Marc-Antoine Vassenet

Transonic limit of traveling waves of the Euler-Korteweg system

Transonic limit of traveling waves of the Euler-Korteweg system

Introduction

The Euler Korteweg system, in dimension d, reads

     ∂ t ρ + div(ρu) = 0, ∂ t u + u.∇u + ∇g(ρ) = ∇ K(ρ)∆ρ + 1 2 K (ρ)|∇ρ| 2 , t ∈ R, x ∈ R d ,
(E-K) where ρ > 0 is the density of the fluid, u ∈ R d is its velocity field, the right hand side of the second line is the capillary tensor. The functions K, g are defined on R + * and smooth, the function K is positive. When the velocity is irrotational i.e u = ∇φ for some φ that cancels at infinity, the momentum equation rewrites

∂ t φ + |∇φ| 2 2 + g(ρ) = K(ρ)∆ρ + 1 2 K (ρ)|∇ρ| 2 .
There is a formally conserved energy

E(ρ, φ) = R d K(ρ)|∇ρ| 2 + ρ|∇φ| 2 2 + G(ρ)dx, (1.1)
where G is a primitive of g with a later specified integration constant. Moreover we have a momentum

P (ρ, φ) = R d (ρ -ρ 0 )∂ 1 φdx. (1.2)
which makes sense when ρ -ρ 0 ∈ L 2 , ∇φ ∈ L 2 . The energy makes sense for (ρ, u) localized near the constant state (ρ 0 , 0), which will be our framework. We call traveling wave a solution of (E-K) of the form (ρ(x.n -ct, x ⊥ ), u(x • n -ct, x ⊥ )), x ⊥ = (x 1 , x 2 , ..., x d ) -x • n, where c is the speed of propagation and n the direction of the speed. The direction of the speed does not matter, thus we let n = e 1 . A traveling wave solves    -c∂ 1 ρ + div(ρ∇φ) = 0,

-c∂ 1 φ + g(ρ) -K(ρ)∆(ρ) + |∇φ| 2 2 -K (ρ) |∇ρ| 2 2 = 0.
(1.3)

In [START_REF] Audiard | Small energy traveling waves for the Euler-Korteweg system[END_REF], Audiard proves the existence of traveling waves in dimension two, localized near the constant state (ρ 0 , 0) with g(ρ 0 ) = 0, g (ρ 0 ) > 0. Their speed is close but less than the speed of sound that we define now. When neglecting the capilary tensor and linearizing this system near (ρ 0 , 0) i.e. ρ = ρ 0 + r, we obtain the Euler equation ∂ t r + ρ 0 div(u) = 0, ∂ t u + g (ρ 0 )∇r = 0.

(1.4)

The speed of sound is

c s = ρ 0 g (ρ 0 ).
For simplification, we use the following rescaling (ρ, φ) = ρ 0 ρ r g (ρ 0 ) ρ 0 x , φ r g (ρ 0 ) ρ 0 x , K r (ρ r ) = K(ρ 0 ρ r ) ρ 0 , g r (ρ r ) = g(ρ 0 ρ r ) g (ρ 0 )ρ 0 , c r = c ρ 0 g (ρ 0 ) .

Then (1.3) becomes

   -c r ∂ 1 ρ r + div(ρ r ∇φ r ) = 0, -c r ∂ 1 φ r + g r (ρ r ) -K r (ρ r )∆(ρ r ) + |∇φ r | 2 2 -K (ρ r ) |∇ρ r | 2 2 = 0.
(1.5)

In this system the constant state is 1, g r (1) = 0, g r (1) = 1 and the speed of sound is 1g (1) = 1. We will forget the subscript r and focus on the rescaled system, i.e. we will assume through the rest of the paper that g(1) = 0, g (1) = 1 and c s = 1.

(

Solutions of (1.5) with speed near the speed of sound are known to exist, the precise existence statement of [START_REF] Audiard | Small energy traveling waves for the Euler-Korteweg system[END_REF] is the following:

Theorem 1. ([1] Theorem 1.1, proposition 3.3 and proposition 2.3) In dimension two, under the assumption Γ := 3 + g (1) = 0, there exists p 0 > 0 such that for any 0 ≤ p ≤ p 0 we have (ρ p , φ p ) ∈ ∩ j≥0 1 + H j × Ḣj+1 , solution of (1.5) for some c p > 0, with P (ρ p , φ p ) = p. Moreover there exists α, β, C > 0 such that, for any 0 ≤ p ≤ p 0 p -βp 3 ≤ E(ρ p , φ p ) ≤ p -αp 3 , (1.7)

1 -βp 2 ≤ c p ≤ 1 -αp 2 , (1.8)
||ρ p -1|| ∞ ≤ C E(ρ p , φ p ) for p 1.

(1.9)

||ρ p -1|| ∞ ≥ Cp 2 for p 1.
(1.10)

Remark 1.1. As has been proven for the Schrödinger equation (see [START_REF] Chiron | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity[END_REF]), it is possible that such solitons exist in higher dimensions.

To construct the traveling waves of theorem 1 the author in [START_REF] Audiard | Small energy traveling waves for the Euler-Korteweg system[END_REF] solves a minimization problem. On the space H = (ρ, φ) ∈ (1 + H 1 ) × Ḣ1 the momentum is well-defined, however the energy (1.1) does not make sense. For example the term ρ|∇φ| 2 is not necessarily integrable. The solution is to work with a modified energy E which has nice coercive properties and such that

E = E if |ρ -1|
1. Then the author finds (ρ, φ) solution of the minimization problem inf E(ρ, φ), (ρ, φ) ∈ H : P (ρ, φ) = p , (1.11) for small p, such that the minimizer is smooth and satisfies |ρ -1| 1. Our aim is to describe the asymptotic behaviour, as p → 0, of the traveling waves (ρ p , φ p ). It is instructive to compare our problem to the extensive litterature studying the nonlinear Schrödinger equation. Indeed in the case K = κ/ρ, with κ a positive constant, up to a rescaling there exists a formal correspondance with the nonlinear Schrödinger equation

i∂ t Ψ + ∆Ψ = g |Ψ| 2 )Ψ, (NLS)
using the Madelung transform ρ, ∇φ → Ψ := √ ρe iφ (see [START_REF] Carles | Madelung, Gross-Pitaevskii and Korteweg[END_REF] for more details).

In the case g(ρ) = ρ -1 (NLS) is called the Gross-Pitaevskii equation

i∂ t Ψ + ∆Ψ = Ψ(|Ψ| 2 -1) on R d × R. (GP)
The counterpart of (1.1) is

E(Ψ) = R d |∇Ψ| 2 + 1 2 R d 1 -|Ψ| 2 2 , (1.12)
and that of (1.2) is

P (Ψ) = 1 2 R d Re i∇Ψ(Ψ -1) . (1.13)
We also call traveling wave a solution of (GP) of the form

Ψ(x, t) = v(x 1 -ct, x ⊥ ), x ⊥ = (x 2 , ..., x d ),
where c is the speed of propagation. The traveling waves play an important role in the long time dynamics of (GP) (see e.g [START_REF] Chiron | Stability and instability for subsonic traveling waves of the nonlinear Schrödinger equation in dimension one[END_REF][START_REF] Béthuel | Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation[END_REF][START_REF] Gravejat | Asymptotic stability of the black soliton for the Gross-Pitaevskii equation[END_REF][START_REF] Chiron | A uniqueness result for the two vortex travelling wave in the nonlinear schrodinger equation[END_REF][START_REF] Chiron | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity[END_REF]). The profile v solves the equation

-ic∂ 1 v + ∆v + v(1 -|v| 2 ) = 0. (TWc)
Using the Madelung transform, the associated speed of sound for (GP), around the constant solution ψ = 1, is √ 2 (a rescaling changes the quantity 1g (1) into 1). The transonic limit was first studied by physicists (see [START_REF] Jones | Motions in a bose condensate. v. stability of solitary wave solutions of non-linear Schrödinger equations in two and three dimensions[END_REF][START_REF] Jones | Motions in a bose condensate. IV. axisymmetric solitary waves[END_REF]). In the onedimensional case, equation (TWc) is integrable with elementary computations. Solutions to (TWc) are related to the soliton of the Korteweg-de Vries equation

∂ t ψ + ψ∂ 1 ψ + ∂ 3 1 ψ = 0. (KdV)
Indeed in the transonic limit c → √ 2, the traveling waves converge, up to rescaling, to the (KdV) soliton (see [START_REF] Béthuel | Existence and properties of travelling waves for the Gross-Pitaevskii equation[END_REF][START_REF] Chiron | Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one[END_REF]). A similar result exists in the twodimensional case: for any p > 0, there exists a non-constant finite energy solution v p to (TWc) with P (v p ) = 1 2 R d Re i∇Ψ(Ψ -1) = p (see [START_REF] Béthuel | Travelling waves for the Gross-Pitaevskii equation[END_REF] theorem 1 and the survey [START_REF] Béthuel | Existence and properties of travelling waves for the Gross-Pitaevskii equation[END_REF] for properties of traveling waves). The convergence, in the transonic limit, of those minimizing traveling waves for the two-dimensional Gross-Pitaevskii equation towards a ground state of the Kadomtsev-Petviashvili equation was obtained by Bethuel-Gravejat-Saut (see [START_REF] Béthuel | On the KP I transonic limit of two-dimensional Gross-Pitaevskii travelling waves[END_REF]). The Kadomtsev-Petviashvili equation

∂ t ψ + ψ∂ 1 ψ + ∂ 3 1 ψ -∂ -1 1 ∂ 2 2 ψ = 0, (KPI)
is a higher dimensional generalization of the Korteweg de Vries equation with energy

E KP (ψ) = 1 2 R 2 (∂ 1 ψ) 2 + 1 2 R 2 (∂ -1 1 (∂ 2 ψ)) 2 - 1 6 R 2 (ψ) 3 . (1.14)
It is a well-known asymptotic model for the propagation of weakly transverse dispersive waves [START_REF] Borisovich | On the stability of solitary waves in weakly dispersing media[END_REF]. Solitary waves are localized solutions to (KPI) of the form ψ(x, t) = ω(x 1 -σt, x 2 ), where ω belongs to the energy space for (KPI), i.e. the space Y (R 2 ) (see [START_REF] De | Solitary waves of generalized Kadomtsev-Petviashvili equations[END_REF]) defined as the closure of

∂ 1 C ∞ c (R 2 ) for the norm ∂ 1 f Y (R 2 ) = ( ∇f 2 L 2 (R 2 ) + ∂ 2 1 f 2 L 2 (R 2 ) ) 1 2
.

The equation of a solitary wave ω of speed σ = 1 is given by

∂ 1 ω -ω∂ 1 ω -∂ 3 1 ω + ∂ -1 1 ∂ 2 2 ω = 0. (SW)
Given any σ > 0, the scale change ∼ ω(x, y) = σω x √ σ, yσ transforms any solution of equation (SW) into a solitary wave ∼ ω of speed σ. The strategy in [START_REF] Béthuel | On the KP I transonic limit of two-dimensional Gross-Pitaevskii travelling waves[END_REF] is to rewrite (GP) as an hydrodynamical system using Madelung transform, then rewrite the new equationss as a Kadomtsev-Petviashvili equation with some remainder. This transonic limit convergence result has been generalized in dimension two and three by Chiron and Mariş in [START_REF] Chiron | Rarefaction pulses for the nonlinear schrödinger equation in the transonic limit[END_REF] for a large class of nonlinearities. In the same spirit, it is proved in [START_REF] Chiron | Long wave asymptotics for the Euler-Korteweg system[END_REF] that solutions of (E-K) with well-prepared initial data converge, in a long wave asymptotic regime, to a solution of the Kadomtsev-Petviashvili (Korteweg de Vries in the one-dimensional case) equation (See also [START_REF] Bethuel | On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation II[END_REF][START_REF] Bethuel | On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation I[END_REF][START_REF] Chiron | The KdV/KP-I limit of the nonlinear Schrödinger equation[END_REF][START_REF] Chiron | Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations[END_REF]).

The main focus of this paper is the convergence in the transonic limit of the Euler Korteweg two dimensional traveling waves to a ground state of (KPI). We also obtain with a more elementary argument, in dimension one, the convergence of the Euler-Korteweg soliton toward the soliton to the Korteweg-de-Vries equation.

Heuristic Let us do the following formal computation. We let

ρ p (x 1 , x 2 ) = 1 + ε 2 p η p (z 1 , z 2 ), φ p = ε p θ p (z 1 , z 2 ), with z 1 = ε p x 1 , z 2 = ε 2 p x 2 and c p = 1 -ε 2 p . Then the first line of (1.5) rewrites -c p ∂ 1 η p + ∂ 2 1 θ p + ε 2 p ∂ 2 2 θ p + η p ∂ 2 1 θ p + ∂ 1 η p ∂ 1 θ p = O(ε 4 p ),
furthermore, by Taylor expansion we have g = ε 2 p η p + g (1)ε 4 p η 2 p 2 + O(ε 4 p ), then the second lines of (1.5) rewrites

-c p ∂ 1 θ p + η p + ε 2 p g (1)η 2 p + (∂ 1 θ p ) 2 2 -K(1)∂ 2 1 η p = O(ε 4 p ).
At first order we have

∂ 1 θ p = η p + O(ε 2 p )
, so that these functions should have the same limit. Then, multiplying the first equation by c p and applying the operator ∂ 1 to the second equation, we obtain

∂ 1 η p + ∂ 2 2 ∂ -1 1 η p + (3 + g (1))η p ∂ 1 η p -K(1)∂ 3 1 η p = O(ε 2 p ).
Finally, we let

η p = - 1 3 + g (1) N p x 1 K(1) , x 2
K(1) .

