
|φ| 2σ dy φ then u solves (5.1). Furthermore,

∥u -R∥ H 1 = exp - i 2 |φ| 2σ dy φ -exp i 2 |h| 2σ dy h H 1 ≲ C(∥φ∥ H 1 , ∥h∥ H 1 )∥φ -h∥ H 1 ≲ ∥ φ∥ H 1 ≲ e -λt ,
Thus for t large enough, we have

∥u -R∥ H 1 ⩽ Ce -λt , (5.54)
for λ = 1 16 v * and C = C(ω 1 , ..., ω K , c 1 , ..., c K ). This completes the proof of Theorem 5.1. Remark 5.5. In the case σ = 1, the integrals in (5.16) disappear. In the case, σ = 2, the integrals (5.16) reduce into x -∞ Im(ψ 2 φ 2 ) dy, we do not need to use the inequality (5.56). Thus, by similar arguments as in the proof of Theorem 5.1 we may prove that there exist multi-solitons solutions of (5.1) when σ = 1 or σ = 2.

5.3

Some technical lemmas

Properties of solitons

In this section, we give the proof of (5.20). We have the following result. Lemma 5.6. There exist C > 0 and a constant λ > 0 such that for t > 0 large enough, the estimate (5.20) uniformly holds in time.

Proof. First, we need some estimates on the prole. We have

|R j (t, x)| = |ψ ω j ,c j (t, x)| = |ϕ ω j ,c j (x -c j t)| = |φ ω j ,c j (x -c j t)| ≈   4ω j -c 2 j 2 √ ω j cosh(σh j (x -c j t)) - c j 2 √ ω j   1 2σ ≲   4ω j -c 2 j 2 √ ω j cosh(σh j (x -c j t)) - |c j | 2 √ ω j cosh(σh j (x -c j t))   1 2σ ≲ 4ω j -c 2 j (2 √ ω j -|c j |) cosh(σh j (x -c j t)) 1 2σ ≲ 2 √ ω j + |c j | cosh(σh j (x -c j t)) 1 2σ ≲ ω j ,|c j | e -h j 2 |x-c j t| , Furthermore, ∂ x φ ω j ,c j (y) ≈ h 2 j 2 √ ω j 1 2σ
-sinh(σh j y) cosh(σh j y) -

c j √ ω j 1+ 1 2σ . Thus, |∂ x φ ω j ,c j (y)| ≲ h 2 j 2 √ ω j 1 2σ | sinh(σh j y)| 1 - |c j | √ ω j 1+ 1 2σ cosh(σh j y) 1+ 1 2σ ≲ ω j ,|c j | 1 cosh(σh j y) 1 2σ ≲ ω j ,|c j | e -h j 2 |y| ,
Using the above estimates, we have

|∂ x R j (t, x)| = |∂ x ψ ω j ,c j (t, x)| = |∂ x ϕ ω j ,c j (x -c j t)| = |∂ x φ ω j ,c j (x -c j t) + iφ ω j ,c j (x -c j t)∂ x θ ω j ,c j (x -c j t)| ≲ |∂ x φ ω j ,c j (x -c j t)| + |φ ω j ,c j (x -c j t)||∂ x θ ω j ,c j (x -c j t)| ≲ ω j ,|c j | |∂ x φ ω j ,c j (x -c j t)| + e -h j 2 |x-c j t| ≲ ω j ,|c j | e -h j 2 |x-c j t| .
By similar arguments, we have

|∂ 2 x R j (t, x)| + |∂ 3 x R j (t, x)| ≲ ω j ,|c j | e -h j 2 |x-c j t| ,
For convenience, we set

χ = -i|R| 2σ ∂ x R + iΣ j |R j | 2σ ∂ x R j , f (R, R, ∂ x R) = i|R| 2σ ∂ x R, g(R, R, ∂ x R, ∂ x R, ∂ 2 x R) = i∂ x (|R| 2σ ∂ x R), r(R, ∂ x R, .., ∂ 3 x R, ∂ x R, ∂ 2 x R) = i∂ 2 x (|R| 2σ ∂ x R). Fix t > 0, for each x ∈ R, choose m = m(x) ∈ {1, 2, ..., K} so that |x -c m t| = min j |x -c j t|. For j ̸ = m we have |x -c j t| ⩾ 1 2 (|x -c j t| + |x -c m t|) ⩾ 1 2 |c j t -c m t| = t 2 |c j -c m |.
Thus, we have

|(R -R m )(t, x)| + |∂ x (R -R m )(t, x)| + |∂ 2 x (R -R m )(t, x)| + |∂ 3 x (R -R m )(t, x)| ⩽ j̸ =m (|R j (t, x)| + |∂ x R j (t, x)| + |∂ 2 x R j (t, x)| + |∂ 3 x R j (t, x)|) ≲ ω 1 ,..,ω K ,|c 1 |,..,|c K | δ m (t, x) := j̸ =m e -h j 2 |x-c j t| . Recall that v * = inf j̸ =k h j |c j -c k |.
We have

|(R -R m )(t, x)| + |∂ x (R -R m )(t, x)| + |∂ 2 x (R -R m )(t, x)| + |∂ 3 x (R -R m )(t, x)| ≲ δ m (t, x) ≲ e -1 4 v * t .
We see that f, g, r are polynomials in R, ∂

x R, ∂ 2 x R, ∂ 3 x R, ∂ x R and ∂ 2 x R. Denote A = sup |u|+|∂xu|+|∂ 2 x u|+|∂ 3 x u|⩽ j ∥R j ∥ H 4 (|df | + |dg| + |dr|).
We have

|χ| + |∂ x χ| + |∂ 2 x χ| ⩽ |f (R, R, ∂ x R) -f Rm,∂xRm,Rm | + |g(R, R, ∂ x R, ..) -g(R m , R m , ∂ x R m , ..)| + |r(R, ∂ x R, .., ∂ 3 x R, R, ..) -r(R m , ∂ x R m , .., ∂ 3 x R m , R m , ..)| + Σ j̸ =m (f (R j , R j , ∂ x R j ) + g(R j , ∂ x R j , ∂ 2 x R j , R j , ∂ x R j ) + r(R j , ..., ∂ 3 x R j , R j , ..., ∂ 2 x R j )) ≲ A(|R -R m | + |∂ x (R -R m )| + |∂ 2 x (R -R m )| + |∂ 3 x (R -R m )|) + AΣ j̸ =m (|R j | + |∂ x R j | + |∂ 2 x R j | + |∂ 3 x R j |) ≲ 2AΣ j̸ =m (|R j | + |∂ x R j | + |∂ 2 x R j | + |∂ 3 x R j |) ≲ 2Aδ m (t, x). In particular, ∥χ∥ W 2,∞ ≲ e -1 4 v * t . (5.55) Moreover, ∥χ∥ W 2,1 ≲ Σ j (∥|R j | 2σ ∂ x R j ∥ L 1 + ∥∂ x (|R j | 2σ ∂ x R j )∥ L 1 + ∥∂ 2 x (|R j | 2σ ∂ x R j )∥ L 1 ) ≲ Σ j (∥R j ∥ ( H 1 2σ + 1) + ∥R j ∥ 2σ+1 H 2 + ∥R j ∥ 2σ+1
H 3 ) < ∞. Thus, using Hölder inequality we obtain

∥χ∥ H 2 ≲ ω 1 ,..,ω K ,|c 1 |,..,|c K | e -1 8 v * t . It follows that if t ≫ max{ω 1 , ..., ω K , |c 1 |, ..., |c K |} is large enough then ∥χ∥ H 2 ⩽ e -1 16 v * t .
Setting λ = 1 16 v * , we obtain the desired result.

Some useful estimates

Lemma 5.7. Let x ⩾ 0. Then there exists C = C(x) such that

(a + b) x -a x ⩽ C(x)(b x + ba x-1 ). ( 5
g(z) = z x , ∀z ∈ R.
We have g is class C 1 . Thus, there exists ξ ∈ (a, a + b) such that

|(a + b) x -a x | = |g(a + b) -g(a)| = |bg ′ (ξ)| = bxξ x-1 < xb(a + b) x-1 . If x-1 ⩽ 1 then (a+b) x-1 ⩽ a x-1 +b x-1 and hence we choose C(x) = x. If x-1 > 1 then by Jensen's inequality for convex function f (z) = z x-1 we have a + b 2 x-1 ⩽ a x-1 + b x-1 2 .
We obtain

(a + b) x -a x < xb(a + b) x-1 ⩽ 2 x-2 xb(a x-1 + b x-1 ).
Choosing C(x) = 2 x-2 x, we obtain the desired result.

Proof

G(φ, v) = Q(φ, v)
Let G(φ, v) be dened as in (5.32) and Q be dened as in (5.18). Then we have the following result.

Lemma 5.8.

Let v = ∂ x φ -i 2 |φ| 2 φ.
Then the following equality holds:

G(φ, v) = Q(φ, v).
Proof. We have

P (φ, v) = iσ|φ| 2(σ-1) φ 2 v -σ(σ -1)φ x -∞ |φ| 2(σ-2) Im(v 2 φ 2 ) dy, Q(φ, v) = -iσ|φ| 2(σ-1) v 2 φ -σ(σ -1)v x -∞ |φ| 2(σ-2) Im(v 2 φ 2 ) dy G(φ, v) = ∂ x P (φ, v) - i 2 (σ + 1)|φ| 2σ P (φ, v) + i 2 σ|φ| 2(σ-1) φ 2 P (φ, v) -iσ|φ| 2(σ-1) φ 2 ∂ 2 x φ - i 2 (σ + 1)∂ x φ∂ x (|φ| 2σ ) + σ(σ + 1)|∂ x φ| 2 |φ| 2(σ-1) φ +σ(σ -1)(∂ x φ) 2 |φ| 2(σ-2) φ 3 . The term contains x -∞ |φ| 2(σ-2) Im(v 2 φ 2 ) dy in the expression of G(φ, v) is the fol- lowing. -σ(σ -1)∂ x φ x -∞ |φ| 2(σ-2) Im(v 2 φ 2 ) dy - i 2 (σ + 1)|φ| 2σ (-1)σ(σ -1)φ x -∞ |φ| 2(σ-2) Im(v 2 φ 2 ) dy + i 2 σ|φ| 2(σ-1) φ 2 (-1)σ(σ -1)φ x -∞ |φ| 2(σ-2) Im(v 2 φ 2 ) dy = -σ(σ -1) x -∞ |φ| 2(σ-2) Im(v 2 φ 2 ) dy ∂ x φ - i 2 (σ + 1)|φ| 2σ φ + i 2 σ|φ| 2σ φ = -σ(σ -1) x -∞ |φ| 2(σ-2) Im(v 2 φ 2 ) dy ∂ x φ - i 2 |φ| 2σ φ = -σ(σ -1)v x -∞ |φ| 2(σ-2) Im(v 2 φ 2 ) dy,
which equals to the term contains

x -∞ |φ| 2(σ-2) Im(v 2 φ 2 ) dy in the expression of Q(φ, v). We only need to check the equality of the remaining terms. The remaining terms of G(φ, v) is the following.

