Victor Vilaça

D A Rocha

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE CUBIC COUPLED SCHRÖDINGER SYSTEMS IN ONE SPACE DIMENSION

Keywords: 2010 Mathematics Subject Classification, 35Q55, Modified Scattering, Nonlinear Schrödinger equation

In this paper, we study a coupled nonlinear Schrödinger system with small initial data in the one dimension Euclidean space. Such a system appears in the context of the coupling between two different optical waveguides. We establish an asymptotic nonlinear behavior and a decay estimate for solutions of this system. The proof uses a recent work of Kato and Pusateri.

Introduction

We consider the following system   

i∂ t u + ∂ xx u = |v| 2 u, (t, x) ∈ [1, +∞) × R, i∂ t v + ∂ xx v = |u| 2 v, u(1, x) = u 1 (x), v(1, x) = v 1 (x). (1.1)
1.1. Motivation and background. An interesting area within the study of the asymptotic behavior of small solutions of nonlinear dispersive PDEs is the search of nonlinear global dynamics. In the well-known case of the cubic Schrödinger equation, different methods have been treated by several authors. On the one hand, the asymptotic behavior depends on the geometry of the spatial domain.

• For Euclidean spaces, we mainly have phase corrections results. For example, Kato and Pusateri treat the case x ∈ R in [START_REF] Kato | A new proof of long-range scattering for critical nonlinear Schrödinger equations[END_REF] using only tools from the analysis in Fourier space. For general Euclidean spaces, we can look at [START_REF] Hayashi | Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations[END_REF] where Hayashi and Naumkin deal with x ∈ R n . • In the case of compact domains, the usual example is the study of the torus T d . We can for example mention the work of Colliander, Keel, Staffilani, Takaoka and Tao who showed some first results in the search of unboundedness of Sobolev norms for small initial data in the spatial space T 2 in [START_REF] Colliander | Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation[END_REF]. • Combining these two kinds of spaces, we can study the product spaces R n ×T d . The first to consider such spaces were Tzvetkov and Visciglia. They obtained some scattering behaviors in R n × M, where M is a compact manifold (see [START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF]), or in R n × T where they extended their point of view and studied the well-posedness of their equation (see [START_REF] Tzvetkov | Well-posedness and scattering for nls on R d × T in the energy space[END_REF]). For the case R × T d , we can cite the work of Hani, Pausader, Tzvetkov and Visciglia, who showed in 2013 (see [START_REF] Hani | Modified scattering for the cubic Schrödinger equation on product spaces and applications[END_REF]) a real growth of Sobolev norms by adding a direction of diffusion to the compact problem. For that purpose, they exhibited a modified scattering of the solution of the Schrödinger equation to the solution of the resonant system associated. Contrary to the Euclidean case, this is an effective scattering, and not only a phase correction.

On the other hand, after the geometry of the spatial domain, we can deal with the equation itself. Using the method presented in [START_REF] Hani | Modified scattering for the cubic Schrödinger equation on product spaces and applications[END_REF], similar results have been obtained by adding a potential (see [START_REF] Grébert | Modified scattering for the cubic Schrödinger equation on product spaces: the nonresonant case[END_REF]), a harmonic trapping (see [START_REF] Hani | Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping[END_REF]), or by considering different derivatives along the Euclidean direction and the periodic one (see [START_REF] Xu | Unbounded sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation[END_REF]). In the last mentioned article, Xu exhibits a scattering between the Schrödinger equation i∂ t U + ∆ R U -|∇ T |U = |U | 2 U in the spatial domain R × T and the cubic Szegő equation.

In this article, we focus on a coupled Schrödinger system. Such a system occurs while, for example, looking for the coupling between two different optical waveguides, that can be provided by a dual-core single-mode fiber (see [START_REF] Agrawal | Contemporary nonlinear optics[END_REF]). Other examples are given by two orthogonally polarized components traveling at different speeds because of different refractive indices associated with them; or by two distinct pulses with different carrier frequencies but with the same polarization. A surprising fact about these examples is that they lead to a system close to the Schrödinger system (1.1), obtained by interchanging the spatial and temporal coordinates.

From our point of view, such a system is interesting. Indeed this coupling effect can provide more nonlinear asymptotic behavior than a single equation. For example, Grébert, Paturel and Thomann exhibited some beating effects (see [START_REF] Grébert | Beating effects in cubic Schrödinger systems and growth of Sobolev norms[END_REF]), that is to say some energy exchanges between different modes of the solutions, although they were just dealing with the spatial domain T. Another example of study of Schrödinger systems is given by Kim in [START_REF] Kim | A note on decay rates of solutions to a system of cubic nonlinear Schrödinger equations in one space dimension[END_REF]. Considering relations on the masses in the equations, the author obtains L ∞ decay of the solution for the spatial domain R.

The goal of this article is to present a self contained proof of nonlinear behavior in the spatial domain R. Following the method of Kato and Pusateri in [START_REF] Kato | A new proof of long-range scattering for critical nonlinear Schrödinger equations[END_REF], we will first prove a L ∞ decay of the solutions. Then, we will highlight an asymptotic behavior, in the sense that we will be looking for a pair (W f , W g ) depending only on the space variable in a space to be determined, such that:

ŵf (t, .) -W f -→ t→∞ 0, ŵg (t, .) -W g -→ t→∞ 0, (1.2)
where

| ŵf (t, ξ)| = |û(t, ξ)| and | ŵg (t, ξ)| = |v(t, ξ)|.
This is a result of phase correction scattering, and it is interesting to see what effect the coupling has in this case, to get informations on W f and W g , and to approximate the speed of this convergence. Moreover, we exhibit an asymptotic formula for large time of the solutions.

1.2. Notations and norms. We define the spatial Fourier transform in Schwartz space, for ϕ ∈ S(R), by

F (ϕ)(ξ) = φ(ξ) := 1 √ 2π R e -ixξ ϕ(x)dx.
We use the notation f g to denote that there exists a positive constant M such that f ≤ M g. This notation will be useful, allowing us to avoid dealing with all the constants in the different inequalities. When a function depends on two variables t and x, we denote by ϕ(t) the function ϕ(t) : x → ϕ(t, x). Working with small initial data, we can expect the nonlinearity to stay small, and thus the linear dynamics to be dominant. This is the reason why it is interesting to consider the profile of a solution, which is the backwards linear evolution of a solution of the nonlinear equation. We define f (respectively g) the profile of the solution u (respectively v) by

f (t, x) := e -it∂xx u(t, x) g(t, x) := e -it∂xx v(t, x). (1.3)
We will see in Subsection 2.1 that after considering Duhamel's formula of the solutions, the equations in f and g are easier to treat than the ones with u and v. This is the main reason why the profiles have been introduced.