Then N p is an (approximate) solution to (SW).

Main result Let (ρ p , φ p ) be the solution given by theorem 1, we consider

η p (x 1 , x 2 ) = ρ p (x 1 , x 2 ) -1, (1.15)
and the rescaled functions .

         θ p (x 1 , x 2 ) = - 1
γε p K(1) φ p K(1)x 1 ε p , K(1)x 2 ε 2 p , N p (x 1 , x 2 ) = - 1 γε 2 p η p K(1)x 1 ε p , K (1)
(1.17)

Our main theorem is Theorem 2. Under the assumption Γ := 3 + g (1) = 0, let (p n ) n∈N such that, p n → 0. Then, there exists a ground state N 0 of (KPI), such that, up to a subsequence, we have

N pn → N 0 in W k,q (R 2 ), when n → ∞. (1.18)
and

∂ 1 θ pn → N 0 in W k,q (R 2 ), when n → ∞. (1.19)
for any k ∈ N and any 1 < q ≤ ∞.

Remark 1.2. The proof in the manuscript follows the same lines of [START_REF] Béthuel | On the KP I transonic limit of two-dimensional Gross-Pitaevskii travelling waves[END_REF], but is more involved at a technical level because of the extension to arbitrary nonlinearities g and K. It provides an extension of recent results on the nonlinear Schrödinger equations with non-zero condition at infinity towards the wider, but still physically relevant, class of the Euler-Korteweg systems. See [START_REF] Chiron | Smooth branch of rarefaction pulses for the nonlinear Schrödinger equation and the Euler-Korteweg system in 2d[END_REF] for the existence of smooth branch of travelling waves for the Euler-Korteweg equation converging to the first lump in the transonic limit. It is not known that the solitons of Theorem 1 are the same as those in [START_REF] Chiron | Smooth branch of rarefaction pulses for the nonlinear Schrödinger equation and the Euler-Korteweg system in 2d[END_REF], thus the result of theorem 2 is not contained in [START_REF] Chiron | Smooth branch of rarefaction pulses for the nonlinear Schrödinger equation and the Euler-Korteweg system in 2d[END_REF] (and conversely) . To obtain the convergence of the full sequence, it is sufficient that the limit N 0 is unique but it is a difficult and open problem (see [START_REF] Liu | Nondegeneracy, Morse index and orbital stability of the KP-I lump solution[END_REF]).

Remark 1.3. A condition similar to Γ = 0 is highlighted in [START_REF] Chiron | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity[END_REF] for the Schrödinger equation. As has been proven for the Schrödinger equation (see [START_REF] Chiron | Rarefaction pulses for the nonlinear schrödinger equation in the transonic limit[END_REF]), it is possible that a similar result exist in dimension three. Let us recall that the existence of solitons in dimension 3 is an open problem for Euler-Korteweg.

We will also prove a similar result in the one-dimensional case. That is, the (E-K) solitons converge, up to rescaling, to the (KdV) soliton, in the transonic limit (see the appendix A for a precise statement). Moreover as the computation are simpler in dimension 1 we are able to compute the transonic limit for Γ = 0 and a new nondegeneracy condition (see proposition 1.6). In this case, the limit is not a solution of (KdV) but of (gKdV).

     -cρ + (ρu) = 0, ( 1
)
cu + u 2 2 + g = K(ρ)ρ + 1 2 K (ρ)ρ 2 (2), x ∈ R, (E-K) with ρ -1 L ∞ → ε→0 0. Moreover if we let ρ -1 = -ε 2 γr ε εx K(1) , (1.20
)

then for any k ∈ N r (k) ε -N (k) L ∞ (R) -→ ε→0 0,
where

N (x) = 3 ch 2 ( x 2 ) , (1.21)
is the classical soliton to the Korteweg-de-Vries equation

-ψ + ψψ + ψ = 0. (KdV)
Remark 1.5. The argument proposed in [START_REF] Chiron | Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one[END_REF] for the Schrödinger equation should extend to our framework, nevertheless we propose an alternative proof in section A.

It is also possible to describe the case Γ = 0: Proposition 1.6. Under the conditions g(1) = 0, g (1) = 1, Γ = 0, Γ := g (1) -12 < 0,

There exists (ρ ± , u ± ) global solution of (E-K) with ρ ± -1 L ∞ → ε→0 0 . Moreover if we let ρ ± -1 = εγ r ± ε εx K (1)
,

with γ = 1 √ 12-g (1)
then for any k ∈ N

r ±(k) ε -(w ±(k) ) L ∞ (R) -→ ε→0 0.
where w ± (x) = ± √ 12 ch(x) are the two opposite soliton of the focusing modified Korteweg de Vries equation

ψ - 1 2 ψ 2 ψ = ψ . (mKdV)
Organization of the article In section 2 we introduce the notations and recall the properties of solitary waves solutions to (KPI). In section 3 we prove that N p and ∂ 1 θ p converge and have the same limit. In section 4, we prove that Sobolev bounds for N p give bound for θ p . In section 5, using the Taylor expansion in (1.5) with respect to ε we obtain the (SW) equation with some remainder. Using Fourier transform we obtain Sobolev bounds for N p and ∂ 1 θ p , in section 6 and 7. Finally, we end the proof of theorem 2 in section 8.

2 Notations, functional spaces and properties of solution to (KPI)

Functional spaces Let p ∈ [1, +∞], k ∈ N, 0 < α < 1 and Ω be an smooth open subset of R d . We denote by W m,p (Ω), H m (Ω) = W m,2
(Ω) the usual Sobolev spaces. For s ≥ 0, we define

H s (R d ) = u ∈ L 2 (R d ) u 2 H s (R d ) = R d (1 + |ξ| 2 ) s |û(ξ)| 2 dξ < ∞ .
We define

Ḣs (R d ) = u û ∈ L 1 loc (R d ), R d |ξ| 2s |û(ξ)| 2 dξ < ∞ . C 0,α (Ω)
is the space of bounded α-Hölder continuous function on Ω. We define

C k 0 (R d ) = {u ∈ C k (R d ) : lim |x|→∞ |∂ α u(x)| = 0, ∀α ∈ N d , |α| ≤ k}, with the norm ||u|| C k 0 (R d ) = |α|≤k sup R d |∂ α u|.
Sobolev embedding We recall the Sobolev embeddings,

∀ kq 1 < d, W k,q1 (R d ) → L q2 (R d ), q 2 = dq 1 d -kq 1 , (2.1)
∀ d < q < ∞, W 1,q (Ω) → C 0,1-d q (Ω). (2.2) ∀(k -k )q > d, W k,q (R d ) → C k 0 (R d ). (2.3) Moreover if Ω is bounded the embedding W 1,q (Ω) → C 0,1-d α (Ω), d < α < q, ( 2.4)
is compact. In particular

H d 2 (R d ) → L q (R d ), 2 ≤ q < ∞, (2.5)
∀ 0 ≤ s < d 2 , H s (R d ) → L q (R d ), 2 ≤ q ≤ 2d d -2s , (2.6
)

∀s > d 2 + k, H s (R d ) → C k 0 (R d ).
(2.7)

Basic convolution results let (f, g) ∈ H s (R d ) × L 1 (R d ), we have ||f * g|| H s (R d ) ≤ ||f || H s (R d ) ||g|| L 1 (R d ) . (2.8)
We recall a result on Fourier multipliers due to Lizorkin.

Theorem 3. ([29]

). Let K be a bounded function in C 2 (R 2 \{0}) and assume that

ξ k1 1 ξ k2 2 ∂ k1 1 ∂ k2 2 K(ξ) ∈ L ∞ (R 2 ), for any integer 0 ≤ k 1 , k 2 ≤ 1 such that k 1 + k 2 ≤ 2. Then, K is a multiplier from L q (R 2 ) to L q (R 2 )
for any 1 < q < ∞, i.e. there exists a constant C(q), depending only on q, such that

||K * f || L q (R 2 ) ≤ C(q)M ( K)||f || L q (R 2 ) , ∀f ∈ L q (R 2 ).
where we denote

M ( K) = sup{|ξ 1 | k1 |ξ 2 | k2 |∂ k1 1 ∂ k2 2 K(ξ)|, ξ ∈ R 2 , 0 ≤ k 1 ≤ 1, 0 ≤ k 2 ≤ 1, 0 ≤ k 1 +k 2 ≤ 2}.
Existence and properties of solitary wave solutions to (KPI) We recall some results on the Kadomtsev-Petviashvili Equation. A ground state is a solitary wave that minimizes the action

S(ω) = E KP (ω) + σ 2 R 2 ω 2 ,
among all non-constant solitary waves of speed σ. The constant S KP denotes the action S(ω) of the ground states ω of speed σ = 1. We will denote by G σ the set of the ground state of speed σ. The ground states solutions are characterized as minimizers of energy constrained by constant L 2 -norm. Let µ > 0, then the minimization problem

E KP (µ) = inf E KP (ω), ω ∈ Y (R 2 ), R 2 |ω| 2 = µ , (P KP (µ))
has at least one solution. Moreover there exists σ such that the set of minima E KP (µ) is exactly equal to G σ (see De Bouard and Saut [START_REF] De | Remarks on the stability of generalized KP solitary waves[END_REF]). For σ = 1 we have µ = µ * = 3S KP . Since it was proved by making use of the concentrationcompactness principle of P.L. Lions (see [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case[END_REF]), we have the compactness of minimizing sequences.

Theorem 4. ( [START_REF] De | Remarks on the stability of generalized KP solitary waves[END_REF]) Let µ > 0, and let (ω n ) n∈N be a minimizing sequence to

(P KP (µ)) in Y (R 2 ).
Then, there exists some points (a n ) n∈N and a function N ∈ Y (R 2 ) such that up to some subsequence,

ω n (. -a n ) → N in Y (R 2 ), as n → ∞.
N is solution to the minimization problem (P KP (µ)) and thus is a ground state for (KPI).

Using a scaling argument it is possible to compute E KP .

Lemma 2.1.

([3]) Let N ∈ Y (R 2 ). Given any σ > 0, the function N σ (x 1 , x 2 ) = σN ( √ σx 1 , σx 2
) is a minimizer for E KP ( √ σµ * ) if and only if N is a minimizer for E KP (µ * ). In particular we have

E KP (µ) = - µ 3 54S 2 KP , ∀µ > 0.
Moreover, N σ and N are ground states for (KPI) , with respective speed σ, and 1. We have the relation

σ = µ 2 (µ * ) 2 .
Finally, in our proof, we will have sequences (ω n ) n∈N which are not minimizing sequences for (P KP (µ)) but satisfy

E KP (ω n ) → E KP (µ), and R 2 ω 2 n → µ, as n → ∞, (2.9)
for some positive number µ.

Proposition 2.2. ([3]

) Let µ > 0, and (ω n ) n∈N denote a sequence of functions in Y (R 2 ) satisfying (2.9) for µ. Then, there exists some sequence (a n ) n∈N and a ground state solution N , with speed σ = µ 2 (µ * ) 2 such that, up to some subsequence,

ω n (. -a n ) → N in Y (R 2 ), as n → ∞. (2.

10)

Reformulation In [START_REF] De | Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves[END_REF], the authors use a new formulation for the solitary wave equation. Applying operator ∂ 1 to (SW) we have

∂ 4 1 ω -∆ω + 1 2 ∂ 2 1 (ω 2 ) = 0. Let K 0 (ξ) = ξ 2 1 |ξ| 2 + ξ 4 1 , ξ ∈ R 2 . (2.11)
and ω ∈ Y (R 2 ). Then, ω is a solution to (SW) if and only if

ω = 1 2 K 0 * ω 2 .
(2.12)

3 Weak convergence in L 2
The aim of this section is to prove that N p and ∂ 1 θ p have the same limit, and compute the convergence speed of N p -∂ 1 θ p towards 0.

Rescaling and energy

Let (ρ p , φ p ) the solutions given by theorem 1. As in [START_REF] Béthuel | On the KP I transonic limit of two-dimensional Gross-Pitaevskii travelling waves[END_REF], we consider rescaled functions and use anisotropic space variables. let

ε p = 1 -c 2 p and γ = 1 g (1) + 3 , (3.1)
θ p (x 1 , x 2 ) = - 1 γε p K(1) φ p K(1)x 1 ε p , K(1)x 2 ε 2 p , (3.2)
and

N p (x 1 , x 2 ) = - 1 γε 2 p η p K(1)x 1 ε p , K(1)x 2 ε 2 p . (3.3)
We can assume, up to a translation, using (1.10)

∃C > 0, N p (0) > C, ∀p 0 > p > 0. (3.4) Proposition 3.1. Let (p n ) n∈N a sequence such that p n → 0. Then, there exists N 0 ∈ L 2 (R 2
) such that, up to a subsequence,

N pn N 0 in L 2 (R 2 ) (3.5) ∂ 1 θ pn N 0 in L 2 (R 2 ) (3.6)
Moreover there exists some positive constant C, not depending on p, such that

R 2 (N p -∂ 1 θ p ) 2 dx ≤ Cε 1 2
p .