iσ∂ x (|φ| 2(σ-1) φ 2 v) -σ(σ -1)|φ| 2(σ-2) φIm(v 2 φ 2 ) - i 2 (σ + 1)|φ| 2σ (iσ|φ| 2(σ-1) φ 2 v) + i 2 σ|φ| 2(σ-1) φ 2 (-iσ|φ| 2(σ-1) φ 2 v) -iσ|φ| 2(σ-1) φ 2 ∂ 2 x φ (5.57) - i 2 (σ + 1)∂ x φ∂ x (|φ| 2σ ) + σ(σ + 1)|∂ x φ| 2 |φ| 2(σ-1) φ +σ(σ -1)(∂ x φ) 2 |φ| 2(σ-2) φ 3 . (5.58) Noting that ∂ x (|φ| 2 ) = 2Re(vφ) and v = ∂ x φ -i 2 |φ| 2σ φ, we have the term (5.57) = iσ∂ x (|φ| 2(σ-1) )φ 2 v + iσ|φ| 2(σ-1) 2φ∂ x φv + iσ|φ| 2(σ-1) φ 2 ∂ x v -σ(σ -1)|φ| 2(σ-2) φ2Re(vφ)Im(vφ) + 1 2 σ|φ| 4σ-2 φ 2 v + σ 2 |φ| 4σ-2 φRe(φv) -iσ|φ| 2(σ-1) φ 2 ∂ 2 x φ = 2iσ(σ -1)|φ| 2(σ-2) Re(vφ)φ 2 v + 2iσ|φ| 2(σ-1) φ∂ x v + iσ|φ| 2(σ-1) φ 2 ∂ x (v -∂ x φ) -2σ(σ -1)|φ| 2(σ-2) φRe(vφ)Im(vφ) + 1 2 σ|φ| 4σ-2 φ 2 v + σ 2 |φ| 4σ-2 φRe(φv) = 2σ(σ -1)|φ| 2(σ-2) Re(vφ)φ(iφv -Im(vφ)) + 2iσ|φ| 2(σ-1) φ∂ x v + iσ|φ| 2(σ-1) φ 2 ∂ x i 2 |φ| 2σ φ + 1 2 σ|φ| 4σ-2 φ 2 v + σ 2 |φ| 4σ-2 φRe(φv) = 2iσ(σ -1)|φ| 2(σ-2) φ(Re(vφ)) 2 + 2iσ|φ| 2(σ-1) φ∂ x φv - 1 2 σ|φ| 2(σ-1) φ 2 (2σ|φ| 2(σ-1) Re(vφ) + |φ| 2σ ∂ x φ) + 1 2 σ|φ| 4σ-2 φ 2 v + σ 2 |φ| 4σ-2 φRe(φv) = 2iσ(σ -1)|φ| 2(σ-2) φ(Re(vφ)) 2 + 2iσ|φ| 2(σ-1) φ∂ x φv - 1 2 σ|φ| 4σ-2 φ 2 ∂ x φ + 1 2 σ|φ| 4σ-2 φ 2 v = 2iσ(σ -1)|φ| 2(σ-2) φ(Re(vφ)) 2 + 2iσ|φ| 2(σ-1) φ∂ x φv + 1 2 σ|φ| 4σ-2 φ 2 (v -∂ x φ) = 2iσ(σ -1)|φ| 2(σ-2) φ(Re(vφ)) 2 + 2iσ|φ| 2(σ-1) φ∂ x φv + i 4 σ|φ| 6σ φ.
Moreover, using Re(∂ x φφ) = Re(vφ) we have the term (5.58)

= -i 2 σ(σ + 1)|∂ x φ| 2 |φ| 2(σ-1) φ + σ(σ + 1)|φ| 2(σ-1) ∂ x φ(∂ x φφ + ∂ x φφ) +σ(σ -1)(∂φ) 2 |φ| 2(σ-2) φ 3 = -i 2 2σ|∂φ| 2 |φ| 2(σ-1) φ + σ(σ -1)|φ| 2(σ-2) ∂ x φφ 2 (∂ x φφ + ∂ x φφ) +2σ(σ + 1)|φ| 2(σ-1) ∂ x φRe(vφ) = -i 2 2σ|∂φ| 2 |φ| 2(σ-1) φ + 2σ(σ -1)|φ| 2(σ-2) ∂ x φφ 2 Re(vφ) +2σ(σ + 1)|φ| 2(σ-1) ∂ x φRe(vφ) = -i σ|∂φ| 2 |φ| 2(σ-1) φ + σ(σ -1)|φ| 2(σ-2) ∂ x φφ 2 Re(vφ) +σ(σ + 1)|φ| 2(σ-1) ∂ x φRe(vφ) = -i σ|∂φ| 2 |φ| 2(σ-1) φ + σ(σ -1)|φ| 2(σ-2) Re(vφ)φ(∂ x φφ + ∂ x φφ) +2σ|φ| 2(σ-1) ∂ x φRe(vφ) = -i σ|∂φ| 2 |φ| 2(σ-1) φ + 2σ(σ -1)|φ| 2(σ-2) (Re(vφ)) 2 φ = -2iσ(σ -1)|φ| 2(σ-2) φ(Re(vφ)) 2 -iσ|∂ x φ| 2 |φ| 2(σ-1) φ -2iσ|φ| 2(σ-1) ∂ x φRe(vφ).
Combining the above expressions we obtain the remaining term of G(φ, v)

= 2iσ|φ| 2(σ-1) φ∂ x φv + i 4 σ|φ| 6σ φ -iσ|∂ x φ| 2 |φ| 2(σ-1) φ -2iσ|φ| 2(σ-1) ∂ x φRe(vφ) = 2iσ|φ| 2(σ-1) ∂ x φ(φv -Re(vφ)) + i 4 σ|φ| 6σ φ -iσ|∂ x φ| 2 |φ| 2(σ-1) φ = -2σ|φ| 2(σ-1) ∂ x φIm(φv) + i 4 σ|φ| 6σ φ -iσ|∂ x φ| 2 |φ| 2(σ-1) φ = -σ|φ| 2(σ-1) ∂ x φ(2Im(φv) + i∂ x φφ) + i 4 σ|φ| 6σ φ = -σ|φ| 2(σ-1) ∂ x φ(2Im(φ∂ x φ) + |φ| 2σ+2 + iRe(φ∂ x φ) -Im(φ∂ x φ)) + i 4 σ|φ| 6σ φ = -σ|φ| 2(σ-1) ∂ x φ(|φ| 2σ+2 + iφ∂ x φ) + i 4 σ|φ| 6σ φ = -iσ|φ| 2(σ-1) φ(∂ x φ) 2 -σ|φ| 4σ ∂ x φ + i 4 σ|φ| 6σ φ = -iσ|φ| 2(σ-1) φ v + i 2 |φ| 2σ φ 2 -σ|φ| 4σ v + i 2 |φ| 2σ φ + i 4 σ|φ| 6σ φ = -iσ|φ| 2(σ-1) φv 2 .
This is exactly the remaining terms of Q(φ, v). Thus, G(φ, v) = Q(φ, v).

Existence of a solution of the system

In this section, using similar arguments as in [START_REF] Le Coz | Fast-moving nite and innite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Innite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF], we prove the existence of a solution of (5.26). For convenience, we recall the equation:

η(t) = i ∞ t S(t -s)[f (W + η) -f (W ) + H](s) ds, (5.59)
where

W = (h, k), H = e -λt (m, n), f (φ, ψ) = (P (φ, ψ), Q(φ, ψ)).
We have the following lemma. Lemma 5.9.

Let H = H(t, x) : [0, ∞) × R → C 2 , W = W (t, x) : [0, ∞) × R → C 2
be given vector functions which satisfy for some C 1 > 0, C 2 > 0, λ > 0, T 0 ⩾ 0:

∥W (t)∥ L ∞ ×L ∞ + e λt ∥H(t)∥ L 2 ×L 2 ⩽ C 1 , ∀t ⩾ T 0 , (5.60
)

∥∂W (t)∥ L 2 ×L 2 + ∥∂W (t)∥ L ∞ ×L ∞ + e λt ∥∂H(t)∥ L 2 ×L 2 ⩽ C 2 , ∀t ⩾ T 0 . (5.61)
Consider equation (5.59). There exists a constant λ * independent of C 2 such that if λ ⩾ λ * then there exists a unique solution η of (5.59) on

[T 0 , ∞) × R satisfying e λt ∥η∥ S([t,∞))×S([t,∞)) + e λt ∥∂η∥ S([t,∞))×S([t,∞)) ⩽ 1, ∀t ⩾ T 0 .
Proof. We rewrite (5.59) by η = Φη. We show that, for λ large enough, Φ is a contraction map in the following ball

B = η : ∥η∥ X := e λt ∥η∥ S([t,∞))×S([t,∞)) + e λt ∥∂ x η∥ S([t,∞))×S([t,∞)) ⩽ 1 .
We will use condition λ ≫ 1 in the proof without specifying it.