The norm used will be essential in order to prove the global existence of the solutions. Let n ∈ N, we define the spaces H n,0

x , H 0,n x and L ∞ T corresponding to the following norms:

• ϕ L ∞ x := sup x∈R |ϕ(x)|, • ϕ H n,0 x := n i=0 ∂ i x ϕ L 2 x , • ϕ H 0,n x := n i=0 x i ϕ L 2 x , • ϕ L ∞ T := sup t∈[1,T ] |ϕ(t)|.
With these norms, we follow the method of Kato and Pusateri in [START_REF] Kato | A new proof of long-range scattering for critical nonlinear Schrödinger equations[END_REF] to construct the space

X T := ϕ, ϕ XT := ϕ L ∞ T L ∞ x + t -α ϕ L ∞ T H 1,0 x + t -α ϕ L ∞ T H 0,2n+1 x < ∞ ,
where α > 0 is small enough. We want to obtain a L ∞ decay of the solutions. For that purpose, we need to control the behavior of the cubic nonlinearity, which is why the terms

ϕ L ∞ T L ∞ x and t -α ϕ L ∞ T H 1,0
x are required in the definition of the norm. With these two terms, we can show that we have a unique and global pair of solutions, and that the scattering limits W f , W g of the equations (1.2) are in L ∞ . The last term t -α ϕ L ∞ T H 0,2n+1

x will allow us to show that W f , W g are also in H 0,n . 1.3. Statement of the results. The idea here is to apply the result of [START_REF] Kato | A new proof of long-range scattering for critical nonlinear Schrödinger equations[END_REF] to our coupled Schrödinger system (1.1). As we can expect, we also obtain a global existence, a decay estimate, a scattering result and an asymptotic formula.

Theorem 1.1. Fix n ∈ N and 0 < ν < 1 4 . Assume u 1 H 2n+1,0 x + u 1 H 0,1 x + v 1 H 2n+1,0 x + v 1 H 0,1
x ≤ ε for ε small enough. Let I = [1, +∞) and set F := C I; H 2n+1,0 (I) ∩ H 0,1 (I) . Then (1.1) admits a unique pair of solutions (u, v) ∈ F × F . The pair of solutions satisfies the following decay estimates:

u(t) L ∞ x 1 t 1 2 , v(t) L ∞ x 1 t 1 2
.

Furthermore, there exists a unique pair of functions

(W f , W g ) ∈ (L ∞ ξ ∩ H 0,n ξ ) × (L ∞ ξ ∩ H 0,n ξ ) such that for t ≥ 1,        f (t, .) exp i 2 √ 2π t 1 1 s |v(s, .)| 2 ds -W f L ∞ ξ t -1 4 +ν , ĝ(t, .) exp i 2 √ 2π t 1 1 s |û(s, .)| 2 ds -W g L ∞ ξ t -1 4 +ν , and        f (t, .) exp i 2 √ 2π t 1 1 s |v(s, .)| 2 ds -W f H 0,n ξ t -1 8 + ν 2 , ĝ(t, .) exp i 2 √ 2π t 1 1 s |û(s, .)| 2 ds -W g H 0,n ξ t -1 8 + ν 2 ,
where f and g are defined in (1.3). Finally, we have an asymptotic formula. It exists a unique pair of functions

Γ f , Γ g ∈ L ∞ ξ such that for large time t,        u(t, x) = 1 (2it) 1 2 W f ( x 2t ) exp i x 2 4t - i 2 √ 2π (|W g | 2 ( x 2t ) ln(t) + Γ g ( x 2t )) + O(t -3 4 +ν ), v(t, x) = 1 (2it) 1 2 W g ( x 2t ) exp i x 2 4t - i 2 √ 2π (|W f | 2 ( x 2t ) ln(t) + Γ f ( x 2t )) + O(t -3 4 +ν ).
Remark 1.2. Let's begin with some remarks about this theorem.

• We see that, as we can expect, the role of the coupling appears here as a phase correction term of the scattering result.

• By taking n = 0 in Theorem 1.1, we obtain that W f , W g ∈ L ∞ ξ ∩ L 2 ξ for a small initial data in H 1,0 x ∩ H 0,1 x . This is the coupled equivalent of the Theorem 1.1 of [START_REF] Kato | A new proof of long-range scattering for critical nonlinear Schrödinger equations[END_REF]. In this article, we are able to get the control of the H 0,n norms too.

• The decay estimate is not new here. Indeed, Kim already obtained it in [START_REF] Kim | A note on decay rates of solutions to a system of cubic nonlinear Schrödinger equations in one space dimension[END_REF], but without scattering result on the solutions. • We see that we need a control on the derivatives on u 1 , v 1 to obtain a control on the H 0,n norm of W f , W g . The reason is that W f , W g are the asymptotic behaviors of w f and w g , defined in (2.5), which are, up to a phase correction, the Fourier transforms of u and v. • They are two losses that can be surprising. First, the loss of speed between the L ∞ behavior and the H 0,n one. Secondly, the need for 2n + 1 derivatives in u, v to obtain n multiplications in W f , W g . Both losses are connected to the same phenomenon. The key part of all the estimations will be the good decay in L ∞ of a remainder term R defined in (2.3). To control the H 0,n norm of R by using this L ∞ decay, we use the inequality (4.7), which gives us these losses. These losses may not be sharp, as this control of the H 0,n norm may not be optimal.

The rest of the paper is organized as follows. In Section 2, we will do some preliminary computations and introduce the notion of profile. In Section 3, we will prove a key proposition. Finally, the proof of the Theorem 1.1 will be treated in Section 4.

Preliminaries

In this section, we reformulate the problem by considering Duhamel's formula of the profiles of the solutions. Moreover, we state a proposition, which will be the main ingredient to begin the proof of the Theorem 1.1. Finally, we state and prove a classical estimate of the Schrödinger linear group.

2.1. Introduction of the profile. We apply Duhamel's formula to the system (1.1). If the solution u exists, then u = Φ(u), where Φ(u)(t, x) = e i(t-1)∂xx u 1 (x)i t 1 e i(t-s)∂xx |v(s, x)| 2 u(s, x)ds.