(3.7)

Remark 3.2. In section 7, we will prove that N 0 is a ground state of (KPI).

Proof of (3.5)

an (3.6). Since G (1) = g (1) = 1, G (1) = g(1) = G(1) = 0, there exists δ > 0 such that G(ρ) ≥ (ρ-1) 2 3
for any ρ ∈]1 + δ, 1 -δ[. Then for p small enough, using (1.9) and the definition (1.1), we have

R 2 (η p x 1 , x 2 ) 2 dx ≤ M E(ρ p , φ p ). (3.8)
Thus, we deduce from (1.7) and (1.17) that

R 2 (N p ) 2 = 1 γ 2 ε 4 p R 2 η p K(1)x 1 ε p , K(1)x 2 ε 2 p 2 ≤ 1 K(1)γ 2 ε p R 2 (η p x 1 , x 2 ) 2 ≤ 1 ε p CE(ρ p , φ p ) ≤ C.
(3.9)

Then, using Banach-Alaoglu theorem, there exists a function N 0 ∈ L 2 (R 2 ) such that, up to some subsequence

N pn N 0 in L 2 (R 2 ). (3.10)
The convergence of ∂ 1 θ pn is a consequence of (3.7). In order to complete the proof of proposition 3.1, it only remains to prove (3.7). This requires to use rescaled energy and Pohozaev estimates, so that (3.7) is postponed to section 3.3.

Lemma 3.3. The energy can be expressed in terms of the new functions as

E(ρ p , φ p ) = K(1)γ 2 ε p 2 E 0 (N p , θ p ) + ε 2 p (E 2 (N p , θ p ))) + ε 4 p (E 4 (N p , θ p )) , with E 0 (N p , θ p ) = R 2 N 2 p + (∂ 1 θ p ) 2 , E 2 (N p , θ p ) = 2 R 2 (∂ 1 N p (x)) 2 2 + (∂ 2 θ p (x)) 2 2 - γ 6 3N p (∂ 1 θ p ) 2 + g (1)N 3 p )dx , E 4 (N p , θ p ) = R 2 K 1 -γε 2 p N p (x) 1 K(1) 2 (∂ 2 N p (x)) 2 -γN p (x)(∂ 2 θ p (x)) 2 + N p (x)j p (x) (∂ 1 N p (x)) 2 + (ε p ) 2 (∂ 2 N p (x)) 2 + γ 4 N 4 p (x)l p (x)dx,
with j p and l p some functions smooth and bounded in L ∞ uniformly in p. For the momentum we have

p = ε p γ 2 K(1) R 2 N p (x)∂ 1 θ p (x)dx.
Proof. Since N p ∂ 1 θ p , then passing to the limit in p we have

E 0 (N p , θ p ) 2 R 2 N 2 0 and E 2 (N p , θ p ) 2E KP (N 0 )
. Later, to prove that the weak limit N 0 is a solution to (SW), we will use (P KP (µ)). This is a direct but tedious computation. The functions j p and l p are given by

G(1 + x) = G(1) + G (1)x + G (1) x 2 2 + x 3 l(x) = x 2 2 + x 3 l(x). So G(ρ p (x)) = γ 2 1 2 ε 4 p N 2 p (x ) -γ 3 ε 6 p g (1)N 3 p (x ) 6 + γ 4 ε 8 p N 4 p (x )l(-ε 2 γN p (x )),
where l is the third order remainder of Taylor expansion and

x = ε p x 1 K(1) , ε 2 p x 2 K (1)
.

In view of (1.9), the function l(-ε 2 p γN p (x )) = l(ρ p (x) -1) is bounded independently of p. To simplify we write

G(ρ p (x)) = γ 2 1 2 ε 4 p N 2 p (x ) -γ 3 ε 6 p g (1)N 3 p (x ) 6 + γ 4 ε 8 p N 4 p (x )l p (x ).
Similarly, we write

K 1 -γε 2 p N p (x ) = K(1) -ε 2 p N p (x )j p (x ).

Pohozaev's identities

We estimate now derivative terms in the energy using Pohozaev's identities.

Lemma 3.4. R 2 (∂ 2 φ p ) 2 + (∂ 2 ρ p ) 2 + (∂ 1 ρ p ) 2 dx ≤ Cε 3 p , (3.11)
and p .

R 2 η p (∂ 1 φ p ) 2 dx ≤ Cε 3 2 p , R 2 η p (∂ 2 φ p ) 2 dx ≤ Cε
(3.12)

Proof. Thanks to (1.7), we have

|E(ρ p , φ p ) -P (ρ p , φ p )| < CP (ρ p , φ p ) 3 .
We recall Pohozaev's identities obtained in [START_REF] Audiard | Small energy traveling waves for the Euler-Korteweg system[END_REF] proposition 5.1

E(ρ p , φ p ) = R 2 ρ p (∂ 2 φ p ) 2 + K(ρ p )(∂ 2 ρ p ) 2 dx + c p P (ρ p , φ p ), (3.13)
E(ρ p , φ p ) = R 2 ρ p (∂ 1 φ p ) 2 + K(ρ p )(∂ 1 ρ p ) 2 dx. (3.14)
Moreover we have

c p P (ρ p , φ p ) = R 2 ρ p (∇φ p ) 2 dx. (3.15)
It follows from (1.7) and (1.17) that

R 2 ρ p (∂ 2 φ p ) 2 + K(∂ 2 ρ p ) 2 dx = |E(ρ p , φ p ) -cP (ρ p , φ p )| = |E(ρ p , φ p ) -P (ρ p , φ p ) + (1 -1 -ε 2 p )P (ρ p , φ p )| ≤ |E(ρ p , φ p ) -P (ρ p , φ p )| + ε 2 p 1 -1 -ε 2 p ε 2 p P (ρ p , φ p ) ≤ CP (ρ p , φ p ) 3 ,
and therefore using ||ρ p -1|| ∞ → 0 when p → 0 we get

R 2 (∂ 2 φ p ) 2 + (∂ 2 ρ p ) 2 dx ≤ CP (ρ p , φ p ) 3 . (3.16)
In view of (3.14) and (3.15) we obtain

R 2 K(ρ p )(∂ 1 ρ p ) 2 dx = ρ p (∂ 2 φ p ) 2 dx + E(ρ p , φ p ) -cP (ρ p , φ p ) ,
and then

R 2 (∂ 1 ρ p ) 2 dx ≤ CP (ρ p , φ p ) 3 . (3.17)
Finally combining (3.17) and (3.16) we have

R 2 (∂ 2 φ p ) 2 + (∂ 2 ρ p ) 2 + (∂ 1 ρ p ) 2 dx ≤ CP (ρ p , φ p ) 3 . (3.18)
Moreover by (3.14)

E(ρ p , φ p ) = R 2 ρ p (∂ 1 φ p ) 2 + K(ρ p )(∂ 1 ρ p ) 2 dx, (3.19)
thus

R 2 (∂ 1 φ p ) 2 dx ≤ CE(ρ p , φ p ).
Combining (3.19) with (1.9), we obtain

R 2 η p (∂ 1 φ p ) 2 dx ≤ Cε 3 2 p . (3.20) Similarly R 2 η p (∂ 2 φ p ) 2 dx ≤ Cε 7 2 p , (3.21)
this ends the proof of lemma 3.4.

Energy estimates

We are now in position to conclude the proof of proposition 3.1.

End of proof of proposition 3.1. First we have

R 2 ∂ 1 N p 2 = 1 γ 2 ε 3 p R 2 (∂ 1 ρ p ) 2 , R 2 ∂ 2 θ p 2 = 1 K(1)γ 2 ε 3 p R 2 (∂ 2 φ p ) 2 , R 2 ∂ 2 N p 2 = 1 γ 2 ε 5 p R 2 (∂ 2 ρ p ) 2 , R 2 N p (∂ 1 θ p ) 2 = - 1 K(1)γ 3 ε 3 p η p (∂ 1 φ p ) 2 , R 2 N 3 p = - 1 γ 3 K(1)ε 3 p R 2 η 3 p , R 2 N p (∂ 2 θ p ) 2 = - 1 γ 3 ε 5 p K(1) R 2 η p (∂ 2 φ p ) 2 .
Using (3.18) and (1.17), we have

R 2 ∂ 2 θ p 2 + ∂ 1 N p 2 ≤ C. (3.22)
Then with (1.9), (3.8) and (3.20) we obtain

E 2 (N p , θ p ) ≤ C ε 3 2 p . (3.23)
Similarly, using (3.21), (3.18), (1.9) and by definition of E 4 , we have

E 4 (N p , θ p ) ≤ C ε 3 p . ( 3.24)
We deduce that

E(ρ p , φ p ) - γ 2 ε p K(1) 2 E 0 (ρ p , φ p ) = E(ρ p , φ p ) - γ 2 ε p K(1) 2 R 2 (N p ) 2 + (∂ 1 θ p ) 2 ≤ Cε 3 2 p . (3.25)
Thus, by (1.7) we have

R 2 (N p -∂ 1 θ p ) 2 = E 0 (N p , θ p ) -2 R 2 N p ∂ 1 θ p ≤ 2E(ρ p , φ p ) γ 2 K(1)ε p - 2 ε p γ 2 K(1) p + Cε 1 2 p ≤ C 1 ε 1 2 p .
This completes the proof of estimate (3.7) and Proposition 3.1.

Elliptic estimates on θ p

In the present section, we aim to prove that N p bounds θ p in Sobolev spaces.

More precisely, we prove the following proposition:

Proposition 4.1. Let 1 < q < ∞ there exists a constant C(q) depending on q, but not on p, such that

||∂ 1 θ p || L q (R 2 ) + ε p ||∂ 2 θ p || L q (R 2 ) ≤ C(q)||N p || L q (R 2 ) , (4.1)
for p small enough. Moreover for any α ∈ N 2 let

Γ p (q, α) = ||∂ α ∂ 1 θ p || L q (R 2 ) + ε p ||∂ α ∂ 2 θ p || L q (R 2 ) ,
then there exists C(q, α) such that

Γ p (q, α) ≤ C(q, α) ||∂ α N p || L q (R 2 ) + ε 2 p 0≤β<α ||∂ β N p || L ∞ (R 2 ) Γ p (q, α -β) . (4.2)
First of all in view of theorem 1 and lemma 4.2 in [START_REF] Audiard | Small energy traveling waves for the Euler-Korteweg system[END_REF] we have:

Lemma 4.2. For any k ∈ N, q ∈ [2, ∞[ we have (η p , ∇φ p ) ∈ W k,q (R 2 ) ∩ C ∞ (R 2 ). ( 4

.3)

Moreover for any α ∈ N 2 , there exists C(α) > 0 not depending on p such that

||(∂ α η p , ∂ α ∇φ p )|| L ∞ (R 2 ) ≤ C(α). (4.4)
Later, using Lizorkin theorem, we will have N p , ∂ 1 θ p and ∂ 2 θ p in W k,q (R 2 ), for any 1 < q < 2 and k ∈ N. Thus the quantity Γ p (q, α) is finite for any 1 < q < ∞, α ∈ N 2 . This is the reason why in proposition 4.1, we let 1 < q < ∞.

Proof of proposition 4.1. We have

-c p ∂ 1 ρ p = -div(ρ p ∇φ p ), (4.5)
then

∆φ p = c p ∂ 1 η p -div(η p ∇φ p ),
and for any α

∈ N 2 ∆(∂ α φ p ) = c p ∂ 1 ∂ α η p -div(∂ α (η p ∇φ p )).
Thus, using elliptic estimates (see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]), there exists some constant C(q) such that

||∇(∂ α φ p )|| L q (R 2 ) ≤ C(q) ||∂ α η p || L q (R 2 ) + ||∂ α (η p ∇φ p )|| L q (R 2 ) ). (4.6)
Using (1.9) we have C(q) η p ∇φ p L q (R 2 ) ≤ 1 2 ∇φ p L q (R 2 ) so that from (4.6) we have

||∇φ p || L q (R 2 ) ≤ C(q)||η p || L q (R 2 ) . (4.7)
Next using Leibniz formula we obtain

||∂ α (η p ∇φ p )|| L q (R 2 ) ≤ C(q, α) ||∂ α η p || L q (R 2 ) ||∇φ p || L ∞ (R 2 ) + 0≤β<α ||∂ β η p || L ∞ (R 2 ) ||∂ α-β (∇φ p )|| L q (R 2 ) . (4.8)
Then, using (4.4) and (4.6), we have

||∂ α (∇φ p )|| L q (R 2 ) ≤ C(q, α) ||∂ α η p || L q (R 2 ) + 0≤β<α ||∂ β η p || L ∞ (R 2 ) ||∂ α-β (∇φ p )|| L q (R 2 ) . (4.9)
We observe that (as a direct consequence of the rescaling (3.2), (3.3)) there exists some positive constants C 1 (q, α), C 2 (q, α) and C 3 (q, α), such that

||∂ α N p || L q (R 2 ) = C 1 (q, α) ε 2+α1+2α2-3 q p ||∂ α η p || L q (R 2 ) , ||∂ α ∂ 1 θ p || L q (R 2 ) = C 2 (q, α) ε 2+α1+2α2-3 q p ||∂ α ∂ 1 φ p || L q (R 2 ) , ||∂ α ∂ 2 θ p || L q (R 2 ) = C 3 (q, α) ε 3+α1+2α2-3 q p ||∂ α ∂ 2 φ p || L q (R 2 ) .
Combining with (4.7) and (4.9), we obtain (4.1) and (4.2).