Step

1. Proof Φ maps B into B Let t ⩾ T 0 , η = (η 1 , η 2 ) ∈ B, W = (w 1 , w 2 ) and H = (h 1 , h 2 ). By Strichartz estimates, we have ∥Φη∥ S([t,∞))×S([t,∞)) ≲ ∥f (W + η) -f (W )∥ N ([t,∞))×N ([t,∞)) , (5.62)
+ ∥H∥ L 1 τ L 2 x ([t,∞))×L 1 τ L 2 x ([t,∞)) . (5.63)
For (5.63), using (5.60), we have

∥H∥ L 1 τ L 2 x ([t,∞))×L 1 τ L 2 x ([t,∞)) = ∥h 1 ∥ L 1 τ L 2 x ([t,∞)) + ∥h 2 ∥ L 1 τ L 2 x ([t,∞)) ≲ ∞ t e -λτ dτ ⩽ 1 λ e -λt < 1 10 e -λt .
(5.64) For (5.62), we have

|P (W + η) -P (W )| = |P (w 1 + η 1 , w 2 + η 2 ) -P (w 1 , w 2 )| ≲ |w 1 + η 1 | 2σ-1) (w 1 + η 1 ) 2 w 2 + η 2 -|w 1 | 2(σ-1) w 2 1 w 2 (5.65) + (w 1 + η 1 ) x -∞ |w 1 + η 1 | 2(σ-2) Im((w 2 + η 2 ) 2 (w 1 + η 1 ) 2 ) -w 1 x -∞ |w 1 | 2(σ-2) Im(w 2 2 η 1 2 ) .
(5.66)

Using the assumption σ ⩾ 5 2 and Lemma 5.7 we have the term (5.65)

≲ ||w 1 + η 1 | 2(σ-1) -|w 1 | 2(σ-1) ||w 1 + η 1 | 2 |w 2 + η 2 | + |w 1 | 2(σ-1) |(w 1 + η 1 ) 2 -w 2 1 ||w 2 + η 2 | + |w 1 | 2(σ-1) |w 1 | 2 |η 2 | ≲ (|η 1 | 2(σ-1) + |η 1 ||w 1 | 2(σ-1)-1 )(|W | + |η|) 3 + |w 1 | 2(σ-1) (|w 1 ||η 1 | + |η 1 | 2 )|w 2 + η 2 | + |w 1 | 2σ |η 2 | ≲ (|η| 2(σ-1) + |η||W | 2(σ-1)-1 )(|W | 3 + |η| 3 ) + |W | 2(σ-1) (|W ||η| + |η| 2 )(|W | + |η|) + |W | 2σ |η| ≲ |η|(|η| 2σ-3 + |W | 2σ-3 )(|η| 3 + |W | 3 ) + |η||W | 2(σ-1) (|W | 2 + |η| 2 ) + |W | 2σ |η| ≲ |η|(|η| 2σ + |W | 2σ ) + |η||W | 2σ + |η| 3 |W | 2(σ-1) + |W | 2σ |η| ≲ |η| 2σ+1 + |η||W | 2σ .
Moreover, the term (5.66)

≲ |η 1 | x -∞ |w 1 + η 1 | 2(σ-2) |w 2 + η 2 | 2 |w 1 + η 1 | 2 dy + |w 1 | x -∞ (|w 1 + η 1 | 2(σ-2) -|w 1 | 2(σ-2) )|w 2 + η 2 | 2 |w 1 + η 1 | 2 dy + |w 1 | x -∞ |w 1 | 2(σ-2) |Im((w 2 + η 2 ) 2 -w 2 2 )(w 1 + η 1 ) 2 | dy + |w 1 | x -∞ |w 1 | 2(σ-2) |Im(w 2 2 ((w 1 + η 1 ) 2 -η 1 2 ))| dy ≲ |η| x -∞ |W | 2σ + |η| 2σ dy + |W | x -∞ (|η 1 | 2(σ-2) + |η 1 ||w 1 | 2σ-5 )(|W | 4 + |η| 4 ) dy + |W | x -∞ |W | 2(σ-2) (|η 2 | 2 + |w 2 ||η 2 |)(|W | 2 + |η| 2 ) dy + |W | x -∞ |W | 2(σ-2) |w 2 | 2 (|η 1 | 2 + |η 1 ||w 1 |) dy ≲ |η| x -∞ |W | 2σ + |η| 2σ dy + |W | x -∞ |η|(|W | 2σ + |η| 2σ ) dy + |W | x -∞ |W | 2(σ-2) |η|(|W | 3 + |η| 3 ) dy + |W | x -∞ |W | 2(σ-2) |W | 2 |η|(|W | + |η|) dy ≲ |η| x -∞ |W | 2σ + |η| 2σ dy + |W | x -∞ |η||W | 2σ-1 + |η| 2σ dy.
Thus, we obtain

|P (W + η) -P (W )| ≲ |η| 2σ+1 + |η||W | 2σ + |η| x -∞ |W | 2σ + |η| 2σ dy + |W | x -∞ |η||W | 2σ-1 + |η| 2σ dy. |Q(W + η) -Q(W )| ≲ |η| 2σ+1 + |η||W | 2σ + |η| x -∞ |W | 2σ + |η| 2σ dy + |W | x -∞ |η||W | 2σ-1 + |η| 2σ dy.
Hence, using σ ⩾ 5 2 , we have: We have

∥f (W + η) -f (W )∥ N ([t,∞))×N ([t,∞)) ≲ ∥P (W + η) -P (W )∥ L 1 τ L 2 x ([t,∞)) + ∥Q(W + η) -Q(W )∥ L 1 τ L 2 x ([t,∞)) ≲ ∥|η| 2σ+1 ∥ L 1 τ L 2 x ([t,∞)) + ∥|η| x -∞ |W | 2σ + |η| 2σ dy∥ L 1 τ L 2 x ([t,∞)) + ∥|W | x -∞ |η||W | 2σ-1 + |η| 2σ dy∥ L 1 τ L 2 x ([t,∞)) ≲ ∥|η|∥ L ∞ L 2 x ([t,∞)) ∥|η|∥ 4 L 4 τ L ∞ x ([t,∞)) + ∥|η|∥ L 1 τ L 2 x ([t,∞)) x -∞ |W | 2σ + |η| 2σ dy L ∞ τ L ∞ x ([t,∞)) + ∥|W |∥ L ∞ τ L 2 x ([t,∞)) ∥ x -∞ |η||W | 2σ-1 + |η| 2σ dy∥ L 1 τ L ∞ x ([t,∞)) ≲ e -5λt + ∥|η|∥ L 1 τ L 2 x ([t,∞)) ∥|W | 2σ + |η| 2σ ∥ L ∞ τ L 1 x + ∥W ∥ L ∞ t L 2 x ∥η∥ L 1 τ L 2 x ([t,∞)) ∥|W | 2σ-1 + |η| 2σ-1 ∥ L ∞ τ L 2 x ([t,∞)) ≲ e -5λt + ∥|η|∥ L 1 τ L 2 x ([t,∞)) = e -5λt
∥∂ x Φη∥ S([t,∞))×S([t,∞)) ≲ ∥∂ x (f (W + η) -f (W ))∥ N ([t,∞))×N ([t,∞)) (5.68) + ∥∂ x H∥ L 1 τ L 2 x ([t,∞))×L 1 τ L 2 x ([t,∞)) . (5.69)
For (5.69), using (5.61) we have

∥∂ x H∥ L 1 τ L 2 x ([t,∞))×L 1 τ L 2 x ([t,∞)) ≲ ∞ t e -λτ dτ = 1 λ e -λt < 1 10 e -λt , (5.70)
For (5.68), we have

∥∂ x (f (W + η) -f (W ))∥ N ([t,∞))×N ([t,∞)) = ∥∂ x (P (W + η) -P (W ))∥ N ([t,∞)) + ∥∂ x (Q(W + η) -Q(W ))∥ N ([t,∞)) . Furthermore, |∂ x (P (W + η) -P (W ))| ≲ |∂ x (|w 1 + η 1 | 2(σ-1) (w 1 + η 1 ) 2 (w 2 + η 2 ) -|w 1 | 2(σ-1) w 2 1 w 2 )| (5.71) + ∂ x (w 1 + η 1 ) x -∞ |w 1 + η 1 | 2(σ-2) Im((w 2 + η 2 ) 2 (w 1 + η 1 ) 2 ) dy -∂ x w 1 x -∞ |w 1 | 2(σ-2) Im(w 2 2 w 2 1
) dy (5.72)

+ (w 1 + η 1 )|w 1 + η 1 | 2(σ-2) Im((w 2 + η 2 ) 2 (w 1 + η 1 ) 2 ) -w 1 |w 1 | 2(σ-2) Im(w 2 2 w 1 ) . (5.73)
For (5.71), we have the term (5.71)

≲ (|η| + |η| 2σ + |∂ x η|)(|W | + |W | 2σ + |η| + |η| 2σ + |∂ x η|) Thus, ∥the term (5.71)∥ L 1 τ L 2 x ([t,∞)) ≲ ∥|η| + |∂η|∥ L 1 τ L 2 x ≲ 1 λ e -λt < 1 10 e -λt .
For (5.72), using Lemma 5.7, we have

∥ the term (5.72)∥ L 1 τ L 2 x ([t,∞)) ≲ ∥∂η 1 ∥ L 1 τ L 2 x ([t,∞)) ∥ x -∞ |w 1 + η 1 | 2(σ-2) Im((w 2 + η 2 ) 2 (w 1 + η 1 ) 2 ) dy∥ L ∞ t L ∞ x + ∥∂ x w 1 ∥ L ∞ t L 2 x × × x -∞ (|w 1 + η 1 | 2(σ-2) Im((w 2 + η 2 ) 2 (w 1 + η 1 ) 2 ) -|w 1 | 2(σ-2) Im(w 2 2 w 2 1 )) dy L 1τ L ∞ x ≲ ∥∂η 1 ∥ L 1 τ L 2 x ([t,∞)) ∥|w 1 + η 1 | 2(σ-2) Im((w 2 + η 2 ) 2 (w 1 + η 1 ) 2 )∥ L ∞ t L 1 x + ∥|w 1 + η 1 | 2(σ-2) Im((w 2 + η 2 ) 2 (w 1 + η 1 ) 2 ) -|w 1 | 2(σ-2) Im(w 2 2 w 2 1 )∥ L 1 τ L 1 x ≲ ∥∂η 1 ∥ L 1 τ L 2 x ([t,∞)) + ∥|η|∥ L 1 τ L 2 x ([t,∞)) ⩽ ∞ t e -λτ dτ ≲ 1 λ e -λt < 1 10 e -λt ,
For (5.73), using Lemma 5.7, we have

∥the term (5.73)∥ L 1 τ L 2 x ([t,∞)) ≲ ∥|η|∥ L 1 τ L 2 x ([t,∞)) ⩽ ∞ t e -λτ dτ ≲ 1 λ e -λt < 1 10 e -λt ,
Combining the above estimates, we obtain

∥∂ x (P (W + η) -P (W ))∥ N ([t,∞)) ⩽ ∥∂ x (P (W + η) -P (W ))∥ L 1 τ L 2 x ([t,∞)) ⩽ 3 10 e -λt , (5.74
)

∥∂ x (Q(W + η) -Q(W ))∥ N ([t,∞)) ⩽ 3
S([t,∞))×S([t,∞)) + ∥∂ x Φη∥ S([t,∞))×S([t,∞)) ⩽ 9 10 e -λt , (5.77)
Thus, for λ large enough ∥Φη∥ X < 1.