To get the existence of the solution, we have to apply the fixed point theorem. We filter out the linear contribution by introducing the profile here. For f defined in (1.3), Duhamel's formula becomes (with the same notation Φ for the profile),

Φ(f )(t, x) = e -i∂xx u 1 (x) -i t 1 e -is∂xx v(s, x)v(s, x)u(s, x)ds.
Using that uv = 1 √ 2π û * v, we compute the spatial Fourier transform of Φ(f ) as follows:

Φ(f )(t, ξ) = e iξ 2 û1 (ξ) -i t 1 e isξ 2 vvu(s, ξ)ds = e iξ 2 û1 (ξ) -i t 1 R 2 e isξ 2 û(s, α) v(s, η -α)v(s, ξ -η)dηdαds = e iξ 2 û1 (ξ) -i t 1 R 2 e isξ 2 û(s, α)v(s, α -η)v(s, ξ -η)dηdαds.
Now, we reintroduce f = e -it∂xx u and g = e -it∂xx v in the above equation to get:

Φ(f )(t, ξ) = e iξ 2 û1 (ξ) -i t 1 R 2 e is(ξ 2 -α 2 +(α-η) 2 -(ξ-η) 2 ) f (s, α)ĝ(s, α -η)ĝ(s, ξ -η)dηdαds = e iξ 2 û1 (ξ) -i t 1 R 2 e 2isη(ξ-α) f (s, α)ĝ(s, α -η)ĝ(s, ξ -η)dηdαds.
After the change of variables α ↔ ξσ, we get

Φ(f )(t, ξ) = e iξ 2 û1 (ξ) - i 2π e 2isησ ĝ(s, ξ -σ)ĝ(s, ξ -η -σ) f (s, ξ -η)dσdηds. (2.1)
We define

F (s, η, σ, ξ) := ĝ(s, ξ -σ)ĝ(s, ξ -η -σ) f (s, ξ -η). (2.2)
Using the Plancherel formula on (2.1) we get

Φ(f )(t, ξ) = e iξ 2 û1 (ξ) - i 2π t 1 F η,σ e 2isησ F -1 η,σ [F (s, η, σ, ξ)] dσdηds. Using δa = 1 √ 2π e -iax , e iax = √ 2πδ a and the fact that f λ (ξ) = 1 λ f ( ξ λ ) where f λ (x) = f (λx), we have F η,σ e 2isησ (a, b) = F η √ 2πδ 2sη (b) (a) = 1 2s e -i ab 2s .
Therefore, with the definition of F in (2.2),

Φ(f )(t, ξ) = e iξ 2 û1 (ξ) -i 4π t 1 1 s e -i ab 2s F -1 η,σ [F (s, η, σ, ξ)] (a, b)dadbds = e iξ 2 û1 (ξ) -i 4π t 1 1 s F -1 η,σ [F (s, η, σ, ξ)] (a, b)dadbds -i 4π t 1 1 s e -i ab 2s -1 F -1 η,σ [F (s, η, σ, ξ)] (a, b)dadbds = e iξ 2 û1 (ξ) -i √ 2π 4π t 1 1 s |ĝ(s, ξ)| 2 f (s, ξ)ds + t 1 R(s, ξ)ds, where R(s, ξ) = R(ĝ, ĝ, f )(s, ξ) := - i 4πs e -i ab 2s -1 F -1 η,σ [F (s, η, σ, ξ)] (a, b)dadb. (2.3)
Thus, we obtain

∂ t Φ(f )(t, ξ) = - i 2 √ 2πt |ĝ(t, ξ)| 2 f (t, ξ) + R(t, ξ). (2.4)
The idea is to use the good behavior of R, which has a good decay in time. To take advantage of this decay, we introduce a last change of variable. We set

B g (t, ξ) = exp i 2 √ 2π t 1 1 s |ĝ(s, ξ)| 2 ds , B f (t, ξ) = exp i 2 √ 2π t 1 1 s f (s, ξ) 2 ds , ŵf (t, ξ) = f (t, ξ)B g (t, ξ), ŵg (t, ξ) = ĝ(t, ξ)B f (t, ξ).
(2.5)

The main property of this change of variables is the conservation of the modulus of f :

| ŵf (t, ξ)| = | f (t, ξ)|.
Thus, we have f = Φ(f ) if and only if ŵf is a solution of the following differential equation:

∂ t ŵf (t, ξ) = B g (t, ξ)R(t, ξ).
Applying Duhamel's formula a second time, we define the Duhamel form for ( ŵf , ŵg ) (still using the same notation Φ) by:

       Φ 1 ( ŵf , ŵg )(t, ξ) = ŵ1 f (ξ) + t 1 B g (s, ξ)R(ĝ, ĝ, f )(s, ξ)ds, Φ 2 ( ŵf , ŵg )(t, ξ) = ŵ1 g (ξ) + t 1 B f (s, ξ)R( f , f , ĝ)(s, ξ)ds, (2.6)
where ŵ1 f (ξ) = ŵf (1, ξ) and ŵ1 g (ξ) = ŵg (1, ξ). They are the equations that we will treat to get the existence of the solutions. For that purpose, we are now able to state the key result of the article.

Proposition 2.1. Assume ŵ1 f H 1,0 ξ + ŵ1 f H 0,2n+1 ξ + ŵ1 g H 1,0 ξ + ŵ1 g H 0,2n+1 ξ ≤ ε, then there exists a positive constant M such that Φ 1 ( ŵf , ŵg ) XT ≤ ε + M ŵf XT ŵg 2 XT (1 + ŵf 4 XT )(1 + ŵg 4 XT ), Φ 2 ( ŵf , ŵg ) XT ≤ ε + M ŵg XT ŵf 2 XT (1 + ŵg 4 XT )(1 + ŵf 4 XT ).
(2.7) Remark 2.2. There are two main differences between this estimation and the Proposition 1.3 of Kato and Pusateri in [START_REF] Kato | A new proof of long-range scattering for critical nonlinear Schrödinger equations[END_REF]. First, we are dealing here with a result about w, and not about u directly. The reason is in the equation (2.4). As long as we have no proof of the existence of f , we cannot take Φ(f ) = f and do a legit change of variables in this equation. But the existence of f is equivalent to the existence of w, which is why we focus on w here. Secondly, the right-hand side in [START_REF] Kato | A new proof of long-range scattering for critical nonlinear Schrödinger equations[END_REF] is a cubic term. The apparition of (1 + ŵf

4 XT )(1 + ŵg 4 XT )
terms in this paper is due to the choice of the X T -norm. In particular, the terms ∂ ξ f (s) L 2 ξ and ∂ ξ ĝ(s) L 2 ξ are not controlled by the definition of the X T -norm. In fact, as soon as we consider small solutions, the right-hand side is very close to a cubic term.

2.2.