Remark 4.3. Assuming we have bounds for N p in W k,q for any k ∈ N, 1 < q ≤ ∞ then, by induction using proposition 4.1 and Sobolev embedding, we can bound θ p in W k,q for any k ∈ N, 1 < q ≤ ∞.

5 Convolution equation

Reformulation

The aim of this section is to rewrite equation (E-K) as a convolution equation for N p , similar to (2.12). The first two lemmas are direct computations.

Lemma 5.1. We have

∂ 1 N p -∂ 2 1 θ p = ε 2 p (L 1 (N p , θ p ) + R(N p , θ p )), (5.1
)

with L 1 (N p , θ p ) = 1 ε 2 p (1 -c p )∂ 1 N p + ∂ 2 2 θ p , R(N p , θ p ) = -γ∂ 1 (N p ∂ 1 θ p ) -γε 2 p ∂ 2 (N p ∂ 2 θ p ).
Lemma 5.2. The function N p and θ p satisfy

N p (x) -c p ∂ 1 (θ p (x)) = + ε 2 p γg (1) 2 N 2 p -ε 2 p γ 2 N 3 p l 1 + K 1 -γε 2 p N p (x) ∂ 2 1 N p K(1) + ε 2 p ∂ 2 2 N p K (1)
+ γ (∂ 1 θ p ) 2 2 + γ ε 2 p (∂ 2 θ p ) 2 2 -γ K 1 -γε 2 p N p (x) 2K(1) ε 2 p ∂ 1 N p 2 + ε 4 p ∂ 2 N p ) 2 , (5.2)
where l 1 is defined by

g(ρ p (x)) = -γε 2 p N p (x ) + ε 4 p g (1) 2 γ 2 N 2 p (x ) -γ 3 ε 6 p N 3 p (x )l 1 (-ε 2 p γN p (x )).
Remark 5.3. We recall that

x = ε p x 1 K(1) , ε 2 p x 2 K (1)
.

In view of (1.9), the function l 1 (-ε 2 p γN p (x )) = l 1 (ρ p (x) -1) is smooth and bounded in L ∞ independently of p. To simplify we write

g(ρ p (x)) = -γε 2 p N p (x ) + ε 4 p g (1) 2 γ 2 N 2 p (x ) -γ 3 ε 6 p N 3 p (x )l 1 (x ).
Now we combine these two lemmas to obtain a Kadomtsev-Petviashvili equation with some remainder. Proposition 5.4.

∂ 4 1 N p -∆N p -L(N p , θ p ) = -∂ 2 1 f p -ε 2 p i+j=2 ∂ i 1 ∂ j 2 R i,j εp , (5.3
)

with L(N p , θ p ) = -(2ε 2 p ∂ 2 1 ∂ 2 2 N p + ε 4 p ∂ 4 2 N p ), (5.4
)

1 γ R 2,0 εp (N p , θ p ) = c p -1 ε 2 p N p ∂ 1 θ p -γN 3 p l 1 + 1 2 (∂ 2 θ p ) 2 - K (ρ p ) K(1) (∂ 1 N p ) 2 -ε 2 p K (ρ p ) K(1) ∂ 2 N p ) 2 - h 1 γε 4 p h 2 ,
(5.5)

1 γ R 0,2 εp (N p , θ p ) = g (1) 2 N 2 p + ∂ 1 θ p 2 2 + ε 2 p -γN 3 p l 1 + (∂ 2 θ p ) 2 2 - K (ρ p )(∂ 1 N p ) 2 2K(1) -ε 4 p K (ρ p )(∂ 2 N p ) 2 2K(1) - 1 γε 2 p h 1 h 2 , (5.6
) 1 γ R 1,1 εp (N p , θ p ) = c p N p ∂ 2 θ p , (5.7)
f p = γ N p ∂ 1 θ p + g (1) 2 N 2 p + 1 2 (∂ 1 θ p ) 2 .
(5.8)

Where h 1 and h 2 are some functions defined later.

Remark 5.5. We observe that

γ 1 + g (1) 2 + 1 2 = γ 3 + g (1) 2 = 1 2 .
(5.9)

Then passing, formally in (5.3), to the limit in p we have

∂ 4 1 N 0 -∆N 0 + 1 2 ∂ 2 1 (N 2 0 ) = 0, (5.10)
i.e. N 0 is a solution of (KPI).

Proof. Let 1 K(1) h 1 (x) = 1 K 1 -γε 2 p N p (x) - 1 K(1) = h 3 (-ε 2 p γN p (x))ε 2 p N p (x), (5.11)
and

h 2 = N p -c p ∂ 1 θ p -ε 2 p γ g (1) 2 N 2 p -γ 2 ε 2 p N 3 p l + γ(∂ 1 θ p ) 2 2 + γ ε 2 p (∂ 2 θ p ) 2 2 -γ K 1 -ε 2 p N p 2K(1) ε 2 p ∂ 1 N p 2 + ε 4 p ∂ 2 N p ) 2 .
(5.12)

First of all, multiplying (5.2) by K(1)

1

K(1) + 1 K 1-γε 2 p Np -1 K(1)
we obtain

N p -c p ∂ 1 θ p = +ε 2 p γ g (1) 2 N 2 p -γ 2 ε 2 p N 3 p l + ∂ 2 1 N p + ε 2 p ∂ 2 2 N p + γ (∂ 1 θ p ) 2 2 + γ ε 2 p (∂ 2 θ p ) 2 2 -γ K 1 -γε 2 p N p 2K(1) ε 2 p ∂ 1 N p 2 + ε 4 p ∂ 2 N p ) 2 -h 1 h 2 .
(5.13)

We have computing

-(∂ 2 1 + ε 2 p ∂ 2
2 )(5.13) + c∂ 1 (5.1)

∂ 4 1 N p -∆N p = -∂ 2 1 γ N p ∂ 1 θ p + g (1) 2 N 2 p + 1 2 (∂ 1 θ p ) 2 + L(N p , θ p ) -ε 2 p i+j=2 ∂ i 1 ∂ j 2 R i,j εp , (5.14)
this completes the proof of Claim (5.3).

We now recast (5.3) as a convolution equation.

Proposition 5.6. Let

K i,j εp (ξ) = ξ i 1 ξ j 2 |ξ| 2 + ξ 4 1 + 2ε 2 p ξ 2 1 ξ 2 2 + ε 4 p ξ 4 2 , (5.15)
we have

N p = K 2,0 εp * f p + i+j=2 ε 2 p K i,j εp * R i,j εp .
(5.16)

Proof. It is a direct computation.

Next we define

Q(ξ) = |ξ| 2 + ξ 4 1 + 2ε 2 p ξ 2 1 ξ 2 2 + ε 4 p ξ 4 2 .
(5.17)

We need to etablish that the remainder term R εp are small enough in some Sobolev spaces.

Proposition 5.7. There exists some positive constant C, not depending on p, such that

R 2 |R 1,1 εp | ≤ C, R 2 |f p | ≤ C, (5.18) R 2 |R 0,2 εp | ≤ C, R 2 |R 2,0 εp | ≤ C ε 2 p . ( 5

.19)

Proof. Proposition 5.7 is a direct consequence of (1.9), identities in subsection 3.3, estimates in subsection 3.2, (3.9) and Hölder's inequality. For example we have

R 2 ∂ 2 θ p 2 = 1 K(1)γ 2 ε 3 p R 2 (∂ 2 φ p ) 2 ≤ C, thus R 2 |R 1,1 εp | ≤ C N p L 2 (R 2 ) ∂ 2 θ p L 2 (R 2 ) ≤ C.
Similarly we obtain the estimate on f p and (5.19).

Kernel estimates

In order to use (5.16) we need to control K i,j εp . In this section we use computations of [START_REF] Béthuel | On the KP I transonic limit of two-dimensional Gross-Pitaevskii travelling waves[END_REF] section 5.

Proposition 5.8. ([3] lemma 5.1) Let 0 ≤ s < 1, there exists a constant C(s) depending possibly on s, but not on p, such that

||K 2,0 εp || Ḣs (R 2 ) ≤ C(s)(1 + ε 1 2 -2s p ), ||K 1,1 εp || Ḣs (R 2 ) ≤ C(s)(1 + ε -1 2 -2s p ), (5.20) ||K 0,2 εp || Ḣs (R 2 ) ≤ C(s)(1 + ε -3 2 -2s p
).

(5.21)

Thus we have

||K 2,0 εp || H s (R 2 ) + ε p ||K 1,1 εp || H s (R 2 ) + ε 2 p ||K 0,2 εp || H s (R 2 ) ≤ C(s), (5.22)
for any 0 ≤ s ≤ 1 4 . Proposition 5.9. ([3] lemma 5.2) let 1 < q < ∞ and 0 ≤ i, j ≤ 4 integers such that 2 ≤ i + j ≤ 4 we denote by

κ i,j = max {i + 2j -4, 0} .
Then there exists a constant C(q), depending possibly on q, but not on p such that

||K i,j εp * f || L q (R 2 ) ≤ C(q) ε κi,j p ||f || L q (R 2 ) ,
for any f ∈ L q (R 2 ) and ε p > 0.

Remark 5.10. As a consequence N p and all its derivatives, belong to L q (R 2 ) for any 1 < q < 2. Indeed, we have

N p = K 2,0 εp * f p + i+j=2 ε 2 p K i,j εp * R i,j εp .
(5.23)

Thus, using lemma 4.2 we obtain

∂ α N p = K 2,0 εp * ∂ α f p + i+j=2 ε 2 p K i,j εp * ∂ α R i,j εp .
Then, combining the definition of R i,j εp , lemma 4.2 with the proposition above, we have ∂ α N p ∈ L q for any 1 < q < 2. Therefore using (4.1), ∂ 1 θ p and ∂ 2 θ p belong to L q (R 2 ) for any 1 < q < 2. Finally (5.2) and (5.1) give, respectively, ∂ 1 θ p ∈ W k,q and ∂ 2 θ p ∈ W k,q for any k ∈ N.

Bounds in Sobolev spaces

This section is devoted to the proof by induction of the following proposition: Proposition 6.1. There exists p 0 > 0 such that for any 0 ≤ p ≤ p 0 , α ∈ N 2 and 1 < q < ∞, there exists C(q, α) such that

||∂ α N p || L q (R 2 ) + ||∂ 1 ∂ α N p || L q (R 2 ) + ||∂ 2 ∂ α N p || L q (R 2 ) +||∂ 2 1 ∂ α N p || L q (R 2 ) + ε p ||∂ 1 ∂ 2 ∂ α N p || L q (R 2 ) + ε 2 p ||∂ 2 2 ∂ α N p || L q (R 2 ) ≤ C(q, α). (6.1)
We have the following consequence.

Theorem 5. There exists p 0 > 0 such that for any 0 ≤ p ≤ p 0 , k ∈ N and 1 < q ≤ ∞ there exists C(k, q) such that

||N p || W k,q (R 2 ) + ||∂ 1 θ p || W k,q (R 2 ) + ε p ||∂ 2 θ p || W k,q (R 2 ) ≤ C(k, q). (6.2)
Proof of theorem 5 assuming proposition 6.1. By proposition 6.1, we have for any k ∈ N, 1 < q < ∞, there exists a constant C(k, q) not depending on p such that ||N p || W k,q (R 2 ) ≤ C(k, q). (6.3) Thus using Sobolev embedding (2.7), we obtain

∀k ∈ N, ||N p || C k 0 (R 2 ) ≤ C(k, q). (6.4)
By proposition 4.1 we have

Γ p (q, α) ≤ C(q, α) 1 + ε 2 p 0≤β<α Γ p (q, α -β) . (6.5)
Combining (4.1) and (6.3) , the term Γ p (q, (0, 0)) is bounded independently of p. Then by induction and (6.5) the quantity Γ p (q, α) is bounded independently of p, for any 1 < q < ∞, α ∈ N 2 . Finally, using Sobolev embedding (2.7) this result is also true for q = ∞.

Preliminary We begin the proof of proposition 6.1. First of all, we have: Lemma 6.2. For any 1 < q < ∞, there exists a constant C(q), independent of p, such that

||N p (∂ 1 θ p ) 2 || L q (R 2 ) + ε 2 p ||N p (∂ 2 θ p ) 2 || L q (R 2 ) ≤ C(q)||N p || 3 L 3q (R 2 ) , (6.6)
and

||(∂ 1 θ p ) 2 || L q (R 2 ) + ε p ||N p ∂ 2 θ p || L q (R 2 ) + ε 2 p ||(∂ 2 θ p ) 2 || L q (R 2 ) + ||N p ∂ 1 θ p || L q (R 2 ) ≤ C(q)||N p || 2 L 2q (R 2 ) . (6.7)
Proof. Hölder inequality and (4.1) leads to

||N p (∂ 1 θ p ) 2 || L q (R 2 ) + ε 2 p ||N p (∂ 2 θ p ) 2 || L q (R 2 ) ≤ C(q)||N p || L 3q (R 2 ) ||∂ 1 θ p || 2 L 3q (R 2 ) + ε 2 p ||∂ 2 θ p || 2 L 3q (R 2 ) ≤ C(q)||N p || 3 L 3q (R 2 ) ,
a similar computations gives us (6.7).