This implies that Φ maps B into B.

Step 2. Φ is a contraction map on B By using (5.60), (5.61) and a similar estimate of (5.77), we can show that, for any η ∈ B and κ ∈ B we have

∥Φη -Φκ∥ X ⩽ 1 2 ∥η -κ∥ X .
for λ large enough. From Banach xed point theorem, there exists a unique solution in B of (5.59) and thus a solution of (5.26). This completes the proof of Lemma 5.9. Introduction

In this chapter, we are interested in the following triple power nonlinear Schrödinger equation:

iu t + ∆u + a 1 |u|u + a 2 |u| 2 u + a 3 |u| 3 u = 0, (t, x) ∈ R × R n , (6.1)
where a 1 , a 2 , a 3 ∈ R and n ∈ {1, 2, 3}.

The standing waves of (6.1) are solutions of the form u ω (t, x) = e iωt ϕ ω (x), where ϕ ω solves:

-

ωϕ ω + ∆ϕ ω + a 1 |ϕ ω |ϕ ω + a 2 |ϕ ω | 2 ϕ ω + a 3 |ϕ ω | 3 ϕ ω = 0. (6.2)
In [START_REF] Liu | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF], the authors study existence and stability of standing waves of (6.1) in one dimension. Existence of standing waves is obtained by ODE arguments. By studying the properties of the nonlinearity, the authors give domains of parameters for existence and nonexistence of standing waves. Stability results are obtained by studying the sign of an integral found by Iliev and Kirchev [START_REF] Iliev | Stability and instability of solitary waves for onedimensional singular Schrödinger equations[END_REF], based on the criteria of stability of Grillakis, Shatah and Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Shatah | Instability of nonlinear bound states[END_REF].

In the special case ω = 0, the prole ϕ 0 , which for convenience we denote by ϕ, satises:

∆ϕ + a 1 |ϕ|ϕ + a 2 |ϕ| 2 ϕ + a 3 |ϕ| 3 ϕ = 0. (6.
3)

The equation (6.3) can be rewritten as S ′ (ϕ) = 0 where S is dened by

S(v) := 1 2 ∥∇v∥ 2 L 2 - a 1 3 ∥v∥ 3 L 3 - a 2 4 ∥v∥ 4 L 4 - a 3 5 ∥v∥ 5 L 5 . (6.4) Dene X := Ḣ1 (R n ) ∩ L 3 (R n ), and ∥u∥ X := ∥∇u∥ L 2 + ∥u∥ L 3 , (6.5)
d := inf{S(v) : v ∈ X \ {0}, S ′ (v) = 0}. (6.6)
The algebraic standing waves are standing waves with algebraic decay. In this paper, we are only interested in a special kind of algebraic standing waves which are minimizers of the problem (6.6). Throughout this paper, for convenience, we dene an algebraic standing wave as a solution of (6.3) solving problem (6.6). Thus, the function ϕ is an algebraic standing wave of (6.1) if ϕ ∈ G, where G is dened by

G := {v ∈ X \ {0} : S ′ (v) = 0, S(v) = d}. (6.7)
The instability of algebraic standing waves was studied in [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF] for double power nonlinearities. Using similar arguments as in [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF], we study existence and instability of algebraic standing waves for the nonlinear Schrödinger equation with triple power nonlinearities (6.1).

First, we study the existence of algebraic standing waves of (6.1). As in [START_REF] Liu | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF], we will use the abbreviation D: defocusing when a i < 0 and F: focusing when a i > 0. In Section 6.2, we prove the following result. Proposition 6.1. Let n = 1. The equation ( 6.3) has an unique even positive solution ϕ in the space H 1 (R) in the following cases: DFF, DDF, DFD and a

1 = a 3 = -1, a 2 > 8 √ 15 .
Moreover, all solutions of (6.3) are of the form e iθ ϕ(x -x 0 ) for some θ, x 0 ∈ R. They are all algebraic standing waves of (6.1).

In high dimensions, the situation is more complex than in the one dimension. The solutions of (6.3) are very diverse. It is not easy to describe all such solutions as in the dimension one. Thus, classifying the algebraic standing waves of (6.1) is not easy problem. It turns out that a radial positive solutions of (6.3) is also an algebraic standing wave of (6.1). To study the positive radial solutions of (6.3), we prove the following result in Section 6.2. Proposition 6.2. Let n = 2, 3 and DDF or DFF. Then there exists a unique radial positive solution of (6.3).

Before stating the next results, we need some denitions. Firstly, we dene the Nehari functional as follows:

K(v) := ⟨S ′ (v), v⟩ = ∥∇v∥ 2 L 2 -a 1 ∥v∥ 3 L 3 -a 2 ∥v∥ 4 L 4 -a 3 ∥v∥ 5 L 5 . (6.8)
The rescaled function is dened by:

v λ (x) := λ n 2 v(λx). (6.9)
The following is Pohozhaev functional:

P (v) := ∂ λ S(v λ )| λ=1 = ∥∇v∥ 2 L 2 - na 1 6 ∥v∥ 3 L 3 - na 2 4 ∥v∥ 4 L 4 - 3na 3 10 ∥v∥ 5 L 5 . (6.10)
The Nehari manifold is dened by:

K := {v ∈ X \ {0} : K(v) = 0}.
Moreover, we consider the following minimization problem:

µ := inf {S(v) : v ∈ K} . (6.11)
The following is the set of minimizers of problem (6.11):

M := {v ∈ K : S(v) = µ}. (6.12)

Finally, we dene a specic set which uses in our proof:

B := v ∈ H 1 (R n ) : S(v) < µ, P (v) < 0 . (6.13)
It turns out that the solution of (6.3) given by Proposition 6.2 satises a variational characterization and each algebraic standing wave of (6.1) is up to phase shift and translation of this special solution. More precise, in Section 6.3, we prove the following result. Proposition 6.3. Let n = 1, 2, 3 and DDF or DFF. Then the radial positive solution ϕ of (6.3) given by Proposition 6.1 and Proposition 6.2 satises

S(ϕ) = µ.
where S and µ are dened as in (6.4), (6.11) respectively. Moreover, all algebraic standing waves of equation ( 6 (2) By using similar arguments as in [36, Proof of Proposition 3.5], we prove that the algebraic standing waves in higher dimensions (n = 2, 3) are also in H 1 (R n ).

(3) By scaling invariance of (6.1), we may assume |a 1 | = |a 3 | = 1 without loss of generality. This assumption will be made throughout the rest of this paper.

Before stating the main result, we dene the orbital stability and orbital instability of standing waves. Denition 6.5. Let u ω (t, x) = e iωt ϕ ω (x) be a standing wave solution of (6.1). We say that this solution is orbitally stable if for all ε > 0 there exists δ > 0 such that for each u 0 ∈ H 1 (R n ) such that ∥u 0 -φ ω ∥ H 1 < δ then the associated solution u of (6.1) is global and satises inf θ∈R, y∈R n ∥u(t) -e iθ φ ω (• -y)∥ H 1 < ε.

Otherwise, u ω is orbitally unstable.

Our main result is the following. Theorem 6.6. Let n = 1, 2, 3. Assume that the parameters of (6.1) satisfy DDF or DFF when n = 2, 3 or DFF and a 2 < 32 15 √ 6 when n = 1. Then the algebraic standing wave ϕ given as in Proposition 6.1 and Proposition 6.3 is orbitally unstable in H 1 (R n ).

The rest of this paper is organized as follows. In Section 6.2, we nd the region of parameters a 1 , a 2 , a 3 in which there exist solutions of the elliptic equation (6.3). Specially, in one dimension, all solution of (6.3) are algebraic standing waves. In Section 6.3, we establish the variational characterization of solutions given in Section 6.2. The existence of algebraic standing waves in high dimensions is also proved in section 6.3. In Section 6.4, we prove instability of algebraic standing waves.

6.2

Existence of solution of the elliptic equation First, we nd the region of parameters a 1 , a 2 , a 3 in which there exist solutions of (6.3).

In dimension one

Let n = 1. To study the existence of algebraic standing waves, we use the following lemma (see [START_REF] Berestycki | Nonlinear scalar eld equations. I. Existence of a ground state[END_REF], [79, Proposition 2.1]) Lemma 6.7. Let g be a locally Lipschitz continuous function with g(0) = 0 and let G(t) = t 0 g(s) ds. A necessary and sucient condition for the existence of a solution ϕ of the problem ϕ ∈ C 2 (R), lim x→±∞ ϕ(x) = 0, ϕ(0) > 0, ϕ xx + g(ϕ) = 0, (6.14) is that c = inf {t > 0 : G(t) = 0} exists, c > 0, g(c) > 0.

Using Lemma 6.7, we have the following result. Lemma 6.8. Let g(u) = a 1 u 2 + a 2 u 3 + a 3 u 4 be such that g satises the assumptions of Lemma 6.7 for some a 1 , a 2 , a 3 ∈ R. Then there exists a positive solution ϕ of (6.14). Moreover, all complex valued solutions of (6.14) are of form:

e iθ 0 ϕ(x -x 0 ),
for some θ 0 , x 0 ∈ R.

Proof. By Lemma 6.7, there exists a real valued solution ϕ of (6.14). We have ϕ xx + a 1 ϕ 2 + a 2 ϕ 3 + a 3 ϕ 4 = 0. (6.15)

Since lim x→±∞ ϕ(x) = 0, there exists x 0 such that ϕ x (x 0 ) = 0. Multiplying two sides of (6.15) by ϕ x and noting that lim x→∞ ϕ(x) = 0 we obtain We see that ϕ is not vanishing on R. Indeed, if ϕ(x 1 ) = 0 for some x 1 ∈ R then ϕ x (x 1 ) = 0 by (6.16). Thus, ϕ ≡ 0 by uniqueness of solutions of (6.16) which is a contradiction. Then, we can assume that ϕ > 0.