Properties of the linear Schrödinger group. Before dealing with the computations on the X T norm, we introduce the following result of independent interest, about the linear Schrödinger group. This lemma will be used in both proofs of Proposition 2.1 and Theorem 1.1. Lemma 2.3. For all ϕ ∈ S(R) and for all 0 < β < 1 4 , we have

e it∂xx ϕ L ∞ x 1 |t| 1 2 φ L ∞ ξ + 1 |t| 1 2 +β φ H 1,0 ξ .
Proof. The following proof is based on the proof of the more general Lemma in [START_REF] Hayashi | Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations[END_REF]. The idea is to write e it∂xx ϕ as an integral with a kernel. Recall that for

K t (x) = (4iπt) -1 2 e -x 2 4it
, we have e it∂xx ϕ = K t * ϕ. Thus,

e it∂xx ϕ(x) = R ϕ(y)K t (x -y)dy = (4iπt) -1 2 R ϕ(y)e -(x-y) 2 4it dy = (4iπt) -1 2 e i x 2 4t R ϕ(y)e i y 2 4t e -i xy 2t dy = (4iπt) -1 2 e i x 2 4t R ϕ(y)e -i xy 2t dy + R ϕ(y)(e i y 2 4t -1)e -i xy 2t dy .
We define A ϕ such that

e it∂xx ϕ(x) := (2it) -1 2 e i x 2 4t φ( x 2t ) + A ϕ (t, x). (2.8)
The first part of the sum gives us the L ∞ part of the Lemma. We now deal with the term A ϕ by using the fact that (with 0 < β < 1 4 ):

e ix -1 = 2 sin( x 2 ) x β .
(2.9)

This trivial inequality gives us for

A A ϕ (t) L ∞ x |t| -1 2 |x| 2β |t| -β ϕ L 1 x |t| -1 2 -β |x| 2β x x -1 ϕ L 1 x |t| -1 2 -β |x| 2β x -1 L 2 x x ϕ L 2 x |t| -1 2 -β φ H 1,0 ξ .
This concludes the proof of the lemma.

Proof of the Proposition 2.1

In this section, we prove the Proposition 2.1, following the proof of the similar proposition in [START_REF] Kato | A new proof of long-range scattering for critical nonlinear Schrödinger equations[END_REF]. We want to deal with the X T norms of ŵf and ŵg . Here we write the computations for ŵf , obviously, the results are the same mutatis mutandis for ŵg . We split the proof in three parts because of the several estimations needed in the X T norm.

3.1. The decay of R, one L ∞ estimate. We first focus on R. We recall that

R(s, ξ) = R(ĝ, ĝ, f )(s, ξ) = - i 4πs e -i ab 2s -1 F -1 η,σ [F (s, η, σ, ξ)] (a, b)dadb, with F (s, η, σ, ξ) := ĝ(s, ξ -σ)ĝ(s, ξ -η -σ) f (s, ξ -η).
We have the following lemma:

Lemma 3.1. For all 0 < 3α < δ < 1 4 we have |R(s, ξ)| s -1-δ ĝ 2 H 1,0 ξ f H 1,0 ξ . Remark 3.2.
This lemma is the key of the estimates we will deal with.

• The important result here is this small δ that ensures the integrability of R(s, .) L ∞ ξ . Moreover, this provides a good decay that will erase the behavior of the other parts, because 3α < δ, and ensures the existence of the solutions of our system (1.1).

• We see here why we choose the time 1 for initial data in the system (1.1). Indeed, here is the main problem with the integral in 0. Almost all the 1 t terms can be controlled in 0 by changing the coefficients in the X T norm (for example by changing the t in 1 + t), but not this 1 s 1+δ here. This is a consequence of the behavior of the solution of the linear Schrödinger equation near 0. Our purpose here is to highlight an asymptotic behavior, this is the reason why we just avoid this problem by taking a strictly positive time for the initial data.

• This lemma is satisfied for all 0 < 3α < δ < 1 4 . The goal after the lemma is to take α small enough, and δ as big as possible. Keeping this in mind, in the Subsection 4.4, it will be natural to consider that we can choose α and δ such that 0 < 4α < δ < 1 4 . Proof. By definition we have

|R(s, ξ)| 1 s e -i ab 2s -1 F -1 η,σ [F (s, η, σ, ξ)] (a, b) dadb.
Once again we use (2.9) with 0 < δ < 1 4 to obtain

|R(s, ξ)| s -1-δ |a| δ |b| δ F -1 η,σ [F (s, η, σ, ξ)] (a, b) dadb.
Let's focus on F -1 η,σ F ,

F -1 η,σ [F (s, η, σ, ξ)] (a, b) = F -1 η,σ ĝ(s, ξ -σ)ĝ(s, ξ -η -σ) f (s, ξ -η) (a, b) = F -1 η f (s, ξ -η)F -1 σ ĝ(s, ξ -σ)ĝ(s, ξ -η -σ) (b) (a). But F -1 (f g) = 1 √ 2π F -1 (f ) * F -1 (g), therefore, F -1 η,σ [F (s, η, σ, ξ)] (a, b) = F -1 η f (s, ξ -η) 1 √ 2π F -1 σ ĝ(s, ξ -σ) * F -1 σ ĝ(s, ξ -η -σ)(b) (a). Moreover, F -1 σ [ĝ(ξ -σ)](b) = e iξy g(-b) and F -1 σ [ĝ(ξ -σ)](b) = e iξb g(b), F -1 η,σ [F (s, η, σ, ξ)] (a, b) = 1 √ 2π F -1 η f (s, ξ -η) e ibξ g(s, -b) * b e ib(ξ-η) g(s, b) (b) (a) = 1 √ 2π F -1 η f (s, ξ -η) R e iyξ g(s, -y)e i(b-y)(ξ-η) g(s, b -y)dy (a) = 1 √ 2π R e ibξ g(s, -y)g(s, b -y)F -1 η e -i(b-y)η f (s, ξ -η) (a)dy. But F -1 η [e -ibη f (ξ -η)](a) = e iξ(a-b) f (b -a), F -1 η,σ [F (s, η, σ, ξ)] (a, b) = 1 √ 2π R e ibξ g(s, -y)g(s, b -y)e iξ(y+a-b) f (s, b -y -a)dy.
Finally, with the change of variables x ↔ (by) we obtain

F -1 η,σ [F (s, η, σ, ξ)] (a, b) = 1 √ 2π R e ibξ g(s, x -b)g(s, x)e iξ(a-x) f (s, x -a)dx = 1 √ 2π e i(a+b)ξ R e -ixξ g(s, x -b)g(s, x)f (s, x -a)dx.
Therefore, back on R, we have

|R(s, ξ)| s -1-δ |a| δ |b| δ |g(s, x -b)| |g(s, x)| |f (s, x -a)| dxdadb. Using that |a| δ |b| δ (|x -a| δ + |x| δ )(|x -b| δ + |b| δ ), we obtain |R(s, ξ)| s -1-δ |x -b| δ |g(s, x -b)| |g(s, x)| |x -a| δ |f (s, x -a)| dxdadb +s -1-δ |g(s, x -b)| |x| δ |g(s, x)| |x -a| δ |f (s, x -a)| dxdadb +s -1-δ |x -b| δ |g(s, x -b)| |x| δ |g(s, x)| |f (s, x -a)| dxdadb +s -1-δ |g(s, x -b)| |x| 2δ |g(s, x)| |f (s, x -a)| dxdadb s -1-δ |x| δ g L 1 x g L 1 x |x| δ f L 1 x + g L 1 x |x| δ g L 1 x |x| δ f L 1 x + |x| δ g L 1 x |x| δ g L 1 x f L 1 x + |x| 2δ g L 1 x g L 1 x g L 1 x .
Using the Cauchy-Schwarz inequality and the fact that δ < 1 4 , we get

x 2δ f L 1 x ≤ x 2δ x -1 L 2 x x f L 2 x f H 1,0 ξ . Therefore, |R(s, ξ)| s -1-δ f H 1,0 ξ ĝ 2 H 1,0 ξ .
3.2. Some L 2 estimates. By the previous estimation, we see that we first need to control the f