Lemma 6.3. For any 1 < q < ∞, there exists a constant C(q), such that

h 1 h 2 ε 2 p L q (R 2 ) ≤ C(q) ||N p || 2 L 2q (R 2 ) + ε 2 p ||N p || 3 L 3q (R 2 ) + ε 4 p ||N p || L ∞ (R 2 ) ||∂ 1 N p || 2 L 2q (R 2 ) + ε 6 p ||N p || L ∞ (R 2 ) ||∂ 2 N p || 2 L 2q (R 2 ) , (6.8
)

||f p || L q (R 2 ) + ε p ||R 1,1 εp || L q (R 2 ) ≤ C(q)||N p || 2 L 2q (R 2 ) , (6.9)
||R 2,0 εp || L q (R 2 ) ≤ C(q) ε 2 p ||N p || 2 L 2q + ε 2 p ||N p || 3 L 3q + ε 2 p ||∂ 1 N p || 2 L 2q + ε 4 p ||∂ 2 N p || 2 L 2q + h 1 (x)h 2 (x) ε 2 p L q (R 2 ) , (6.10)
||R 0,2 εp || L q (R 2 ) ≤ C(q) ||N p || 2 L 2q (R 2 ) + ε 2 p ||N p || 3 L 3q (R 2 ) + ε 2 p ||∂ 1 N p || 2 L 2q (R 2 ) + ε 4 p ||∂ 2 N p || 2 L 2q (R 2 ) + h 1 (x)h 2 (x) ε 2 p L q (R 2 ) . ( 6

.11)

Where h 1 , h 2 are defined in proposition 5.4.

Proof. The proof is a direct computation combined with lemma 6.2, estimate (1.9) and observing that

(h 1 h 2 ) ε 2 p L q (R 2 ) ≤ C(q) N 2 p L q (R 2 ) + N p ∂ 1 θ p L q (R 2 ) + ε 2 p N 3 p L q (R 2 ) + ε 2 p N p (∂ 1 θ p ) 2 L q (R 2 ) + ε 4 p N 4 p L q (R 2 ) + ε 4 p (∂ 2 θ p ) 2 N p L q (R 2 ) + ε 4 p (∂ 1 N p ) 2 N p L q (R 2 ) + ε 6 p (∂ 2 N p ) 2 N p L q (R 2 ) .
Lemma 6.4. For any 1 < q < ∞, there exists a constant C(q), such that

||N p || L q (R 2 ) + ||∂ 1 N p || L q (R 2 ) + ||∂ 2 N p || L q (R 2 ) +||∂ 2 1 N p || L q (R 2 ) + ε p ||∂ 1 ∂ 2 N p || L q (R 2 ) + ε 2 p ||∂ 2 2 N p || L q (R 2 ) ≤ C(q) ||N p || 2 L 2q (R 2 ) + ε 2 p ||N p || 3 L 3q (R 2 ) + ε 2 p ||∂ 1 N p || 2 L 2q (R 2 ) + ε 4 p ||∂ 2 N p || 2 L 2q (R 2 ) + ε 4 p ||∂ 1 N p || 2 L 2q (R 2 ) ||N p || L ∞ (R 2 ) + ε 6 p ||∂ 2 N p || 2 L 2q (R 2 ) ||N p || L ∞ (R 2 ) . (6.12)
Proof. First we observe that

∂ α K j,k εp = ı α1+α2 K j+α1,k+α2 εp , (6.13)
for any |α| ≤ 2. Applying the operator ∂ α to (5.16) we obtain

||∂ α N p || L q (R 2 ) ≤ ||∂ α K 2,0 εp * f p || L q (R 2 ) + ε 2 p i+j=2 ||∂ α K i,j εp * R i,j εp || L q (R 2 ) , (6.14)
then, with proposition 5.9, we deduce

||N p || L q (R 2 ) ≤ ||K 2,0 εp * f p || L q (R 2 ) + i+j=2 ε 2 p ||K i,j εp * R i,j εp || L q (R 2 ) ≤ C(q) ||f p || L q (R 2 ) + ε 2 p i+j=2 ||R i,j εp || L q (R 2 ) , ||∂ 1 N p || L q (R 2 ) ≤ ||∂ 1 K 2,0 εp * f p || L q (R 2 ) + i+j=2 ε 2 p ||∂ 1 K i,j εp * R i,j εp || L q (R 2 ) ≤ C(q) ||f p || L q (R 2 ) + ε 2 p ||R 2,0 εp || L q (R 2 ) + ||R 1,1 εp || L q (R 2 ) + ε p ||R 0,2 εp || L q (R 2 ) .
Similar estimates hold for

∂ 2 N p , ∂ 2 1 N p , ∂ 1 ∂ 2 N p and ∂ 2 2 N p .
Finally using lemma 6.3, we obtain lemma 6.4. Lemma 6.5. Let 2 ≤ q ≤ ∞, there exists a positive constant C(q), depending possibly on q, but not on p, such that

||∂ 1 N p || L q (R 2 ) + ε p ||∂ 2 N p || L q (R 2 ) ≤ C(q)ε 6 q -3 p , (6.15)
and

||N p || L q (R 2 ) ≤ C(q), ∀2 ≤ q ≤ 8 3 . (6.16)
Moreover, for 8 3 < q < 8, there exists C(q) such that the following bound holds.

ε 2 3 p ||N p || L q (R 2 ) ≤ C(q). (6.17)
Proof. Using (4.4), we have

||∂ 1 N p || L ∞ (R 2 ) ≤ C ε 3 p , and
||∂ 2 N p || L ∞ (R 2 ) ≤ C ε 4 p . (6.18)
Combining with (3.18) and (3.22) we obtain (6.15) using standard interpolations inequalities. Applying (2.8) on (5.16) for any 0 ≤ s ≤ 1 4 we obtain

||N p || H s (R 2 ) ≤||K 2,0 εp || H s (R 2 ) ||f p || L 1 (R 2 ) + ε 2 p ||R 2,0 εp || L 1 (R 2 ) + ε 2 p ||R 1,1 εp || L 1 (R 2 ) ||K 1,1 εp || H s (R 2 ) + ε 2 p ||R 0,2 εp || L 1 (R 2 ) ||K 0,2 εp || H s (R 2 ) . (6.19)
By proposition 5.7, lemma 6.3 and claim (5.22), we deduce

||N p || H s (R 2 ) ≤ C(s). (6.20)
Thus, by Sobolev embedding (2.6) we have

||N p || L q (R 2 ) ≤ C(q), ∀2 ≤ q ≤ 8 3 . (6.21)
Let ν > 0 combining (6.16) and (6.15) there exists a constant C(ν) and q > 2 such that

||N p || W 1,q (R 2 ) ≤ C(ν) 1 + ε -1-ν p .
Thus, by Sobolev embedding (2.2), we obtain

||N p || L ∞ (R 2 ) ≤ C(ν) 1 + ε -1-ν p .
Combining with (6.16) we have (6.17) by interpolation between Lebesgue spaces.

We are now able to prove the first step of our induction.

The first step Let 1 < q < ∞, there exists a constant C(q), such that

||N p || L q (R 2 ) + ||∂ 1 N p || L q (R 2 ) + ||∂ 2 N p || L q (R 2 ) + ||∂ 2 1 N p || L q (R 2 ) + ε p ||∂ 1 ∂ 2 N p || L q (R 2 ) + ε 2 p ||∂ 2 2 N p || L q (R 2 ) ≤ C(q). (6.22)
Proof. First of all for any 1 < q ≤ 4 3 we have, using lemma 6.4, inequalities (6.17), (6.15) and (6.16),

||N p || L q (R 2 ) + ||∂ 1 N p || L q (R 2 ) + ||∂ 2 N p || L q (R 2 ) + ||∂ 2 1 N p || L q (R 2 ) + ε p ||∂ 1 ∂ 2 N p || L q (R 2 ) + ε 2 p ||∂ 2 2 N p || L q (R 2 ) ≤ C(q). (6.23)
Then, by Sobolev embedding (2.1) and standard interpolations inequalities,

||N p || L q (R 2 ) + ε p ||∂ 1 N p || L q (R 2 ) + ε 2 p ||∂ 2 N p || L q (R 2 ) ≤ C(q), (6.24)
for any 1 < q ≤ 4. Then, for any 1 < q ≤ 2 by lemma 6.4, we obtain

||N p || L q (R 2 ) + ||∂ 1 N p || L q (R 2 ) + ||∂ 2 N p || L q (R 2 ) + ||∂ 2 1 N p || L q (R 2 ) + ε p ||∂ 1 ∂ 2 N p || L q (R 2 ) + ε 2 p ||∂ 2 2 N p || L q (R 2 ) ≤ C(q). (6.25)
We deduce, by Sobolev embedding (2.1), that

||N p || L q (R 2 ) + ε p ||∂ 1 N p || L q (R 2 ) + ε 2 p ||∂ 2 N p || L q (R 2 ) ≤ C(q), (6.26)
∀ 1 < q < ∞. Finally using lemma 6.4 we obtain claim (6.22).

Inductive step

We fix k ∈ N. In this part we assume that (6.1) is true for any 1 < q < ∞ and any α ∈ N 2 such that |α| ≤ k. We will prove that (6.1) holds for any |α| ≤ k + 1.

Lemma 6.6. There exists p 0 > 0 such that for any 0 ≤ p ≤ p 0 , 1 < q < ∞ and α ∈ N 2 such that |α| ≤ k + 1, then there exists a constant C(q, α) such that

||∂ α ∂ 1 θ p || L q (R 2 ) + ε p ||∂ α ∂ 2 θ p || L q (R 2 ) ≤ C(q, α). (6.27)
Proof. Let α ∈ N 2 , |α| ≤ k + 1. Using Sobolev embedding (2.3) and our hypothesis, we have

||N p || C k 0 (R 2 ) ≤ C(k), (6.28)
where C(k) is a constant not depending on p. Then, by (4.2), we obtain

||∂ α ∂ 1 θ p || L q (R 2 ) + ε p ||∂ α ∂ 2 θ p || L q (R 2 ) ≤ C(q, α) ||∂ α N p || L q (R 2 ) + ε 2 p 0≤β<α ||∂ α-β ∂ 1 θ p || L q (R 2 ) + ε p ||∂ α-β ∂ 2 θ p || L q (R 2 ) . (6.29)
If we denote

S q k = |α|≤k+1 ||∂ α ∂ 1 θ p || L q (R 2 ) + ε p ||∂ α ∂ 2 θ p || L q (R 2 ) ,
summing the previous inequalities, we obtain

S q k ≤ C(q, α) ε 2 p S q k + |α|≤k+1 ||∂ α N p || L q (R 2 ) .
We conclude by using (6.1).

Lemma 6.7. There exists a positive constant C(q, α) depending possibly on q and α, but not on p, such that

||∂ α f p || L q (R 2 ) + ||∂ α R 0,2 εp || L q (R 2 ) + ε p ||∂ α R 1,1 εp || L q (R 2 ) + ε 2 p ||∂ α R 2,0 εp || L q (R 2 ) ≤ C(q, α), (6.30)
for any 1 ≤ q < ∞ , |α| ≤ k + 1.

Proof. We will detail the computation for h 1 h 2 defined in proposition 5.4. First of all we have using (5.11), (5.12)

∂ α (h 1 h 2 ) ε 2 p L q (R 2 ) ≤ C ∂ α (δN 2 p ) L q (R 2 ) + ∂ α (δN p ∂ 1 θ p ) L q (R 2 ) + ε 2 p ∂ α (δN 3 p ) L q (R 2 ) + ε 2 p ∂ α (N p (∂ 1 θ p ) 2 δ) L q (R 2 ) + ε 4 p ∂ α (N 4 p lδ) L q (R 2 ) + ε 4 p ∂ α ((∂ 2 θ p ) 2 N p δ) L q (R 2 ) + ε 4 p ∂ α ((∂ 1 N p ) 2 N p δK ) L q (R 2 ) + ε 6 p ∂ α ((∂ 2 N p ) 2 N p δK ) L q (R 2 ) ,
where δ = δ(-ε 2 p γN p ), K = K (-ε 2 p γN p ), l = l(-ε 2 p γN p ). Using our hypothesis and the chain rule we have for any

β ∈ N 2 , |β| ≤ k + 1, 1 < q < ∞ ∂ β (K (-ε 2 p γN p )) L q (R 2 ) + ∂ β (δ(-ε 2 p γN p )) L q (R 2 ) + ∂ β (l(-ε 2 p γN p )) L q (R 2 ) ≤ C.
Thus using Leibniz formula, our hypothesis and (6.27) we have

∂ α (h 1 h 2 ) ε 2 L q (R 2 ) ≤ C.
Applying the operator ∂ α on (5.5), (5.6), (5.7) and using our hypothesis and (6.27) we obtain claim (6.30).