The value ϕ(x 0 ) is a positive solution of G(u) = a 1 3 u 3 + a 2 4 u 4 + a 3 5 u 5 = 0. Since g satises the condition in Lemma 6.7, it follows that G(u) = 0 has a rst positive solution c such that g(c) > 0. If ϕ(x 0 ) ̸ = c then G has another positive zero d > c such that d = ϕ(x 0 ). By continuity of ϕ, there exists x 1 > x 0 such that ϕ(x 1 ) = c and by (6.16) ϕ x (x 1 ) = 0. This conclusion implies that every positive solution of (6.15) has a critical point such that the value of solution at this point equals to c.

Let u be a complex valued solution of (6.14). We prove that u = e iθ 0 ϕ(x-x 0 ), for some θ 0 , x 0 ∈ R. We use similar arguments as in [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Theorem 8.1.4 Using lim x→±∞ u(x) = 0 we have K = 0. In particular, |u| > 0. Indeed, if u vanishes then u x vanish at the same point, hence, u ≡ 0. Therefore, we may write u = ρe iθ , where ρ > 0 and ρ, θ ∈ C 2 (R). Substituting u = ρe iθ in (6.14) we have 2ρ x θ x + ρθ xx = 0 which implies there exists K ∈ R such that ρ 2 θ x = K and so θ x = K ρ 2 . Moreover, since |u x | is bounded, it follows that ρ 2 θ 2 x is bounded. Thus,

K2 ρ 2
is bounded. Since ρ(x) → 0 as x → ∞, we have K = 0. Thus, since ρ > 0 we have θ ≡ θ 0 for some θ 0 ∈ R. Thus u = e iθ 0 ρ. Since ρ is a positive solution of (6.15), there exists x 2 ∈ R such that ρ(x 2 ) = c and ρ x (x 2 ) = 0. Thus, by uniqueness of solution of (6.15), there exists x 3 ∈ R such that ρ(x) = ϕ(x -x 3 ) and u = e iθ 0 ϕ(x -x 3 ). This implies the desired result.

Moreover, we have the following result. Lemma 6.9. Let g and ϕ be as in Lemma 6.8. Then ϕ ∈ H 1 (R).

Proof. Firstly, since g satises the assumption of Lemma 6.7, we have a 1 < 0 (see the arguments in the proof of Proposition 6.1). As in the proof of Lemma 6.8, up to a translation, we may assume that ϕ x (0) = 0 and let c = ϕ(0). Then ϕ is an even function of x. Furthermore, ϕ satises

1 2 ϕ 2
x + G(ϕ) = 0. (6.17)

Moreover, ϕ xx (0) = -g(ϕ(0)) = -g(c) < 0. Therefore, there exists a > 0 such that ϕ x < 0 on (0, a). We claim that a = ∞. Otherwise, there would exists b > 0 such that ϕ x < 0 on (0, b) and ϕ x (b) = 0. Thus, ϕ(b) < c is a positive zero of G. This is a contradiction since c is the rst positive solution of G. Hence, ϕ x < 0 on (0, ∞). Thus, there exists 0 ⩽ l < c such that lim x→∞ ϕ(x) = l. In particular, there exists x m → ∞ such that ϕ x (x m ) → 0 as m → ∞. Passing to the limit in (6.17) we have G(l) = 0 and hence l = 0 by denition of c. Therefore ϕ decreases to 0, as x → ∞. Thus, from (6.17), for |x| large enough, we have

ϕ 2 x ≈ - a 1 3 ϕ 3 .
Then -ϕ x ≈ cϕ 3 2 , for some c > 0.

Thus, for |x| large enough, we have

0 ⩾ ϕ x + cϕ 3 2 .
It follows that ϕ ⩽ 1 (cx+d) 2 for some c, d > 0. Hence ϕ ∈ L 1 (R) ∩ L ∞ (R), especially ϕ ∈ L 2 (R). Combining this and (6.17), we obtain that ϕ x ∈ L 2 (R). Thus, ϕ ∈ H 1 (R), this completes the proof of Lemma 6.9. Now, we comeback to the proof of Proposition 6.1.

Proof of Proposition 6.1. A solution of (6.3) in the space X satises u xx + g(u) = 0, u ∈ C 2 (R), and lim x→±∞ u(x) = 0, (

From Lemma 6.7, the necessary condition for existence of solutions of (6.18) is a 1 < 0. Indeed, let c is the rst positive root of G(u) then G ′ (c) = g(c) > 0. Thus, G do not change sign on (0, c) and is increasing in a neighborhood of c. It follows that G < 0 on (0, c) and hence a 1 < 0.

To conclude the existence of solution of (6.18), we consider the three cases DDF, DFF, DFD. In the case DDD we have G < 0 on (0, ∞), therefore there is no solution of (6.18). In the case DDF (i.e a 1 = -1, a 2 < 0, a 3 = 1), we have , and g(c) = c 2 (c 2 + a 2 c -1). It easy to check that c is larger than the largest root of x 2 + a 2 x -1. Thus, g(c) > 0. It follows that in case DDF, there exists a solution of (6.18). By similar arguments, in the case DFF, (6.18) has a solution. In the case DFD, (6.18) has a solution if and only if a 2 > 8 √ 15 .

Let ϕ be a solution of (6.18). From Lemma 6.8 all solution of (6.18) are of the form e iθ ϕ(x -x 0 ), and belong to H 1 (R) by Lemma 6.9. Thus, they are all algebraic standing waves of (6.1). This completes the proof of Proposition 6.1.

In higher dimensions

In this section, we prove existence and uniqueness of a radial positive solution of (6.3) when a 1 = -1, a 3 = 1 and n = 2, 3. The existence result is a consequence of the following theorem. (the positive zero of G), β = ∞ and 4 < l < 5 when n = 3 and l > 4 when n = 2. Thus, in high dimensions (n = 2, 3), there exists a decreasing radial positive solution of (6.3).

The uniqueness of a radial positive solution is obtained by following result. Theorem 6.11 ([102],Theorem 1). Let us consider, for n ⩾ 2, the following equation ∆u + g(u) = 0, (6.21) where g satises the following conditions:

(a) g is continuous on [0, ∞) and g(0) = 0, (b) g is a C 1 -function on (0, ∞), (c) There exists a > 0 such that g(a) = 0 and g(u) < 0 for 0 < u < a, g(u) > 0 for u > a. Thus, there exists a unique radial positive solution of (6.3) by Theorem 6.11. This completes the proof of Proposition 6.2.

Variational characterization

Let n = 1, 2, 3. In this section, we prove Proposition 6.3. By the assumption of Proposition 6.3, we may pick a 1 = -1 and a 3 = 1. We recall that S, K, P are dened in (6.4), (6.8) and (6.10).

Let M and K be dened as (6.12) and (6.8). First, as in [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF], we prove that M is not empty. We set

J(v) = 1 4 ∥∇v∥ 2 L 2 + 1 12 ∥v∥ 3 L 3 + 1 20 ∥v∥ 5 L 5 ,
which is well dened on X. The functional S is rewritten as We can rewrite µ as µ = inf{J(v) : v ∈ K}. (6.23) Lemma 6.12. Let v ∈ H 1 (R n ). If K(v) < 0 then µ < J(v). In particular,

S(v) = 1 2 K(v) - 1
µ = inf{J(v) : v ∈ X \ {0}, K(v) ⩽ 0}. (6.24)
Proof. Since K(v) < 0 and K(λv) > 0 if λ > 0 small enough, there exists λ 1 ∈ (0, 1)

such that K(λ 1 v) = 0. Therefore, by (6.23) and since the function λ → J(λv) on (0, ∞) is increasing, we have µ ⩽ J(λ 1 v) < J(v).

This completes the proof.

Lemma 6.13. The following is true:

µ > 0.
Proof. Let v ∈ K. By using the Gagliardo-Nirenberg inequalities, for some θ ∈ (0, 5)

and θ ∈ (0, 4), we have ∥v∥ , for some C, C > 0. Hence, ∥∇v∥ L 2 or ∥v∥ 3 L 3 bounded below by some constant. In two cases, J(v) is bounded below by some constant. Combining with (6.23) we have the conclusion.

We need the following results. Lemma 6.14 ([2,[START_REF] Lieb | On the lowest eigenvalue of the Laplacian for the intersection of two domains[END_REF]). Let p ⩾ 1. Let (f n ) be a bounded sequence in Ḣ1 (R n ) ∩ L p+1 (R n ). Assume that there exists q ∈ (p, 2 * -1) such that lim sup n→∞ ∥f n ∥ L q+1 > 0. Then there exist (y n ) ⊂ R n and f ∈ Ḣ1 (R n ) ∩ L p+1 (R n ) \ {0} such that (f n (• -y n )) has a subsequence that converges to f weakly in Ḣ1 (R n ) ∩ L p+1 (R n ). Lemma 6.15 ([14]). Let 1 ⩽ r < ∞. Let (f n ) be a bounded sequence in L r (R n ) and f n → f a.e in R n as n → ∞. Then

∥f n ∥ r L r -∥f n -f ∥ r L r -∥f ∥ r L r → 0,
as n → ∞. Combining with (6.29), we deduce γ = 0. Thus, S ′ (ψ) = 0. Hence, v solves the following equation

(-∆ + |φ| -a 2 |φ| 2 -|φ| 3 )v = 0.
Since v is nonnegative and not identically equal to zero, using [77, Theorem 9.10], we infer that v is positive function. Furthermore, since K(|ψ|) ⩽ K(ψ) and S(|ψ|) ⩽ S(ψ), it follows from Lemma 6.12 we have K(|ψ|) = K(ψ) and S(|ψ|) = S(ψ).

Then, ∥∇|ψ|∥ L 2 = ∥∇ψ∥ L 2 . By [START_REF] Lieb | Analysis[END_REF]Theorem 7.8], there exists a constant c such that w = cv for some c ⩾ 0. Since v is continuous and positive, Reφ and Imφ do not change sign. Then, there exist constants λ = ±1 and η ∈ R such that Reφ = λv and Imφ = ηv. Taking θ ∈ R such that e -iθ = λ+iη |λ+iη| , we have e iθ φ = e iθ (λ + iη)v = |λ + iη|v. This completes the step 1.

Step 2. Radial symmetry of minimizer. Since [75, Theorem 1], there exists y ∈ R n such that e iθ φ(• -y) is a radial and decreasing function.