H 1,0 ξ
term, which will occur in a lot of our computations. By definition,

f (t) H 1,0 ξ = f (t) L 2 ξ + ∂ ξ f (t) L 2 ξ = ŵf (t) L 2 ξ + ∂ ξ ( ŵf B g )(t) L 2 ξ ≤ ŵf (t) L 2 ξ + ∂ ξ ŵf (t) L 2 ξ + ŵf (t) L ∞ ξ ∂ ξ B g (t) L 2 ξ ≤ ŵf (t) H 1,0 ξ + ŵf (t) L ∞ ξ t 1 s -1 ∂ ξ ŵg . ŵg (s) L 2 ξ ds.
Thus, using the definition of the X T -norm, we obtain

f (t) H 1,0 ξ t α ŵf XT (1 + ŵg 2 XT ). (3.1)
We now have all the tools for studying R(s) L2 ξ . By construction, we have that

R(s, ξ) = i √ 2π 4π 1 s |ĝ(s, ξ)| 2 f (s, ξ) -iF ξ e -is∂xx (|e is∂xx g(s, x)| 2 e is∂xx f (s, x)) . (3.2)
Using that e is∂xx ϕ L 2 ξ = ϕ L 2 ξ and the conservation of the L 2 norm of Fourier's transform, we get

R(s) L 2 ξ 1 s ĝ(s) 2 L ∞ ξ f (s) L 2 ξ + e is∂xx g(s) 2 L ∞ x f (s) L 2
x . Thus, we use the Lemma 2.3 with α < β and the fact that

(a + b) 2 a 2 + b 2 , R(s) L 2 ξ s -1 ĝ(s) 2 L ∞ ξ f (s) L 2 ξ + s -2β ĝ(s) 2 H 1,0 ξ f (s) L 2 ξ s -1 ŵg (s) 2 L ∞ ξ ŵf (s) L 2 ξ + s 2(α-β) ŵg (s) 2 XT (1 + ŵf (s) 2 XT ) 2 ŵf (s) L 2 ξ . Thus we get R(s) L 2 ξ s -1+α ŵg 2 XT ŵf XT (1 + ŵf 4 XT ), (3.3)
Now, let us deal with the derivative with respect to ξ. In the computation to obtain (3.1), we already showed that

∂ ξ B g (t) L 2 ξ t α ŵg 2 XT . (3.4)
For R, we go back once again to its construction. We have

R(s, ξ) = -i 2π e 2isησ ĝ(s, ξ -σ)ĝ(s, ξ -η -σ) f (s, ξ -η)dσdη + i √ 2π 4π 1 s |ĝ(s, ξ)| =: I(ĝ, ĝ, f )(s, ξ) + 1 s N (ĝ, ĝ, f )(s, ξ). (3.5)
Then, by linearity,

∂ ξ R = I(∂ ξ ĝ, ĝ, f ) + I(ĝ, ∂ ξ ĝ, f ) + I(ĝ, ĝ, ∂ ξ f ) + 1 s N (∂ ξ ĝ, ĝ, f ) + 1 s N (ĝ, ∂ ξ ĝ, f ) + 1 s N (ĝ, ĝ, ∂ ξ f ). (3.6)
Obviously, the three terms in I and N play the same role. We will focus on one of each.

I(∂ ξ ĝ, ĝ, f )(s) L 2 ξ F ξ e -is∂xx (e is∂xx xg(s, x)e -is∂xx g(s, x)e is∂xx f (s, x)) L 2 ξ e is∂xx f (s) L ∞ x e is∂xx g(s) L ∞ x xg(s) L 2 x s -1 ( f (s) L ∞ ξ + s -β f (s) H 1,0 ξ )( ĝ(s) L ∞ ξ + s -β ĝ(s) H 1,0 ξ ) ∂ ξ ĝ(s) L 2 ξ s -1+α ŵf XT (1 + ŵg 2 XT ) ŵg 2 XT (1 + ŵf 4 XT ).
For N , we have

s -1 N (∂ ξ ĝ, ĝ, f )(s) L 2 ξ s -1 f (s) L ∞ ξ ĝ(s) L ∞ ξ ∂ ξ ĝ(s) L 2 ξ s -1+α ŵf XT ŵg 2 XT (1 + ŵf 2 XT
). Thus, we can conclude that

∂ ξ R(s, ξ) L 2 ξ s -1+α ŵf XT ŵg 2 XT (1 + ŵg 2 XT )(1 + ŵf 4 XT ).
(3.7)

The computations are almost the same for the H 0,2n+1 ξ part. Since |B(s, ξ)| = 1, we just focus on R. With the decomposition (3.5), we have

ξ 2n+1 R(s, ξ) ≤ ξ 2n+1 I(ĝ, ĝ, f )(s, ξ) + 1 s ξ 2n+1 N (ĝ, ĝ, f )(s, ξ) .
For the I term, using that by convexity of x → x 2n+1 on R + , we have

|ξ| 2n+1 |ξ -σ| 2n+1 + |ξ -η| 2n+1 + |ξ -σ -η| 2n+1 . (3.8)
We obtain

ξ 2n+1 I(ĝ, ĝ, f )(s, ξ) ≤ I(ξ 2n+1 ĝ, ĝ, f )(s, ξ) + I(ĝ, ξ 2n+1 ĝ, f )(s, ξ) + I(ĝ, ĝ, ξ 2n+1 f )(s, ξ) .
Let's deal for example with the third part, we have

I(ĝ, ĝ, ξ 2n+1 f )(s, ξ) L 2 ξ e is∂xx g(s) 2 L ∞ x ∂ 2n+1 x f (s) L 2 x s -1 ( ĝ(s) L ∞ ξ + s -β ĝ(s) H 1,0 ξ ) 2 ξ 2n+1 f (s) L 2 ξ s -1+α ŵf XT ŵg 2 XT (1 + ŵf 4
XT ). Doing the same for the two others parts, we conclude for the I part by

ξ 2n+1 I(ĝ, ĝ, f )(s) L 2 ξ s -1+α ŵf XT ŵg 2 XT (1 + ŵg 2 XT )(1 + ŵf 4 XT ).
The N term is simpler, cause we remark that ξ 2n+1 N (ĝ, ĝ, f )(s, ξ) = N (ξ 2n+1 ĝ, ĝ, f )(s, ξ) . Thus we have that

s -1 ξ 2n+1 N (ĝ, ĝ, f )(s) L 2 ξ s -1 f (s) L ∞ ξ ĝ(s) L ∞ ξ ξ 2n+1 ĝ(s) L 2 ξ s -1+α ŵf XT ŵg 2 XT . Finally we conclude for R that ξ 2n+1 R(s) L 2 ξ s -1+α ŵg 2 XT ŵf XT (1 + ŵg 2 XT )(1 + ŵf 4 XT ).
(3.9)

Conclusion of the proof.