Finally let |α| ≤ k + 1 using (5.16) we have

∂ α N p = K 2,0 εp * ∂ α f p + i+j=2 ε 2 p K i,j εp * ∂ α R i,j εp . (6.31)
Then using proposition 5.9, there exists C(q) such that

||∂ α N p || L q (R 2 ) ≤ C(q) ||∂ α f p || L q (R 2 ) + i+j=2 ε 2 p ||∂ α R i,j εp || L q (R 2 ) , , (6.32
)

||∂ α ∂ 1 N p || L q (R 2 ) ≤ C(q) ||∂ α f p || L q (R 2 ) + ε 2 p ||∂ α R 2,0 εp || L q (R 2 ) + ||∂ α R 1,1 εp || L q (R 2 ) + ε p ||∂ α R 0,2 εp || L q (R 2 ) , (6.33)
and

||∂ α ∂ 2 N p || L q (R 2 ) + ||∂ α ∂ 2 1 N p || L q (R 2 ) + ε p ||∂ α ∂ 1 ∂ 2 N p || L q (R 2 ) + ε 2 p ||∂ α ∂ 2 2 N p || L q (R 2 ) ≤ C(q) ||∂ α f p || L q (R 2 ) + ε 2 p ||∂ α R 2,0 εp || L q (R 2 ) + ε p ||∂ α R 1,1 εp || L q (R 2 ) + ||∂ α R 0,2 εp || L q (R 2 ) . (6.34)
We conclude the induction by using (6.30).

Conclusion

We have proved proposition 6.1, theorem 5 follows from our discussion at the beginning of the section.

7 Strong convergence

Strong local convergence

Proposition 7.1. Let (p n ) n∈N such that p n → 0. Then there exists N 0 a non constant solution of (SW) such that, up to a subsequence,

∀1 < q < ∞, ∀k ∈ N, N pn N 0 in W k,q R 2 , when n → ∞. (7.1)
Thus for any 0 ≤ γ < 1 and any compact subset K, we obtain

N pn → N 0 in C 0,γ (K), when n → ∞. (7.2)
Proof. Combining (6.2) with Banach-Alaoglu theorem, there exists a subsequence (p n ) n∈N and a function N 0 such that N pn N 0 in W k,q R 2 for any 1 < q < ∞, k ∈ N. Then (7.2) is a consequence of (7.1) and (2.4). Thus N 0 is non constant using (3.4). We will now prove that N 0 is a solution of (SW). We recall that

f p = γ N p ∂ 1 θ p + g (1) 2 N 2 p + 1 2 (∂ 1 θ p ) 2 , (7.3)
with

γ 1 + g (1) 2 + 1 2 = γ 3 + g (1) 2 = 1 2 , (7.4)
and

∂ 4 1 N p -∆N p = -∂ 2 1 f p + L(N p , θ p ) -ε 2 p i+j=2 ∂ i 1 ∂ j 2 R i,j εp . (7.5)
Using (5.4) and theorem 5 we have

L(N p , θ p ) L 2 (R 2 ) -→ n→∞ 0.
Using (5.6) and theorem 5 we have

ε 2 p ||∂ 2 1 R 2,0 εp || L 1 (R 2 ) -→ n→∞ 0,
then using (6.30) we have

ε 2 p ||∂ 2 2 R 0,2 εp || L 1 (R 2 ) + ε 2 p ||∂ 1 ∂ 2 R 1,1 εp || L 1 (R 2 ) -→ n→∞ 0.
On the other hand, using (7.2), we have

N 2 pn -N 2 0 L ∞ (K) -→ n→∞ 0, for any compact K. Thus N 2 pn -→ n→∞ N 2 0 in D (R 2 ).
Combining (3.7) and

||(∂ 1 θ p ) 2 -N 2 p || L 1 (R 2 ) ≤ ||∂ 1 θ p -N p || L 2 (R 2 ) ||∂ 1 θ p + N p || L 2 (R 2 ) , we obtain ||N ∂ 1 θ p -N 2 p || L 1 (R 2 ) -→ p→0 0, ||(∂ 1 θ p ) 2 -N 2 p || L 1 (R 2 ) -→ p→0 0.
Thus, we deduce

1 2 N 2 p -f p L 1 (R 2 ) ≤ |γ|||N p ∂ 1 θ p -N 2 p || L 1 (R 2 ) + |γ| 2 ||(∂ 1 θ p ) 2 -N 2 p || L 1 (R 2 ) -→ p→0 0.
Finally passing to the limit in (7.5) we have

∂ 4 1 N 0 -∆N 0 + 1 2 ∂ 2 1 (N 2 0 ) = 0. ( 7.6)
We will now prove the strong global convergence.

Strong global convergence

We recall that

E(ρ p , φ p ) = K(1)γ 2 ε p 2 E 0 (N p , θ p ) + ε 2 p (E 2 (N p , θ p ))) + ε 4 p (E 4 (N p , θ p )) , with E 0 (N p , θ p ) = R 2 N 2 p + ∂ 1 θ 2 p , E 2 (N p , θ p ) = 2 R 2 (∂ 1 N p (x)) 2 2 + |∂ 2 θ p (x)| 2 2 - γ 6 3N p (∂ 1 θ p ) 2 + g (1)N 3 p )dx , E 4 (N p , θ p ) = R 2 K 1 -γε 2 p N p (x) 1 K(1) 2 |∂ 2 N p (x)| 2 -γN p (x)|∂ 2 θ p (x)| 2 + N p (x)j(x) (∂ 1 N p (x)) 2 + ε 2 p (∂ 2 N p (x)) 2 + γ 4 N 4 p (x)l(x)dx, and
E KP (∂ 1 θ p ) = 1 2 R 2 (∂ 2 1 θ p ) 2 + 1 2 R 2 (∂ 2 θ p ) 2 - 1 6 R 2 (∂ 1 θ p ) 3 .
Proposition 7.2. Let (p n ) n∈N a sequence which converges to 0 and satisfies (7.1) and (7.2). Then, up to a subsequence, there exists a positive constant µ 0 such that

E KP (∂ 1 θ pn ) → E KP (µ 0 ), and
R 2 |∂ 1 θ pn | 2 -→ n→∞ µ 0 > 0. (7.7)
See section 2 for the definition of E KP .

First of all, we prove three lemmas.

Lemma 7.3. There exists a constant C, not depending on p, such that

E(ρ p , φ p ) -p ≤ -1 (K(1)γ 2 ) 2 54S 2 KP p 3 + Cp 4 . (7.8)
Proof. Let ω a ground state of (SW) and p > 0. According to [START_REF] De | Solitary waves of generalized Kadomtsev-Petviashvili equations[END_REF][START_REF] De | Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves[END_REF] ω is smooth, belongs to L q (R 2 ) for any q > 1 its first order derivatives are in L 2 (R 2 ).

There exists v such that ∂ 1 v = ω (see lemma 3.9 [START_REF] Béthuel | Travelling waves for the Gross-Pitaevskii equation[END_REF] and [START_REF] Gravejat | Asymptotics of the solitary waves for the generalised Kadomtsev-Petviashvili equations[END_REF]) with v smooth. Moreover v is in L q (R 2 ), q > 2 and its gradient belongs to L q (R 2 ), q > 1. We let

ρ(x 1 , x 2 ) = 1 -ε 2 γω ε x 1 K(1) , ε 2 x 2 K(1) , (7.9
) φ(x 1 , x 2 ) = -γ K(1)εv ε x 1 K(1) , ε 2 x 2 K(1) , (7.10)
where the constant ε is chosen such that

p = K(1)γ 2 ε R 2 ω 2 .
We observe that ρ ∈ 1 + H 1 (R 2 ) and φ ∈ Ḣ1 (R 2 ). Thus, by computation, we have

E(ρ, φ) = K(1)γ 2 ε R 2 ω 2 + K(1)γ 2 ε 3 E KP (ω) + K(1)γ 2 2 ε 5 E 4 (ω, v).
Then, using lemma 2.1, we obtain

E(ρ, φ) -p ≤ K(1)γ 2 ε 3 E KP (ω) + Cε 5 ≤ K(1)γ 2 p K(1)γ 2 ω 2 3 E KP (ω) + Cp 4 ≤ p 3 (K(1)γ 2 ) 2 E KP (ω) ( ω 2 ) 3 + Cp 4 ≤ p 3 (K(1)γ 2 ) 2 -1 54S 2 KP + Cp 4 .
For p 1, since ω ∈ L ∞ (R 2 ) we have |ρ -1| 1 and |ρ p -1| 1. Thus using (1.11) we have

E(ρ p , φ p ) -p = E(ρ p , φ p ) -p ≤ E(ρ, φ) -p = E(ρ, φ) -p ≤ p 3 (K(1)γ 2 ) 2 -1 54S 2 KP + Cp 4 . Lemma 7.4. We have E(N p , θ p ) -p = K(1)γ 2 ε 3 p E Kp ∂ 1 θ p + o(ε 3 p ), (7.11)
and Using lemma 2.1

- 1 54S 2 KP R 2 (∂ 1 θ p ) 2 3 ≤ E KP (∂ 1 θ p ) ≤ - 1 54S 2 KP R 2 (∂ 1 θ p ) 2 3 + o(1). ( 7
E KP (∂ 1 θ p ) ≥ E KP min R 2 (∂ 1 θ p ) 2 = - 1 54S 2 KP R 2 (∂ 1 θ p ) 2 3 . (7.16)
Then combining (7.11), lemma 7.3 and (7.13) we obtain

E KP (∂ 1 θ p ) ≤ - p 3 (K(1)γ 2 ) 3 54S 2 KP 1 ε 3 + o(1) ≤ -1 54S 2 KP R 2 N p ∂ 1 θ p 3 + o(1) ≤ -1 54S 2 KP R 2 (∂ 1 θ p ) 2 3 + o(1).
This ends the proof of lemma 7.4.

We are now able to prove proposition 7.2.

Proof of proposition 7.2 . Using (7.1) and (7.13) we have Using proposition 7.1 we have ∀1 < q < ∞, ∀k ∈ N, N pn (. -a n ) N 0 in W k,q R 2 , when n → ∞, with N 0 a solution of (SW). Thus by (7.24) N 0 = ∼ N 0 , and ∼ N 0 is a ground state of speed 1. We will now prove the convergence of N pn and ∂ 1 θ pn . Using the continuity of the translation in L q (R 2 ) for any 1 ≤ q < ∞, if a n → a then |f (x + a n )| 2 dx -→ n→∞ 0, (7.28) this concludes the proof of proposition 7.5.

lim inf n→∞ R 2 (∂ 1 θ pn ) 2 ≥ R 2 N 2 0 , (7.17
Proof of theorem 2. Combining proposition 7.5 (L 2 convergence), theorem 5 (boundedness in W k,q ) and an interpolation argument we have

N pn -→ n→∞ N 0 in W k,q (R 2 ). (7.29)
Since the embedding Y (R 2 ) → L 2 (R 2 ) is continuous the same argument gives

∂ 1 θ pn -→ n→∞ N 0 in W k,q (R 2 ). (7.30)
A Soliton in the one dimensional case

We begin with some reminders about nonlinear Schrödinger equations in dimension 1. Traveling wave solutions to (TWc) are related to the soliton of the generic Korteweg-de Vries equation (see [START_REF] Béthuel | Existence and properties of travelling waves for the Gross-Pitaevskii equation[END_REF]). For traveling waves solution to (NLS) the transonic limit can lead to solitons of the modified (KdV) equation or even the generalized (KdV) (see [START_REF] Chiron | Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one[END_REF]). We show in this section that traveling waves of (E-K) exhibit similar properties. The existence of traveling waves in dimension one for the Euler-Korteweg equation follows from basic ode arguments that we sketch here. The Euler-Korteweg system, in dimension one, reads

     -cρ + (ρu) = 0, (1)
-cu + u 2 2 + g = K(ρ)ρ + 1 2 K (ρ)ρ 2 (

Proposition 1 . 4 .