Step 3. Conclusion. Since ϕ and e iθ φ(• -y) are positive radial solutions of (6.3), using Proposition 6.2, we obtain ϕ = e iθ φ(• -y), Thus, S(ϕ) = S(φ) = µ, ϕ ∈ M and each element of M is of form e iθ ϕ(• -x 0 ) for some θ, x 0 ∈ R. It remains to classify all algebraic standing waves of (6.1). We only need to prove that G = M ̸ = ∅, where G and M are dened in (6.7) and (6.12), respectively. We use similar arguments as in [36, Proof of Theorem 2.1]. We divide the proof of this in two steps.

Step 1. M ⊂ G. Let ψ ∈ M. Then, S ′ (ψ) = 0. Now, we show that ψ ∈ G. Let v ∈ X \ {0} such that S ′ (v) = 0. From K(v) = ⟨S ′ (v), v⟩ = 0 and by denition of M, we have S(ψ) ⩽ S(v). Thus, ψ ∈ G and M ⊂ G.

Step 2. G ⊂ M and conclusion. Let ψ ∈ G. Then K(ψ) = ⟨S ′ (ψ), ψ⟩ = 0. As the above, ϕ ∈ M. As in step 1, ϕ ∈ G. Therefore, S(ψ) = S(ϕ) = µ, which implies ψ ∈ M. Thus G ⊂ M, which completes the proof of Proposition 6.3.

It turns out that the algebraic standing waves of (6.1) in high dimensions (n = 2, 3) belongs to H 1 (R n ). To prove this, we need the following lemma (see [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF]Lemma 3.4]). Lemma 6.17. Let φ ∈ C 1 ([0, ∞)) be a positive function. If there exist ρ, A > 0such that φ ′ (r) + Aφ(r) 1+ρ ⩽ 0, for all r > 0, then φ(r) ⩽ 1 ρAr For r > r 0 large enough, we have

|a 2 |ϕ 3 + ϕ 4 ⩽ 1 2 ϕ 2 ,
Since ϕ solves (6.3) and is decreasing as a function of r, this implies ϕ ′′ (r) ⩾ ϕ ′′ (r) + n -1 r ϕ ′ (r) = ϕ 2 -a 2 ϕ 3 -ϕ 4 ⩾ 1 2 ϕ 2 , for r > r 0 .

Multiplying the two sides by ϕ ′ and integrating it on [r, ∞), we get ϕ ′ (r) 2 ⩾ 1 3 ϕ 3 , for r ⩾ r 0 .

Since ϕ ′ < 0 we obtain that ϕ ′ (r) + 1 3 ϕ 3 2 ⩽ 0, for r ⩾ r 0 .

By Lemma 6.17, we deduce that ϕ(r) ⩽ Cr -2 , for r ⩾ r 0 .

Thus, ϕ ∈ L 2 (R n ), for n = 1, 2, 3. From the proof of Proposition 6.3, we have ϕ ∈ M. Hence, |∇ϕ| ∈ L 2 (R n ) and ϕ ∈ H 1 (R n ). This completes the proof.

6.4

Instability of algebraic standing waves

Let n = 1, 2, 3. In this section, we prove Theorem 6.6. Throughout this section, we consider the case DDF or DF F and a 2 small. Then we may pick a 1 = -1 and a 3 = 1. First, we prove the following result by using similar arguments as in [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF] (see also [36, Proof of Proposition 5.1]). Proposition 6.18. Assume that Then the algebraic standing wave ϕ is unstable.

We dene a tube around the standing wave by

N ε := v ∈ H 1 (R n ) : inf (θ,y)∈R×R n ∥v -e iθ ϕ(• -y)∥ H 1 < ε .
Lemma 6.19. Assume (6.30) holds. Then there exist ε 1 , δ 1 ∈ (0, 1) such that: For any v ∈ N ε 1 there exists Λ(v) ∈ (1 -δ 1 , 1 + δ 1 ) such that µ ⩽ S(v) + (Λ(v) -1)P (v).

Proof. First, we recall that S, K and P are dened as in (6.4), (6.8) and (6.10), respectively.

Since ∂ 2 λ S(ϕ λ )| λ=1 < 0, by the continuity of the function

(λ, v) → ∂ 2 λ S(v λ ),
there exist ε 1 , δ 1 ∈ (0, 1) such that ∂ 2 λ S(v λ ) < 0 for any λ ∈ (1 -δ 1 , 1 + δ 1 ) and v ∈ N ε 1 . Moreover, by the denition of P we have S(v λ ) ⩽ S(v) + (λ -1)P (v), (6.31) for λ ∈ (1 -δ 1 , 1 + δ 1 ) and v ∈ N ε 1 . Moreover, consider the map:

(λ, v) → K(v λ ) = λ 2 ∥∇v∥ 2 L 2 + λ n 2 ∥v∥ 3 L 3 -a 2 λ n ∥v∥ 4 L 4 -λ 3n 2 ∥v∥ 5 L 5 .
Note that K(ϕ) = 0 and

∂ λ K(ϕ λ )| λ=1 = 2∥∇ϕ∥ 2 L 2 + n 2 ∥ϕ∥ 3 L 3 -na 2 ∥ϕ∥ 4 L 4 - 3n 2 ∥ϕ∥ 5 L 5 .
Thus,

∂ λ K(ϕ λ )| λ=1 = ∂ λ K(ϕ λ )| λ=1 -5P (ϕ) = -3∥∇ϕ∥ 2 L 2 - n 3 ∥ϕ∥ 3 L 3 + na 2 4 ∥ϕ∥ 4 L 4 .
Proof of Proposition 6.18. By Lemma 6.22, there exists R : (1, ∞) → (0, ∞) such that χ R(λ) ϕ λ → ϕ in H 1 (R n ) as λ ↓ 1. Moreover, χ R(λ) ϕ λ ∈ B ∩ Σ ∩ N ε 1 for λ > 1 close to 1. Thus, by Lemma 6.21, |I ε 1 (χ R(λ) ϕ λ )| < ∞ for λ > 1 close to 1 and since χ R(λ) ϕ λ → ϕ as λ → 1 in H 1 (R n ) we have ϕ is unstable. This completes the proof.

Proof of Theorem 6.6. Using Proposition 6.18, we only need to check the condition (6.30). We have We divide into three cases.

Case n = 1:

In this case, we have

∂ 2 λ S(ϕ λ )| λ=1 = ∥ϕ ′ ∥ 2 L 2 - 1 12 ∥ϕ∥ 3 L 3 - 3 20 ∥ϕ∥ 5 L 5 .
In the case DDF, using K(ϕ) = 0 and P (ϕ) = 0 we have The instability of algebraic standing waves in case n = 3 follows. This completes the proof of Theorem 6.6.

0 = P (ϕ) - 1 4 K(ϕ) = 3 4 ∥ϕ ′ ∥ 2 L 2 - 1 12 ∥ϕ∥ 3 L 3 - 1

  94, Proof of Proposition 1.1]. Proof of Theorem 2.7. Let ϕ be a nonconstant solution of (2.10) such that m = inf x∈R |ϕ(x)| > 0. From (2.10), we have ϕ ∈ X 3 (R). Using the assumptions on ϕ we can write ϕ as ϕ(x) = R(x)e iθ(x)

1 2

 1 are satisfy(3.4). To obtain the existence solution of (3.1), the authors prove that the relation(3.4) satises for all t ∈ [0, T ]. Thus, if we setu(t, x) = exp i x -∞ |h(t, y)| 2 dy h(t, x), then u ∈ C([0, T ], H 1 (R)) solves(3.1). In[START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF], the authors have proved the global well posedness of(3.3) given initial data in H (R). In half line case,[START_REF] Wu | Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space[END_REF] Wu prove existence of blow up solution of (3.3) under Dirichlet boundary condition, given initial data in Σ

  21) and (3.22) we have d(ω) = 2d(ω).

  then by Lemma 3.10 (3), L(v) > 3d(ω). Thus, by (3.26) and (3.25), we have N

  then, the sequence (φ n ) satises the desired property. This completes the proof of Theorem 3

  ) For z = (a, b) ∈ C 2 a vector, we denote |z| = |a| + |b|. (3) We denote a ≲ b, for a, b > 0, if a is smaller than b up to multiplication by a positive constant. Moreover, we denote a ≈ b if a equal to b up to multiplication by a positive constant. (4) We denote a ≲ k b if there exists a constant C(k) depending only on k such that a ⩽ C(k)b.

  ) For a, b ∈ R 2 , we denote |(a, b)| = |a| + |b|. (4) Let a, b > 0. We denote a ≲ b if a is smaller than b up to multiplication by a positive constant and denote a ≲ c b if a is smaller than b up to multiplication by a positive constant depending on c. Moreover, we denote a ≈ b if a equals to b up to multiplication by a positive constant.

. 56 )

 56 for all a, b ⩾ 0. Proof. If x = 0 or x = 1 or b = 0 or a = 0 then (5.56) is true for C(x) = 1. Consider a, b > 0. If 0 < x < 1 then using m x > m for m < 1 and 0 < x < 1 we have b) x < a x + b x , if we choose C(x) = 1 then (5.56) holds. Considering a, b > 0 and x > 1, we set

  .1) are of the forme iθ 0 ϕ(• -x 0 ),for some θ ∈ R and x 0 ∈ R n . Remark 6.4.[START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF] In case DFD, we only obtain the result on existence of algebraic standing waves when n = 1 (see Proposition 6.1). The variational characterization of algebraic standing waves and stability or instability of these solutions are open problems, even in dimension one.

g(s) = -s 2 + a 2 s 3 + s 4 ,

 234

Theorem 6 .

 6 10 ([6],Theorem 1.1). Let g be a locally Lipschitz continuous function from R + to R with g(0) = 0, satisfying (1) α = inf{ζ > 0, g(ζ) ⩾ 0} exists, and α > 0.

( 2 )

 2 There exists a number ζ > 0 such that G(ζ) > 0, whereG(t) = t 0 g(s) ds. Dene ζ 0 = inf{ζ > 0, G(ζ) > 0}.Then, ζ 0 exists, and ζ 0 > α.

( 3 )

 3 lim s↓α g(s) s-α > 0.

( 4 )

 4 g(s) > 0 for s ∈ (α, ζ 0 ]. Let β = inf{ζ > ζ 0 , g(ζ) = 0}. Then, ζ 0 < β ⩽ ∞.