We have to compile our estimates with the norms that defined the X T norm. We remind that

ŵf XT := ŵf L ∞ T L ∞ ξ + t -α ŵf L ∞ T H 1,0 ξ + t -α ŵf L ∞ T H 0,2n+1 ξ , and
Φ 1 ( ŵf , ŵg )(t, ξ) := ŵ1 f (ξ) + t 1 B g (s, ξ)R(s, ξ)ds.
For the L ∞ part, we use the Lemma 3.1, the equation (3.1) and the fact that we have 3αδ < 0,

Φ 1 ( ŵf , ŵg )(t) L ∞ ξ ≤ ŵ1 f L ∞ ξ + t 1 R(s) L ∞ ξ ds ε + ŵf XT ŵg 2 XT (1 + ŵg 2 XT )(1 + ŵf 4 XT ).
For the H 1,0 part, we use the Lemma 3.1, the equations (3.3), (3.4) and (3.7),

t -α Φ 1 ( ŵf , ŵg )(t) H 1,0 ξ ≤ ŵ1 f H 1,0 ξ + t -α t 1 R(s) H 1,0 ξ + R(s) L ∞ ξ ∂ ξ B g (s) L 2 ξ ds ε + ŵf XT ŵg 2 XT (1 + ŵf 4 XT )(1 + ŵg 4
XT ). The H 0,2n+1 part is easier, just using (3.9) we get

t -α Φ 1 ( ŵf , ŵg )(t) H 0,2n+1 ξ ε + ŵf XT ŵg 2 XT (1 + ŵf 4 XT )(1 + ŵg 2 XT ).
Finally, we have, with a constant M > 0,

Φ 1 ( ŵf , ŵg ) XT ≤ ε + M ŵf XT ŵg 2 XT (1 + ŵf 4 XT )(1 + ŵg 4 XT ).
The other equation is obtained by the same way by symmetry of the roles of u and v.

Proof of the Theorem 1.1

We are now able to use the proposition in order to prove the Theorem 1.1. First, this gives us a local existence, by a fixed point argument in a suitable space. Moreover, by a continuity argument, we prove a global one. In the two last parts of this section, we use our estimates to get the L ∞ decay and the scattering.

4.1.

Local existence and uniqueness of the solution. For the introduction of the function Φ, we need to take care of the notations while considering two pairs of solutions. We consider the application Φ : ( ŵf , ŵg ) → (Φ 1 ( ŵf , ŵg ), Φ 2 ( ŵf , ŵg )) with Φ 1 and Φ 2 defined in (2.6). The idea is to apply a fixed point argument to Φ in

E := {( ŵf , ŵg ) ∈ X T , ŵf XT < 2ε, ŵg XT < 2ε} ,
that we endow with the norm ( ŵf , ŵg ) E = ŵf XT + ŵg XT . Using the Proposition 2.1, we have that

Φ 1 ( ŵf , ŵg ) XT ≤ ε + M ŵf XT ŵg 2 XT (1 + ŵf 4 XT )(1 + ŵg 4 XT ), Φ 2 ( ŵf , ŵg ) XT ≤ ε + M ŵg XT ŵf 2 XT (1 + ŵg 4 XT )(1 + ŵf 4 XT ).
For ε small enough,

8M ε 3 (1 + 16ε 4 ) 2 ≤ 1 2 ε. (4.1)
In this case, E is invariant under the action of Φ. For this invariance, we could have chosen just ε instead of 1 2 ε in the RHS of (4.1). The result would have been the same, but we will need a deeper accuracy for the global existence in the next part. We now have to show that Φ is a contraction, therefore we consider two pairs of solutions. Let ( ŵf1 , ŵg1 ), ( ŵf2 , ŵg2 ) ∈ E,

Φ( ŵf1 , ŵg1 ) -Φ( ŵf2 , ŵg2 ) E = Φ 1 ( ŵf1 , ŵg1 ) -Φ 1 ( ŵf2 , ŵg2 ) XT + Φ 2 ( ŵf1 , ŵg1 ) -Φ 2 ( ŵf2 , ŵg2 ) XT .
We do the computations for Φ 1 , they are mutatis mutandis the same for Φ 2 .

(Φ 1 ( ŵf1 , ŵg1 ) -Φ 1 ( ŵf2 , ŵg2 )) (t, ξ) = t 1 B g1 (s, ξ)R 1 (s, ξ)ds - t 1 B g2 (s, ξ)R 2 (s, ξ)ds = t 1 B g1 (R 1 -R 2 )(s, ξ)ds + t 1 R 2 (B g1 -B g2 )(s, ξ)ds.
As we saw in the proof of the Proposition 2.1, the role of f1 (t) -f2 (t) H 1,0 ξ will be predominant, but we have to do some preliminary computations before. We first focus on the coupling effect of B g1 -B g2 , which give us in a second time estimates for f1 -f2 . In the third part of the proof, we will focus on the effect of the R 1 -R 2 terms. To be as clear as possible, we split the proof by announcing the estimates we are going to prove. 2 ) , so we get

(B g1 -B g2 )(t, ξ) = 2i sin t 1 | ŵg1 (s, ξ)| 2 -| ŵg2 (s, ξ)| 2 2s ds e i   t 1 | ŵg1 (s, ξ)| 2 + | ŵg2 (s, ξ)| 2 2s ds   .
Thus, we have the following relations:

• (B g1 -B g2 )(t) L ∞ ξ ln(t)ε ŵg1 -ŵg2 XT . Actually, we have (B g1 -B g2 )(t) L ∞ ξ t 1 s -1 (| ŵg1 (s)| 2 -| ŵg2 (s)| 2 )ds L ∞ ξ t 1 s -1 ŵg1 (s) -ŵg2 (s) L ∞ ξ ( ŵg1 (s) L ∞ ξ + ŵg2 (s) L ∞ ξ )ds. • ∂ ξ (B g1 -B g2 )(t) L 2 ξ t α ε(1 + ε 2 ln(t)) ŵg1 -ŵg2 XT .
For this relation, we need more computations due to the derivative. We have

∂ ξ (B g1 -B g2 )(t) L 2 ξ ≤ ∂ ξ sin t 1 | ŵg1 (s)| 2 -| ŵg2 (s)| 2 2s ds L 2 ξ + sin t 1 | ŵg1 (s)| 2 -| ŵg2 (s)| 2 2s ds ∂ ξ e i   t 1 | ŵg1 (s)| 2 + | ŵg2 (s)| 2 2s ds   L 2
ξ . We consider separately the two parts of the sum. For the first part,