 14 Under the conditionsg(1) = 0, g (1) = 1, Γ = 0.there exists (ρ, u) global solution of

∼ N 0 with speed σ = µ 2 0

 2 (µ * ) 2 , such that ∂ 1 θ pn (. -a n ) -→ n→∞ ∼ N 0 in Y (R 2 ).(7.23)So by (7.13) we haveN pn (. -a n ) -→ n→∞ ∼ N 0 in L 2 (R 2).(7.24)

∂ 1 θ 1 )

 11 pn (. -a) -→ n→∞ N 0 in Y (R 2 ), and N pn (. -a) -→ n→∞ N 0 in L 2 (R 2 ).Thus, it is sufficient to prove that (a n ) n∈N is bounded. By contradiction assume, up to an extraction, that (a n ) n∈N satisfiesa n -→ n→∞ ∞.Then combining (7.2) and (1.10) there exists C > 0 not depending on n such that B(0,1)N 2 pn > 2C,(7.25)thus by(7.24) we haveB(0,1) |N 0 (x + a n ) -N pn (x)| 2 dx -→ |N 0 (x + a n )| 2 dx ≥ C,(7.27)which is absurd since for any f ∈ L 2 we have B(0,1)

  2), x ∈ R. (E-K) We assume g(1) = 0, g (1) = 1 , Γ = g (1) + 3 = 0, γ = 1 g (1) + 3 , c ∈]0, 1[. (A.1)

Figure 3 : 2 -

 32 Figure 3: K=1, Γ = 0, ε=0.5, G(x) = (x-1) 2 2 -(x-1) 3 2 .

  N p -∂ 2 1 θ p || L 2 (R 2 ) ≤ M ε p . + (∂ 1 θ p ) 3 ) + γ -N p (∂ 1 θ p ) 2 + (∂ 1 θ p ) 3 )dx + K(1)γ 2 ε 5 p 2 E 4 (N p , θ p ). Thus E(N p , θ p ) -p = K(1)γ 2 ε 3 p E KP (∂ 1 θ p ) + o(ε 3 p ). (7.15)

	and
			||∂ 1 (7.14)
	Then, we have
	E(N p , θ p ) -p =	K(1)γ 2 ε p 2	R 2	N p -∂ 1 θ p	2 + K(1)γ 2 ε 3 p E KP (∂ 1 θ p )
	+	K(1)γ 2 ε 3 p 2	R 2	(∂ 1 N p (x)) 2 -(∂ 2 1 θ p ) 2 )dx+
		+	γg (1) 3	(-N 3 p
						.12)
	Proof. Combining (5.2) and (7.1) we obtain
			1 ε 2 p R 2		p→0	0,	(7.13)

(N p -∂ 1 θ p ) 2 -→

  Combining lemma 7.4,(7.18) and lemma 2.1 we obtainE KP (∂ 1 θ pn ) → E KPmin (µ 0 ), andR 2 |∂ 1 θ pn | 2 → µ 0 > 0, when n → ∞.We can now obtain global convergence thanks to proposition 2.2.Proposition 7.5. Let (p n ) n∈N a sequence which converges to 0. Up to an extraction, there exists a ground state N 0 such that∂ 1 θ pn → N 0 in Y (R 2 ),andN pn → N 0 in L 2 (R 2 ), when n → ∞. (7.21)Proof. Using proposition 7.2 there exists a constant µ 0 , such that, up to a subsequence,E KP (∂ 1 θ pn ) → E KP min (µ 0 ),andR 2|∂ 1 θ pn | 2 → µ 0 > 0, when n → ∞.

					(7.20)
					)
	thus, up to extraction,
	lim n→∞ R 2	(∂ 1 θ pn ) 2 → µ 0 , when n → ∞,	(7.18)
	where
		µ 0 ≥	R 2	N 2 0 > 0.	(7.19)

(7.22)

Then using proposition 2.2, there exists (a n ) n∈N and a ground state

We will study solitons whose limits are ρ ±∞ = 1; ρ ±∞ = 0, ρ ±∞ = 0, u ±∞ = 0; u ±∞ = 0, u ±∞ = 0.

(A.2)

As in dimension two, we let ε = 1 -c 2 .

(A.

Multiplying by ρ and integrating on [x, +∞[, we have

where G is a primitive of g such that G(1) = 0.

The case Γ = 0 We define the function

and

Then we have:

Proof. The lemma is a direct consequence of the intermediate value theorem and the fact that

Remark A.2. For ε 1,

To continue our reasoning, we need informations on the derivative.

.

Indeed, we will prove, if γ > 0, that the soliton decreases from 1 to ρ m,ε between ] -∞, 0] then increases between [0, ∞[. If γ < 0, the soliton increases from 1 to ρ m,ε between ] -∞, 0] then decreases between [0, ∞[. But if F ε (ρ m,ε ) = 0 then the Cauchy solution of (A.6) such that ρ(0) = ρ m,ε , ρ (0) = 0 is stationary. And in this case there is no soliton that converges to 1.

Proof. Consequence of lemma A.1 and elementary computation.

We can now conclude

Proposition A.4. Under the conditions

For ε 1, there exists ρ solution of (A.6) with

Moreover if γ > 0

• ρ is increasing on [0, +∞[ and even.

If γ < 0

• ρ is decreasing on [0, +∞[ and even.

Proof. Multiplying the equation (A.6) by ρ and integrating on [0, x] we obtain

We have, by (A.11), ρ (0) = F (ρ m ) has the same sign as γ. So near 0, if γ > 0 ρ is increasing. The rest of the proof is deduced using basic ode argument and phase portrait analysis (see figure 1,2).

We let, as in the two-dimensional case

Then we have

the classical soliton to the Korteweg-de-Vries equation (KdV). Then

Figure 2: Graph of ρ(x).

First of all we have

Lemma A.6. For any ε 1, r is even, increasing on ] -∞, 0] then decreasing on [0, +∞[. Moreover

and there exists a constant C, not depending on ε, such that

Proof. The variations of r are an immediate consequence of the proposition A.4. Using (A.13) and lemma A.1 we have for ε 1

Using (A.12) we obtain

this ends the proof.

Proof of proposition A.5. We recall that

After lenghty but simple computations, we find that (r ε , r ε ) is a solution of

,

Moreover N is a solution of

Then as r ε is a solution of

with r ε (0) → 3, we obtain, using the Cauchy-Lipschitz theorem with parameter, that for any compact set [a, b]

We have using proposition A.4 and (A.13) that the functions N, r ε are increasing on ] -∞, 0], decreasing on [0, ∞[ and converge to 0 in +∞ and -∞. Thus we obtain

which concludes the proof.

The following proposition ends the proof of proposition 1.4

Proposition A.7. We have for all k ∈ N

Proof. We already know the result for k = 0, we prove it for k = 1. We have by (A.18) uniform convergence on any compact. As r ε converges uniformly to N and N (x) -→ |x|→∞ 0 then sup

0. Since we have [START_REF] De | Solitary waves of generalized Kadomtsev-Petviashvili equations[END_REF] we obtain that sup

combining with (A.15) we have

The result for higher order follows by use of the ODE and a simple induction argument.

Remark A.8. Using (A.5) and letting

so by the proposition A.7 we have that

On the other hand by using Leibniz formula one has, for all n ∈ N

and therefore by proposition (A.7) we have for all n ∈ N

a result similar to the one obtained in dimension 2.

The case Γ = 0 The proof of proposition 1.6 is similar to what was done earlier (see figure 3 for a phase portrait). let us give a heuristic argument. Substituting For a similar (and more general) result in the case of the non-linear Schrödinger equation, see [START_REF] Chiron | Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one[END_REF].


Patrick G Érard
email: patrick.gerard@universite-paris-saclay.fr

M Bellassoued

N Burq

B Dehman

Sandrine Grellier

Enno Lenz- Mann

Peter Perry

Peter Topalov

AN EXPLICIT FORMULA FOR THE BENJAMIN-ONO EQUATION

Keywords: Mathematics Subject Classification. 37K15 primary, 47B35 secondary Benjamin-Ono equation, explicit formula, Lax pair, Toeplitz operators

We establish an explicit formula for the general solution of the Benjamin-Ono equation on the torus and on the line. Contents 1. Introduction 1 1.1. The Benjamin-Ono equation 1 1.2. The Lax pair 2 1.3. The explicit formula on the torus 3 1.4. The explicit formula on the line 3 1.5. Organization of the paper 4 2. Proof of the explicit formula on the torus 4 3. Proof of the explicit formula on the line 6 4. Final remarks 8 References 9

1. Introduction 1.1. The Benjamin-Ono equation. The Benjamin-Ono equation was introduced by Benjamin [START_REF] Benjamin | Internal waves of permanent form in fluids of great depth[END_REF] (see also , Ono [START_REF] Ono | Algebraic solitary waves in stratified fluids[END_REF]) to model long, one-way internal gravity waves in a two-layer fluid. It reads [START_REF] Benjamin | Internal waves of permanent form in fluids of great depth[END_REF] ∂ t u = ∂ x (|D|u -u 2 ) .

Here u = u(t, x) denotes a real valued function. There is a vast literature about this equation, and we refer to the book by Klein and Saut [START_REF] Klein | Nonlinear dispersive equations, Inverse scattering and PDE methods[END_REF] for a recent survey. We consider both the case of periodic boundary conditions u(t, x + 2π) = u(t, x), which we refer as x ∈ T, and the case where u(t, x) cancels as x tends to ±∞, which we refer as x ∈ R. In both cases, we will restrict ourselves to sufficiently smooth solutions, which can be proved to exist globally by a combination of standard quasilinear scheme and appropriate conservation laws. More precisely, if we denote by H 2 r the Sobolev space of real valued functions with two derivatives in L 2 , one can prove the following result.

Theorem 1 (Saut, 1979 [16]). For every u 0 ∈ H 2 r , there exists a unique solution u ∈ C(R, H 2 r ) of (1) with u(0) = u 0 . Our goal in this paper is to provide an explicit formula of the solution u(t) in terms of the initial datum u 0 .

For this, we need to introduce the Lax pair structure for (1). We denote by Π the orthogonal projector from

L 2 onto L 2 + . Remarkable operators on L 2 + are Toeplitz operators, associated to functions b ∈ L ∞ by the formula ∀f ∈ L 2 + , T b f = Π(bf ) . For every u ∈ L 2
r , we denote by L u the semi-bounded selfadjoint operator defined on

L 2 + by Dom(L u ) = H 1 + := H 1 ∩ L 2 + , L u f = Df -T u f , D := 1 i d dx .
We also consider, for u ∈ H 2 r , the bounded antiselfadjoint operator defined by

B u = i(T |D|u -T 2 u
) . Then one can check the following identity (see e.g. [START_REF] Fokas | Inverse scattering transform for the Benjamin-Ono equation: A pivot to multidimensional problems[END_REF], [START_REF] Wu | Simplicity and finiteness of discrete spectrum of the Benjamin-Ono scattering operator[END_REF], [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], [START_REF] Gérard | The Lax pair structure for the spin Benjamin-Ono equation[END_REF]).

Theorem 2. Under the conditions of Theorem 1, we have

∀t ∈ R , d dt L u(t) = [B u(t) , L u(t) ] .
1.3. The explicit formula on the torus. Let us mention some more properties of the Hardy space on the torus. The Hardy space L 2 + (T) is equipped with the shift operator S := T e ix and with its adjoint S * = T e -ix . With this notation, our main result on the torus reads as follows.

Theorem 3. Let u ∈ C(R, H 2 r (T)) be the solution of the Benjamin-Ono equation on the torus T with u(0) = u 0 . Then u(t) = Πu(t) + Πu(t) -u 0 |1 , with ∀z ∈ D , Πu(t, z) = (Id -ze it e 2itLu 0 S * ) -1 Πu 0 |1 . Remark 1. The above formula can be equivalently stated as a characterization of Fourier coefficients of the solution u,

∀k ≥ 0 , û(t, k) = (e it e 2itLu 0 S * ) k Πu 0 |1 = Πu 0 |(Se -it e -2itLu 0 ) k 1 .
Under this form, it extends to much more singular data, for which the flow of the Benjamin-Ono has been proved to extend continuously. According to [START_REF] Gérard | Sharp wellposedness results of the Benjamin-Ono equation on H s (T, R) and qualitative properties of its solutions[END_REF], this is the case if u 0 belongs to the Sobolev space H s r (T) for every s > -1/2. Indeed, in this case, L u 0 is selfadjoint, semibounded, and the domain of the square root of L u 0 +KId, for K big enough, is the space H 1/2 + (T). Consequently, the operator Se -it e -2itLu 0 acts on H 1/2 + (T), so that the inner product in the second line above is well defined.

1.4. The explicit formula on the line. On L 2 + (R), the shift operator S must be replaced by the Lax-Beurling semi-group (S(η)) η≥0 of isometries defined as

∀f ∈ L 2 + (R) , S(η)f (x) = e iηx f (x)
. We denote by G the adjoint of the operator of multiplication by x on L 2 + (R). Notice that -iG is the infinitesimal generator of the adjoint semi-group of contractions (S(η) * ) η≥0 , so that

∀η ≥ 0 , S(η) * = e -iηG .
It is easy to check that the domain of G consists of those functions f ∈ L 2 + (R) such that the restriction of f to the half-line (0, +∞) belongs to the Sobolev space H 1 (0, +∞), and that

Gf (ξ) = i d dξ [ f (ξ)] 1 ξ>0 .
As a consequence, for every f ∈ Dom(G), one may define

I + (f ) := f (0 + ) .
This definition can be extended to any f ∈ L 2 + such that the restriction of f to some interval (0, δ) belongs to the Sobolev space H 1 (0, δ) for some δ > 0, and we shall use it as well. With this notation, our main result on the line reads as follows.

Theorem 4. Let u ∈ C(R, H 2 r (R)) be the solution of the Benjamin- Ono equation on the line R with u(0) = u 0 . Then u(t) = Πu(t) + Πu(t), with ∀z ∈ C + , Πu(t, z) = 1 2iπ I + [(G -2tL u 0 -zId) -1 Πu 0 ] .
Notice that, in the above formula, the function

f z,t := (G -2tL u 0 -zId) -1 Πu 0
belongs to the domain of G -2tL u 0 -see the end of section 3 for more detail -, and therefore its Fourier transform satisfies f ∈ H 1 (0, δ) for every finite δ > 0, hence one can define I + (f z,t ).