( 5 )

 5 If β = ∞ then g(s) s l = 0, with l < n+2 n-2 , (If n = 2, we may choose for l just any nite real number).Then there exists a numberζ ∈ (ζ 0 , β) such that the solution u ∈ C 2 (R + ) of the Initial Value problem -u ′′ -n-1 r u ′ = g(u), for r > 0, u(0) = ζ, u ′ (0) = 0has the properties: u > 0 on R + , u ′ < 0 on R + and lim r→∞ u(r) = 0.In our case, we have g(s) = -s 2 + a 2 s 3 + s 4 , to check that the function g and G satisfy the conditions of Theorem 6.10 when n = 2, 3 with α =

⩾ n- 2 2n,

 2 for u > 0, u ̸ = a, where G(s) = s 0 f (τ )dτ . Then(6.21) admits at most one radial positive solution. The function g given in (6.19) satises conditions (a), (b), (c) of Theorem 6.11 for a the positive root of g. When n = 2, 3, the condition (d) is satised if only if d ds

γ(2∥∇ψ∥ 2 L 2 + 3∥ψ∥ 3 L 3 -4a 2 ∥ψ∥ 4 L 4 -5∥ψ∥ 5 L 5 ) - 4 (∥∇ψ∥ 2 L 2 + ∥ψ∥ 3 L 3 -a 2 ∥ψ∥ 4 L 4

 22334454223344 ⟨K ′ (ψ), ψ⟩ = ⟨S ′ (ψ), ψ⟩ = K(ψ) = 0. (6.29) Moreover, using K(ψ) = 0 we have⟨K ′ (ψ), ψ⟩ = ∂ λ K(λψ)| λ=1 = ∂ λ K(λψ)| λ=1 -4K(ψ) = -∥ψ∥ 5 L 5 ) = -2∥∇ψ∥ 2 L 2 -∥ψ∥ 3 L 3 -∥ψ∥ 5 L 5 < 0.

.

  Proof of Remark 6.4[START_REF] Bellazzini | Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems[END_REF]. We use similar arguments as in [36, Proof of Proposition 3.5]. Firstly, we denote ϕ(r) as function of ϕ respect to variable r = |x|. Since ϕ is positive decreasing radial function, we have∥ϕ∥ 3 L 3 ⩾ |x|⩽R |ϕ| 3 dx ⩾ |B(R)||ϕ(R)| 3 = CR n |ϕ(R)| 3 ,for all R > 0. Hence, ϕ(x) ⩽ |x| -n 3 ∥ϕ∥ L 3 , for all x ∈ R.

∂ 2 λ

 2 S(ϕ λ )| λ=1 < 0, where v λ (x) := λ n 2 v(λx).(6.30)

∂ 2 λ 2 +

 22 S(ϕ λ )| λ=1 = ∥∇ϕ∥ 2 L

  ϕ λ )| λ=1 = ∂ 2 λ S(ϕ λ )| λ=1 -2P (ϕ)

  in the case f ≡ 0. In Duhamel form, (1.6) is rewritten as follows:

	t
	u(t) = S(t)u 0 +	S(t -s)f (u(s)) ds.	(1.7)
	0
	Formally, under smoothness and boundedness conditions on f and u, a function u
	solves (1.6) if only if u solves (1.7) (see [17, Lemma 4.1.1], [17, Proposition 4.1.6],
	[17, Corollary 4.1.7], [17, Corollary 4.1.8], [17, Proposition 4.1.9]

  Theorem 2.2. Let ϕ ∈ X 4 (R) and u 0 ∈ ϕ + H 2 (R). Then the problem (2.1) has a unique maximal solution u ∈ C((T min , T max ), ϕ + H 2 (R)) which is dierentiable as a function of C((T min , T max ), ϕ + L 2 (R)) and such that u t ∈ C((T min , T max ), L 2 (R)).

  as in (2.37) and (2.39) are Lipschitz continuous on bounded set of

  2 |u| 2 -|u| 4 Im(u∂u) + |u| 4 ∂Im(u∂u) + 2Im(|u| 2 ∂uu t ). Im(|u| 2 u∂u) -∂Im(|u| 2 u∂ t u) .

	(2.55) (2.56) ∂ t (2.57) ∂Im(∂uu) = -1 2 ∂ t (|u| 2 ) -1 4 ∂(|u| 4 ). 2Im(|u| 2 u t ∂u) = Recall that we have Furthermore, 1 2 ∂ Thus, Combining (2.55), (2.56) and (2.57) we obtain

t Im(|u| 2 u∂u) = 4Im(u t |u| 2 ∂u) + ∂Im(|u| 2 u∂ t u).

  .86) Now, using Lemma 2.20 we have k -2 √ B ∈ H 3 (R). Combining with (2.86) we obtain

  by similar arguments as in [23, Proof of Lemma 2], we can prove that

  +

					∞
					e -λτ dτ
					t
	≲ e -5λt +	1 λ	e -λt <	1 10	e -λt ,
	Combining with (5.64) and (5.62), (5.63) we obtain
				∥Φη∥ S([t,∞))×S([t,∞)) <	1 5	e -λt .	(5.67)

  ]. Multiplying the equation by u x and taking real part, we obtain:

	d dx		1 2	|u x | 2 +	a 1 3	|u| 3 +	a 2 4	|u| 4 +	a 3 5	|u| 5 = 0.
	Thus,	1 2	|u x | 2 +	a 1 3	|u| 3 +	a 2 4	|u| 4 +	a 3 5	|u| 5 = K.

  5 L 5 ≲ ∥∇v∥ θ L 2 ∥v∥ 5-θ L 3 ⩽ C 1 ∥∇v∥ 5 L 2 + C 2 ∥v∥ 5 L 3 , ∥v∥ 4 L 4 ≲ ∥∇v∥ θ L 2 ∥v∥ 4- θ L 3 ⩽ C 3 ∥∇v∥ 4 L 2 + C 4 ∥v∥ 4 L 3 , we have 0 = K(v) ⩾ (1-C 1 ∥∇v∥ 3 L 2 -|a 2 |C 3 ∥∇v∥ 2 L 2 )∥∇v∥ 2 L 2 +(1-C 2 ∥v∥ 2 L 3 -|a 2 |C 4 ∥v∥ L 3 )∥v∥ 3 L 3 , It follows that 1 ⩽ C 1 ∥∇v∥ 3 L 2 + |a 2 |C 3 ∥∇v∥ 2 L 2 ⩽ C∥∇v∥ 3 L 2 + 1 2 or 1 ⩽ C 2 ∥v∥ 2 L 3 + |a 2 |C 4 ∥v∥ L 3 ⩽ C∥v∥ 2 L 3 + 1 2

  20 ∥ϕ∥5 L 5 . S(ϕ λ )| λ=1 < 0. This implies the instability of algebraic standing waves in the case DDF. In the case DFF, using (6.35) and the fact thata∥ϕ∥ 3 L 3 + b∥ϕ∥ 5 L 5 ⩾ 2The instability of algebraic standing waves in the case n = 2 follows.Case n = 3:In this case, we have∂ 2 λ S(ϕ λ )| λ=1 = ∥∇ϕ∥ 2

	Case n = 2:
	In this case, we have
		∂ 2 λ S(ϕ λ )| λ=1 = ∥∇ϕ∥ 2 L 2 -	a 2 2	∥ϕ∥ 4 L 4 -	6 5	∥ϕ∥ 5 L 5 .	(6.36)
	Moreover,	0 = P (ϕ) = ∥∇ϕ∥ 2 L 2 +	1 3	∥ϕ∥ 3 L 3 -	a 2 2	∥ϕ∥ 4 L 4 -	3 5	∥ϕ∥ 5 L 5 .
	Replacing a 2 2 ∥ϕ∥ 4 L 4 = ∥∇ϕ∥ 2 L 2 + 1 3 ∥ϕ∥ 3 L 3 -3 5 ∥ϕ∥ 5 L 5 in (6.36), we obtain
			∂ 2 λ S(ϕ λ )| λ=1 = -	1 3	∥ϕ∥ 3 L 3 -	3 5	∥ϕ∥ 5 L 5 < 0.
									L 2 +	1 4	∥ϕ∥ 3 L 3 -	3a 2 2	∥ϕ∥ 4 L 4 -	63 20	∥ϕ∥ 5 L 5 .	(6.37)
	Moreover,
	Thus,	∥ϕ ′ ∥ 2 L 2 = 0 = P (ϕ) = ∥∇ϕ∥ 2 L 2 +	1 9 1 ∥ϕ∥ 3 L 3 + 2 ∥ϕ∥ 3 L 3 -	1 15 3a 2 ∥ϕ∥ 5 L 5 . 4 ∥ϕ∥ 4 L 4 -
	It follows that
	∂ 2 λ S(ϕ λ )| λ=1 =	1 36	∥ϕ∥ 3 L 3 -	1 12	∥ϕ∥ 5 L 5
		=	1 36	∥ϕ∥ 3 L 3 -	1 12	10 3	∥ϕ ′ ∥ 2 L 2 +	1 6	∥ϕ∥ 3 L 3 -	a 2 4	∥ϕ∥ 4 L 4 -P (ϕ)
		= -	5 18	∥ϕ ′ ∥ 2 L 2 -	1 54	∥ϕ∥ 3 L 3 +	5a 2 72	∥ϕ∥ 4 L 4 .	(6.35)
	Thus,
									∂ 2 λ √ ab∥ϕ∥ 4 L 4 for
	all a, b > 0 we have
	∂ 2 λ S(ϕ λ )| λ=1 = -	5 18	1 9	∥ϕ∥ 3 L 3 +	1 15	∥ϕ∥ 5 L 5	-	1 54	∥ϕ∥ 3 L 3 +	5a 2 72	∥ϕ∥ 4 L 4
			= -	4 81	∥ϕ∥ 3 L 3 -	1 54	∥ϕ∥ 5 L 5 +	5a 2 72	∥ϕ∥ 4 L 4
			⩽ -	4 27 √ 6	∥ϕ∥ 4 L 4 +	5a 2 72	∥ϕ∥ 4 L 4 < 0,
	since we have assumed a 2 < 32 15 √ the instability of algebraic standing waves. 6 . Thus, in the case DFF and a 2 < 32 15 √	6 we obtain

ε ∈ (0, ε 0 ). Since h is increasing, we have h(-ε) < h(0) < h(ε). Moreover, by K ω (φ) = 0 and (3.25),(3.26), we see that 3h(0) = 3d(ω) = 3S ω (φ) = N (φ). Thus,

Acknowledgements

I would like to express my sincere gratitude to my supervisor Professor Stefan

The proof of Theorem 4.6 uses similar arguments as in the one of Theorem 4.1.