I 1 = ∂ ξ sin t 1 | ŵg1 (s)| 2 -| ŵg2 (s)| 2 2s ds L 2 ξ t 1 s -1 ∂ ξ (| ŵg1 (s)| 2 -| ŵg2 (s)| 2 )ds L 2 ξ t 1 s -1 ( ∂ ξ ŵg1 (s) L 2 ξ + ∂ ξ ŵg2 (s) L 2 ξ ) ŵg1 (s) -ŵg2 (s) L ∞ ξ + ( ŵg1 (s) L ∞ ξ + ŵg2 (s) L ∞ ξ ) ∂ ξ ( ŵg1 (s) -ŵg2 (s)) L 2 ξ ds t α ε ŵg1 -ŵg2 XT .
For the second one, using the same relation for the coupled part,

I 2 = sin t 1 | ŵg1 (s)| 2 -| ŵg2 (s)| 2 2s ds ∂ ξ e i   t 1 | ŵg1 (s)| 2 + | ŵg2 (s)| 2 2s ds   L 2 ξ t 1 s -1 ŵg1 (s) -ŵg2 (s) L ∞ ξ ( ŵg1 (s) L ∞ ξ + ŵg2 (s) L ∞ ξ )ds × t 1 s -1 ( ∂ ξ ŵg1 (s) L 2 ξ ŵg1 (s) L ∞ ξ + ∂ ξ ŵg2 (s) L 2 ξ ŵg2 (s) L ∞ ξ )ds t α ln(t)ε 3 ŵg1 -ŵg2 XT .
4.1.2. The f1 -f2 estimates. Using the previous estimates, we now deal with the coupling effect of f1 -f2 . First, we see that

f1 -f2 = ŵf1 B g1 -ŵf2 B g2 = ŵf1 (B g1 -B g2 ) + B g2 ( ŵf1 -ŵf2 ).
Thanks to this decomposition, we deal with the L ∞ , H 1,0 and H 0,2n+1 norms of f1 -f2 .

• ( f1 -f2 )(t) L ∞ ξ ln(t)ε 2 ŵg1 -ŵg2 XT + ŵf1 -ŵf2 XT .
We prove this inequality by using the estimation of

(B g1 -B g2 )(t) L ∞ ξ . We have ( f1 -f2 )(t) L ∞ ξ ŵf1 (t) L ∞ ξ (B g1 -B g2 )(t) L ∞ ξ + ( ŵf1 -ŵf2 )(t) L ∞ ξ . • ( f1 -f2 )(t) L 2 ξ t α ln(t)ε 2 ŵg1 -ŵg2 XT + t α ŵf1 -ŵf2 XT .
Indeed, by the exact same way but with the L 2 control of the ŵ terms,

( f1 -f2 )(t) L 2 ξ ŵf1 (t) L 2 ξ (B g1 -B g2 )(t) L ∞ ξ + ( ŵf1 -ŵf2 )(t) L 2 ξ . • ξ 2n+1 ( f1 -f2 )(t) L 2 ξ t α ln(t)ε 2 ŵg1 -ŵg2 XT + t α ŵf1 -ŵf2
XT . This is the same estimation and exactly the same computation because

ξ 2n+1 ( f1 -f2 )(t) L 2 ξ ξ 2n+1 ŵf1 (t) L 2 ξ (B g1 -B g2 )(t) L ∞ ξ + ξ 2n+1 ( ŵf1 -ŵf2 )(t) L 2 ξ . • ∂ ξ ( f1 -f2 )(t) L 2 ξ t α ln(t)ε 2 ŵg1 -ŵg2 XT + t α ŵf1 -ŵf2 XT .
For this one we have

∂ ξ ( f1 -f2 )(t) L 2 ξ ∂ ξ ŵf1 (t) L 2 ξ (B g1 -B g2 )(t) L ∞ ξ + ŵf1 (t) L ∞ ξ ∂ ξ (B g1 -B g2 )(t) L 2 ξ + ∂ ξ B g2 (t) L 2 ξ ( ŵf1 -ŵf2 )(t) L ∞ ξ + ∂ ξ ( ŵf1 -ŵf2 )(t) L 2 ξ .
Here, we can use the previous relations of this part, the equation (3.4), and the fact that ε is small to obtain the desired estimate.

Computing these estimates, we finally have

( f1 -f2 )(t) L ∞ ξ ln(t)ε 2 ŵg1 -ŵg2 XT + ŵf1 -ŵf2 XT , ( f1 -f2 )(t) H 1,0 ξ t α ln(t)ε 2 ŵg1 -ŵg2 XT + t α ŵf1 -ŵf2 XT , ( f1 -f2 )(t) H 0,2n+1 ξ t α ln(t)ε 2 ŵg1 -ŵg2 XT + t α ŵf1 -ŵf2 XT . (4.2) 4.1.3. The R 1 -R 2 estimates. • (R 1 -R 2 )(t) L ∞ ξ t -1-δ+3α ε 2 (ln(t)ε 2 + 1)( ŵg1 -ŵg2 XT + ŵf1 -ŵf2 XT ).
Indeed, by definition of R we have

|(R 1 -R 2 )(t, ξ)| = |R( ĝ1 , ĝ1 , f1 )(t, ξ) -R( ĝ2 , ĝ2 , f2 )(t, ξ)| ≤ |R( ĝ1 -ĝ2 , ĝ1 , f1 )| + |R( ĝ2 , ĝ1 -ĝ2 , f1 )| + |R( ĝ2 , ĝ2 , f1 -f2 )| (t, ξ).
Therefore, using the Lemma 3.1 and the equation (4.2), we obtain the announced result.

• (R 1 -R 2 )(t) L 2 ξ t -1+α ε 2 (ln(t)ε 2 + 1)( ŵf1 -ŵf2 XT + ŵg1 -ŵg2 XT ).
In order to get this estimate, we use the relation (3.5),

(R 1 -R 2 )(t) L 2 ξ ≤ (I( ĝ1 , ĝ1 , f1 ) -I( ĝ2 , ĝ2 , f2 ))(t) L 2 ξ + t -1 (N ( ĝ1 , ĝ1 , f1 ) -N ( ĝ2 , ĝ2 , f2 ))(t) L 2 ξ .
Once again, by linearity, we remark that

I( ĝ1 , ĝ1 , f1 ) -I( ĝ2 , ĝ2 , f2 ) = I( ĝ1 -ĝ2 , ĝ1 , f1 ) + I( ĝ2 , ĝ1 -ĝ2 , f1 ) + I( ĝ2 , ĝ2 , f1 -f2 ), N ( ĝ1 , ĝ1 , f1 ) -N ( ĝ2 , ĝ2 , f2 ) = N ( ĝ1 -ĝ2 , ĝ1 , f1 ) + N ( ĝ2 , ĝ1 -ĝ2 , f1 ) + N ( ĝ2 , ĝ2 , f1 -f2 ).
Here, we just show how to control one term in I, and one in N , all the others computations are the same. For I we use the Lemma 2.3 and the equations (3.1) and (4.2),