Remark 2. At this time, the wellposedness theory for (1) on the line is slightly less advanced than on the torus, see [START_REF] Molinet | The Cauchy problem for the Benjamin-Ono equation in L 2 revisited[END_REF] for a detailed account of this, with extension of the flow map to L 2 r (R). However, one can easily prove -see section 3 below -that the above formula makes sense for u 0 in the space L ∞ (R) ∩ L 2 r (R). 1.5. Organization of the paper. Sections 2 and 3 are respectively devoted to the proofs of Theorems 3 and 4. The main idea is to take advantage of commutation identities between the operators of the shift structure of the Hardy space and the operators L u and B u of the Lax pair, in the spirit of what was done in [START_REF] Gérard | An explicit formula for the cubic Szegő equation[END_REF] for the cubic Szegő equation on the torus. At the end of Section 3, we also provide a short discussion of the meaning of the formula, leading to an extension to initial data in L ∞ (R) ∩ L 2 r (R). Section 4 briefly draws possible applications and extensions to other equations.

Proof of the explicit formula on the torus

The proof is based on the following lemma.

Lemma 1. For every u ∈ H 2 r (T), [S * , B u ] = i((L u + Id) 2 S * -S * L 2 u
) . Let us postpone the proof of Lemma 1 and complete the proof of Theorem 3. Since u(t) is real valued, we have the identity

u(t) = Πu(t) + Πu(t) -u(t)|1 ,
and u(t)|1 = u 0 |1 because of the equation. It remains to identify Πu(t) as a holomorphic function on the disc. For this, we proceed as in [START_REF] Gérard | An explicit formula for the cubic Szegő equation[END_REF], where a similar formula was established for the cubic Szegő equation. We have, for every z ∈ D,

Πu(t, z) = ∞ n=0 z n Πu(t), e inx = (Id -zS * ) -1 Πu(t)|1 .
We denote by U = U(t) the solution of the linear initial value problem in

L (L 2 + (T)), U ′ (t) = B u(t) U(t) , U(0) = Id . Since B u(t) is anti-selfadjoint, U(t)
is unitary, and we can write

(2) Πu(t, z) = (Id -zU(t) * S * U(t)) -1 U(t) * Πu(t)|U(t) * 1 . Let us calculate d dt U(t) * 1 = -U(t) * B u(t) 1 = -iU(t) * [(T |D|u(t) -T 2 u(t) )(1)] = -iU(t) * [DΠu(t) -T u(t) Πu(t)] = -iU(t) * L u(t) Πu(t) = iU(t) * L 2 u(t) (1) = iL 2 u 0 U(t) * 1 , from which we conclude U(t) * 1 = e itL 2 u 0 (1) . Consequently, U(t) * Πu(t) = -U(t) * L u(t) (1) = -L u 0 U(t) * (1) = -L u 0 e itL 2 u 0 (1) = e itL 2 u 0 (1)Πu 0 . Finally, using Lemma 1, d dt U(t) * S * U(t) = U(t) * [S * , B u(t) ]U(t) = U(t) * [i((L u(t) + Id) 2 S * -S * L 2 u(t) ]U(t) = i(L u 0 + Id) 2 U(t) * S * U(t) -iU(t) * S * U(t)L 2
u 0 . from which we infer U(t) * S * U(t) = e it(Lu 0 +Id) 2 S * e -itL 2 u 0 . Plugging the previous identites into (1), we conclude Πu(t, z) = (Id -ze it(Lu 0 +Id) 2 S * e -itL 2 u 0 ) -1 e itL 2 u 0 (1)Πu 0 |e itL 2 u 0 (1) = (Id -ze -itL 2 u 0 e it(Lu 0 +Id) 2 S * ) -1 Πu 0 |1 which yields the claimed formula.

Finally, let us prove Lemma 1. First of all, it easy to check the following commutation identity with the Toeplitz operators,

∀b ∈ L ∞ (T), [T b , S * ] = . |1 S * Πb .
We infer, using the adjoint Leibniz formula

S * D = DS * + S * , [S * , B u ] = i([S * , T |D|u ] -T u [S * , T u ] -[S * , T u ]T u ) = i( . |1 S * DΠu -T u . |1 S * Πu -. |1 S * ΠuT u ) = i( . |1 (DS * Πu -T u S * Πu + S * Πu) -. |T u 1 S * Πu) = i( . |1 (L u S * Πu + S * Πu) + . |L u 1 S * Πu) = i((L u + Id) . |1 S * Πu + ( . |1 S * Πu)L u ) = i((L u + Id)((L u + Id)S * -S * L u ) + ((L u + Id)S * -S * L u )L u ) = i((L u + Id) 2 S * -S * L 2 u
) . The proof of Theorem 3 is complete.

Proof of the explicit formula on the line

We start with the inverse Fourier transform for every f ∈ L 2 + (R), which we can write in the upper-half plane, as an absolutely convergent integral,

∀z ∈ C + , f (z) = 1 2π ∞ 0 e izξ f (ξ) dξ ,
while, in view of the Plancherel theorem, we have, in L 2 (0, +∞),

f (ξ) = lim ε→0 R e -ixξ f (x) 1 + iεx dx = lim ε→0 S(ξ) * f |χ ε ,
where χ ε denotes the following function in L 2 + (R),

χ ε (x) := 1 1 -iεx .
Plugging the second formula into the first one, we infer

f (z) = lim ε→0 1 2π ∞ 0 e izξ S(ξ) * f |χ ε dξ = lim ε→0 1 2π ∞ 0 e izξ e -iξG f |χ ε dξ = lim ε→0 1 2iπ (G -zId) -1 f |χ ε = 1 2iπ I + [(G -zId) -1 f ] .
Since u(t) is real valued, we can write u(t) = Πu(t) + Πu(t), and it remains to characterize Πu(t, z) for z ∈ C + . We are going to proceed as in the previous paragraph, using the family U(t) of unitary operators defined by the linear initial value problem in

L (L 2 + (R)), U ′ (t) = B u(t) U(t) , U(0) = Id .
For every z ∈ C + , we have

Πu(t, z) = lim ε→0 1 2iπ U(t) * (G -zId) -1 Πu(t)|U(t) * χ ε = lim ε→0 1 2iπ (U(t) * GU(t) -zId) -1 U(t) * Πu(t)|U(t) * χ ε .
We have the following lemma.

Lemma 2. For every u ∈ H 2 r (T), [G, B u ] = -2L u + i[L 2 u , G] .
Let us postpone the proof of Lemma 2 and complete the proof of Theorem 4. We calculate

d dt U(t) * GU(t) = U(t) * [G, B u(t) ]U(t) = U(t) * (-2L u(t) + i[L 2 u(t) , G])U(t) = -2L u 0 + i[L 2 u 0 , U(t) * GU(t)] .
Integrating this ODE, we get

U(t) * GU(t) = -2tL u 0 + e itL 2 u 0 Ge -itL 2 u 0 .
Let us determine the other terms in the inner product. We have (see also [START_REF] Sun | Complete integrability of the Benjamin-Ono equation on the multisoliton manifolds[END_REF])

d dt U(t) * Πu(t) = U(t) * (∂ t Πu(t)-B u(t) Πu(t)) = iU(t) * L 2 u(t) Πu(t) = iL 2 u 0 U(t) * Πu(t) ,
from which we infer

U(t) * Πu(t) = e itL 2 u 0 Πu 0 .
Finally, we have

d dt U(t) * χ ε = -U(t) * B u(t) χ ε = -iU(t) * (T |D|u(t) χ ε -T 2 u(t) χ ε )
and the right hand side converges in

L 2 + to -iU(t) * (DΠu(t) -T u(t) Πu(t)) = -iU(t) * L u(t) Πu(t) = -iL u 0 U(t) * Πu(t) = -iL u 0 e itL 2 u 0 Πu 0 = lim ε→0 iL 2 u 0 e itL 2 u 0 χ ε .
By integrating in time, we infer

U(t) * χ ε -e itL 2 u 0 χ ε → 0 in L 2 + .
Plugging these informations into the formula which gives Πu(t, z), we infer

Πu(t, z) = lim ε→0 1 2iπ e itL 2 u 0 Ge -itL 2 u 0 -2tL u 0 -zId -1 e itL 2 u 0 Πu 0 |e itL 2 u 0 χ ε = lim ε→0 1 2iπ (G -2tL u 0 -zId) -1 Πu 0 |χ ε = 1 2iπ I + [(G -2tL u 0 -zId) -1 Πu 0 ] .
It remains to prove Lemma 2. We shall appeal to the following elementary identity, whose proof can be found in [START_REF] Sun | Complete integrability of the Benjamin-Ono equation on the multisoliton manifolds[END_REF], [START_REF] Gérard | The Calogero-Moser derivative nonlinear Schrödinger equation[END_REF].

Lemma 3. For every f ∈ Dom(G), b ∈ H 1 (R), T b f ∈ Dom(G) and [G, T b ]f = i 2π I + (f )Πb .
Using Lemma 3 and the simple observation that [G, D] = iId, we obtain (see also [START_REF] Sun | Complete integrability of the Benjamin-Ono equation on the multisoliton manifolds[END_REF]),

∀f ∈ Dom(G) ∩ Dom(L u ) , [G, L u ]f = if - i 2π I + (f )Πu . We infer, for f ∈ Dom(G) ∩ Dom(L 2 u ), [G, B u ]f = i([G, T |D|u ]f -T u [G, T u ]f -[G, T u ]T u f ) = i 2π (iI + (f )(DΠu -T u Πu) -iI + (T u f )Πu) = i 2π (iI + (f )L u Πu + iI + (L u f )Πu) = i(L u (if -[G, L u ]f ) + iL u f -[G, L u ]L u f ) = -2L u f + i[L 2 u , G]f . The proof of Theorem 4 is complete.
Let us conclude this section by discussing the formula of Theorem 4 for more singular data u 0 . First of all, let us observe that, for every t ∈ R, the operator A t := -i(G -2tL 0 ) is maximally dissipative. Indeed, its expression in the Fourier representation is given by

A t f (ξ) = d dξ f (ξ) + 2itξ f (ξ) ,
and therefore it is easy to check by explicit calculations that

Dom(A t ) = {f ∈ L 2 + (R) : e itξ 2 f ∈ H 1 (0, ∞)} with ∀f ∈ Dom(A t ) , Re A t f |f ≤ 0, and that A t + izId : Dom(A t ) → L 2
+ (R) is bijective for every z ∈ C + . From standard perturbation theory, we infer that, for every bounded antiselfadjoint operator B on L

2 + (R), A t + B is maximally dissipative. In particular, if u 0 ∈ L ∞ (R) ∩ L 2 r (R), -i(G -2tL u 0 ) = A t -2itT u 0
is maximally dissipative, so that the formula of Theorem 4 still holds. In particular, Theorem 4 provides a formula for the extension of the flow map of the Benjamin-Ono equation to H s r (R) for every s > 1/2 [START_REF] Molinet | The Cauchy problem for the Benjamin-Ono equation in L 2 revisited[END_REF].

Final remarks

In the case of finite gap potentials on the torus [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], or multisolitons on the line [START_REF] Sun | Complete integrability of the Benjamin-Ono equation on the multisoliton manifolds[END_REF], formulae of Theorems 3 and 4 take place in finite dimensional vector spaces, and they reduce to calculations on finite dimensional matrices, as already observed in these references.

We expect Theorems 3 and 4 to be useful for the study of long time behaviour of solutions to the Benjamin-Ono equation. This is particular important on the line, where soliton resolution for generic data is still an open problem (see however [START_REF] Ifrim | Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation[END_REF] for partial results in this direction).

Let us now briefly discuss applications of a similar approach to other integrable equations. First of all, it is clear that Theorems 3 and 4 easily extend to the spin Benjamin-Ono system [START_REF] Berntson | Spin generalizations of the Benjamin-Ono equation[END_REF], [START_REF] Gérard | The Lax pair structure for the spin Benjamin-Ono equation[END_REF]. Furthermore, these formulae could probably be very useful in the study of the small dispersion limits of these equations, in particular the half-wave maps equation [START_REF] Gérard | A Lax pair structure for the half-wave maps equation[END_REF], [START_REF] Berntson | Spin generalizations of the Benjamin-Ono equation[END_REF]. We also expect similar formulae to hold for the recently introduced Calogero-Moser equation [START_REF] Gérard | The Calogero-Moser derivative nonlinear Schrödinger equation[END_REF], since the Lax pair of operators for this equation enjoys similar commutation properties with the shift structure of the Hardy space. Finally, as we already observed, a similar formula is known to hold for the cubic Szegő equation on the torus [START_REF] Gérard | An explicit formula for the cubic Szegő equation[END_REF], and it is possible to adapt the approach with the operator G developed in this paper in order to get an explicit formula for the cubic Szegő equation on the line [START_REF] Gérard | [END_REF]. On the other hand, we have no clue whether such explicit formulae could be extended to KdV, cubic NLS or DNLS equations.

1. 2 . 0 f

 20 The Lax pair. On T or R, we denote by L 2 + the Hardy space corresponding to L 2 functions having a Fourier transform supported in the domain ξ ≥ 0. Recall that both Hardy spaces identify to some spaces of holomorphic functions. The space L 2 + (T) identifies to holomorphic functions f on the unit disc D := {z ∈ C : |z| < 1} such that sup r<1 2π (re ix ) 2 dx < +∞, while L 2 + (R) identifies to holomorphic functions on the upper half plane C + := {z ∈ C : Im(z) > 0} such that sup y>0 R |f (x + iy)| 2 dy < +∞ .