To prove Theorem 4.1, our strategy is the following. Let R be the multi-soliton prole. Our aim is to construct a solution of (4.2) which behaves as R at large times. Using the Gauge transform (4.26), we construct a system of equations of (φ, ψ). Let h, k be the prole under the Gauge transform of R. We see that h, k solves the same system as φ, ψ up to exponential decay pertubations. The decay of these terms is showed by using the separation of solitons. Set φ = φ -h and ψ = ψ -k. We see that if u solves (4.2) then ( φ, ψ) solves (4.35). By using the Banach xed point theorem, we show that there exists a solution of this system which decays exponentially fast at innity. Using this property and combining with the condition (4.8), we may prove a relation between φ and ψ. This relation allows us to obtain a solution of (4.2) satisfying the desired property.

This chapter is organized as follows. In the section 4.2, we prove the existence of multi-solitons for the equation (4.2). In the section 4.3, we prove the existence of multi kink-solitons for the equation (4.11). In the section 4.4, we prove some tools which is used in the proofs in the section 4.2 and the section 4.3. More precisely, we prove the exponential decay of the pertubations in the equations of h, k (Lemma 4.11, Lemma 4.14) and the existence of exponential decay solutions of the systems considered in the proofs of the main results in the section 4.2 (Lemma 4.13). Before proving the main results, we recall Strichartz estimates and introduce some notations used in this chapter. We need the following denition of admissible pairs. Denition 4.8. Let N ∈ N * . We say that a pair (q, r) is admissible if

Lemma 4.9. (i) There exists a constant C such that for all φ ∈ L 2 (R N ), we have

for every admissible pair (q, r).

In the case m = 1, we have

In the case m = 0, we have

In all case we have In this chapter, we consider the following generalized derivative nonlinear Schrödinger equation:

where σ ∈ R + is a given constant and u : R t × R x → C. The local well-posedness and global well-posedness of (5.1) was studied in [START_REF] Hayashi | Well-posedness for a generalized derivative nonlinear Schrödinger equation[END_REF] when the initial data is in the Sobolev space H 1 0 (Ω), where Ω is any unbounded interval of R. In this work, Hayashi-Ozawa used an approximation argument. In [START_REF] Santos | Existence and uniqueness of solution for a generalized nonlinear derivative Schrödinger equation[END_REF], Santos proved the local well-posedness for small size initial data in weighted Sobolev spaces. The arguments used in this work follow parabolic regularization approach introduced by Kato [START_REF] Kato | Nonstationary ows of viscous and ideal uids in R 3[END_REF].

The equation (5.1) has a two parameters family of solitons. The stability of the solitons has attracted the attention of many researchers. In [START_REF] Liu | Stability of solitary waves for a generalized derivative nonlinear Schrödinger equation[END_REF], by using the abstract theory of Grillakis-Shatah-Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF], Liu-Simpson-Sulem proved that in the case σ ⩾ 2, the solitons of (5.1) are orbitally unstable; in the case 0 < σ < 1, they are orbitally stable and in the case σ ∈ (1, 2) they are orbitally stable if c < 2z 0 √ ω and orbitally unstable if c > 2z 0 √ ω for some constant z 0 ∈ (0, 1). In the critical case c = 2z 0 √ ω, Guo-Ning-Wu [START_REF] Guo | Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the critical frequency case[END_REF] proved that solitons are always orbitally unstable. In [START_REF] Tang | Stability of the sum of two solitary waves for (gDNLS) in the energy space[END_REF], in the case σ ∈ (1, 2), Tang and Xu proved the stability of the sum of two solitary waves in the energy space using perturbation arguments, modulational analysis and an energy argument as in [START_REF] Martel | Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations[END_REF][START_REF] Martel | Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations[END_REF]. In this chapter, we show the existence of multi-soliton trains in energy space in the case σ ⩾ 5 2 . Before stating the main result, we give some preliminaries on multi-soliton trains of (5.1).

As mentioned in [START_REF] Liu | Stability of solitary waves for a generalized derivative nonlinear Schrödinger equation[END_REF], the equation (5.1) admits a two-parameters family of soli-Using (5.29) and (5.30), we have

We denote by Z 1 , Z 2 , Z 3 , Z 4 the terms (5.45), (5.46), (5.47) and (5.48) respectively. Using (5.49), (5.50), (5.51), (5.29) and (5.30), for N ≫ t, we have

)

Similarly, for N ≫ t, we have

and

Hence, from (5.44), we have

Chapter 6

Instability of algebraic standing waves for nonlinear Schrödinger equations with triple power nonlinearities 6.1

Now, we comeback to prove the set M is not empty. Lemma 6.16. If (v n ) ∈ X is a minimizing sequence for µ, that is,

then there exist (y n ) ⊂ R n , a subsequence (v n j ), and v 0 ∈ X \ {0} such that v n j (• -

From (6.25), we infer that (v n ) is bounded in X. Also, since µ > 0 by Lemma 6.13 and the Gagliardo-Nirenberg inequality ∥v∥ 5 L 5 ≲ ∥∇v∥ 5 L 2 + ∥v∥ 5 L 4 , we have lim sup n→∞ ∥v n ∥ L 4 > 0. Then, by Lemma 6.14 there exist (y n ) ⊂ R n and v 0 ∈ X \{0} and a subsequence of (v n (• -y n )), which we still denote by the same notation, such that v n (• -y n ) ⇀ v 0 weakly in X. we put

We can assume that w n → v 0 a.e in R n and we prove that w n → v 0 strongly in X. By Lemma 6.15, we have

Since J(v 0 ) > 0 by v 0 ̸ = 0, it follows from (6.27) and (6.25) that

From this and (6.24) we have K(w n -v 0 ) > 0 for n large. Thus, since K(v n ) → 0 and (6.28) we obtain K(v n ) ⩽ 0. By (6.24) and weak lower semicontinuity of the norms, we have

Combining with (6.27) imply that J(w n -v 0 ) → 0 thus, w n → v 0 strongly in X. This completes the proof.

Proof of Proposition 6.3. Firstly, we prove the variational characterization of ϕ as follows

This means that ϕ is a minimizer of (6.11). From Lemma 6.16, we have M ̸ = ∅.

Let φ ∈ M. We divide the proof of this to three steps.

Step 1. There exists θ ∈ R such that e iθ φ is a positive function.

We use similar arguments as in [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF]Lemma 2.10]. Put v := |Reφ|, w := |Imφ| and ψ := v + iw. By a phase modulation, we may assume that v ̸ = 0. Since |ψ| = |φ| and |∇ψ| = |∇φ|, we have K(ψ) = K(φ) and S(ψ) = S(φ). Thus, ψ ∈ M. Then, there exists γ ∈ R such that

Thus, in the case a 2 < 0, we have ∂ λ K(ϕ λ )| λ=1 < 0. In the case a 2 ⩾ 0, using P (ϕ) = 0, we have

hence we also have ∂ λ K(ϕ λ )| λ=1 < 0. In all cases, by the implicit function theorem, taking ε 1 and δ 1 small enough, for any v ∈ N ε 1 there exists

such that Λ(ϕ) = 1 and K(v Λ(v) ) = 0. Therefore, by denition of µ as in (6.11) we obtain:

This completes the proof.

Let u 0 ∈ N ε and u(t) be the associated solution of (6.1). We dene the exit time from the tube N ε by

Lemma 6.20. Assume (6.30) holds and let ε 1 be given by Lemma 6.19. Then for any u 0 ∈ B ∩ N ε 1 , where B is dened as in (6.13), there exists m = m(u 0 ) > 0 such that P (u(t)) ⩽ -m for all t ∈ I ε 1 (u 0 ).

Proof. For t ∈ I ε 1 (u 0 ), since u(t) ∈ N ε 1 , it follows from Lemma 6.19 that µ -S(u 0 ) = µ -S(u(t)) ⩽ -(1 -Λ(u(t)))P (u(t)).

In particular, since µ > S(u 0 ) by u 0 ∈ B, we have P (u(t)) ̸ = 0. By continuity of the ow and P (u 0 ) < 0 we obtain

Therefore, we obtain

This completes the proof. Lemma 6.21. Assume (6.30) holds. Then

Proof. Let u(t) be associated solution of u 0 ∈ B ∩ N ε 1 ∩ Σ. By the virial identity and Lemma 6.20 we have

This completes the proof.

Let χ be a smooth cut-o function such that

and for R > 0 dene χ R (x) = χ |x| R . The following is similar as in [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF]Lemma 4.5].

Lemma 6.22. There exists a function

Proof. We divide the proof in three steps.

Step

We have

The term (6.33) converges to zero as λ → 1. To prove the term (6.34) converges to zero as λ → 1, we prove for all ϕ ∈ L p , 1 < p < ∞, then the following holds

Indeed, we only need to consider ϕ is a integrable step function, by density of step function in L p (R n ). It is sucient to consider ϕ = 1 A , for some measurable set A.

We have ϕ(λx) = 1 1 λ A and ∥ϕ(λx) -ϕ(x)∥ p L p = ∥1

this converges to zero when λ converges to 1. Thus, if we consider ∇ϕ as a vector function then the term (6.34) converges to zero as λ converges to 1.

Step 2: χ R(λ) ϕ λ → ϕ as λ → 1 for some function R.

Choosing R : (1, ∞) → (0, ∞) such that R(λ) → ∞ as λ → 1. Thus, for all v ∈ H 1 (R n ), we have χ R(λ) v → v, as λ → 1 and χ R(λ) ϕ λ → ϕ in H 1 (R n ) as λ ↓ 1, since step 1.

Step 3: Conclusion. We claim that ϕ λ ∈ B for λ > 1 close to 1. Since ∂ λ S(ϕ λ )| λ=1 = 0 and ∂ 2 λ S(ϕ λ )| λ=1 < 0, there exists λ 1 > 1 such that ∂ λ S(ϕ λ ) < 0 and S(ϕ λ ) < µ for λ ∈ (1, λ 1 ). We see that P (ϕ λ ) = λ∂ λ S(ϕ λ ) < 0 for λ ∈ (1, λ 1 ). Moreover, taking λ 1 close to 1, we get ϕ λ ∈ N ε 1 for all λ ∈ (1, λ 1 ). Since χ R(λ) has compact support and ∥χ R(λ) ϕ λ -ϕ λ ∥ H 1 → 0 as λ → 1, we have χ R(λ) ϕ λ ∈ B ∩ N ε 1 ∩ Σ for λ close to 1. This completes the proof.