I( ĝ1 -ĝ2 , ĝ1 , f1 )(t) L 2 ξ e it∂xx g 1 (t) L ∞ x e it∂xx f 1 (t) L ∞ x (g 1 -g 2 )(t) L 2 x t -1 ( ĝ1 L ∞ ξ + t -β ĝ1 H 1,0 ξ )( f1 L ∞ ξ + t -β f1 H 1,0 ξ ) ĝ1 -ĝ2 L 2 ξ t -1+α ε 2 ln(t)ε 2 ŵf1 -ŵf2 XT + ŵg1 -ŵg2 XT .
For the N term, by equation (4.2) we have

N ( ĝ1 -ĝ2 , ĝ1 , f1 )(t) L 2 ξ = ( ĝ1 -ĝ2 ) ĝ1 f1 (t) L 2 ξ f1 L ∞ ξ ĝ1 L ∞ ξ ( ĝ1 -ĝ2 ) L 2 ξ t α ε 2 ln(t)ε 2 ŵf1 -ŵf2 XT + ŵg1 -ŵg2 XT .
Combining these results, we have the desired one. For the H 1,0 part, we want to show that

• ∂ ξ (R 1 -R 2 )(t) L 2 ξ t -1+α ε 2 (ln(t)ε 2 + 1)( ŵf1 -ŵf2 XT + ŵg1 -ŵg2 XT ).
To prove this one, we write ∂ ξ (R 1 -R 2 ) by using the equation (3.6), we obtain twelve terms, six terms in I, six in N . We need to couple these terms, using the same method as previously. For example, for I we have

I(∂ ξ ĝ1 , ĝ1 , f1 ) -I(∂ ξ ĝ2 , ĝ2 , f2 ) = I(∂ ξ ( ĝ1 -ĝ2 ), ĝ1 , f1 ) + I(∂ ξ ĝ2 , ĝ1 -ĝ2 , f1 ) + I(∂ ξ ĝ2 , ĝ2 , f1 -f2 ).
Thus, we obtain eighteen terms with a coupled part in each of them. Following the way of the estimation of (R 1 -R 2 )(t) L 2 ξ , and using the equations (3.1) and (4.2), we finally obtain the result. The last estimate we deal with is

• ξ 2n+1 (R 1 -R 2 )(t) L 2 ξ t -1+α ε 2 (ln(t)ε 2 + 1)( ŵg1 -ŵg2 XT + ŵf1 -ŵf2 XT ). This is the same estimation as (R 1 -R 2 )(t) L 2
ξ , and the computations are almost the same too. As in the previous estimate, we use the following decomposition,

|ξ 2n+1 (R 1 -R 2 )| ≤ |ξ 2n+1 R( ĝ1 -ĝ2 , ĝ1 , f1 )| + |ξ 2n+1 R( ĝ2 , ĝ1 -ĝ2 , f1 )| + |ξ 2n+1 R( ĝ2 , ĝ2 , f1 -f2 )|.
Let's just focus on the first term, using the equation (3.5), we have that

|ξ 2n+1 R( ĝ1 -ĝ2 , ĝ1 , f1 )| ≤ |ξ 2n+1 I( ĝ1 -ĝ2 , ĝ1 , f1 )| + t -1 |ξ 2n+1 N ( ĝ1 -ĝ2 , ĝ1 , f1 )|.
For the I term, we use a second time the equation (3.8) to obtain

|ξ 2n+1 I( ĝ1 -ĝ2 , ĝ1 , f1 )| ≤ |I(ξ 2n+1 ( ĝ1 -ĝ2 ), ĝ1 , f1 )| + |I( ĝ1 -ĝ2 , ξ 2n+1 ĝ1 , f1 )| + |I( ĝ1 -ĝ2 , ĝ1 , ξ 2n+1 f1 )|.
Now we just do the same computations that previously. We take the L 2 x part of the norm for the coupling part, and the others ones in L ∞

x . For N , we simply have

|ξ 2n+1 N ( ĝ1 -ĝ2 , ĝ1 , f1 )| = |N ( ĝ1 -ĝ2 , ξ 2n+1 ĝ1 , f1 )|.
ψ(t) = t 1 B g1 (R 1 -R 2 )(s, ξ)ds + t 1 R 2 (B g1 -B g2 )(s, ξ)ds.
Thus, we have

ψ(t) L ∞ ξ ≤ t 1 (R 1 -R 2 )(s) L ∞ ξ ds + t 1 R 2 (s) L ∞ ξ (B g1 -B g2 )(s) L ∞ ξ ds, ψ(t) L 2 ξ ≤ t 1 (R 1 -R 2 )(s) L 2 ξ ds + t 1 R 2 (s) L 2 ξ (B g1 -B g2 )(s) L ∞ ξ ds, ξ 2n+1 ψ(t) L 2 ξ ≤ t 1 ξ 2n+1 (R 1 -R 2 )(s) L 2 ξ ds + t 1 ξ 2n+1 R 2 (s) L 2 ξ (B g1 -B g2 )(s) L ∞ ξ ds, ∂ ξ ψ(t) L 2 ξ ≤ t 1 ∂ ξ B g1 (s) L 2 ξ (R 1 -R 2 )(s) L ∞ ξ ds + t 1 ∂ ξ (R 1 -R 2 )(s) L 2 ξ ds + t 1 R 2 (s) L ∞ ξ ∂ ξ (B g1 -B g2 )(s) L 2 ξ ds + t 1 ∂ ξ R 2 (s) L 2 ξ (B g1 -B g2 )(s) L ∞ ξ ds. Finally, we find, for a constant M > 0, Φ 1 ( ŵf1 , ŵg1 ) -Φ 1 ( ŵf2 , ŵg2 ) XT ≤ M ε 2 (1 + ln(t)ε 2 )( ŵf1 -ŵf2 XT + ŵg1 -ŵg2 XT ).
We chose ε small enough. Thus, for T sufficiently close to 1, M ε 2 (1 + ε 2 ln(T )) ≤ 1 4 , and wet get

Φ( ŵf1 , ŵg1 ) -Φ( ŵf2 , ŵg2 ) E ≤ 1 2 ( ŵf1 , ŵg1 ) -( ŵf2 , ŵg2 ) E . (4.3)
Finally, for such a T , Φ is a contraction. We can apply the fixed point theorem to get the local existence of the solutions. Let's see now of to pass from the local existence to the global one.

Global existence of the solution.

The key argument for the transition from the local existence to the global one is the continuity. First, we show that t → ŵf (t)

L ∞ ξ + t -α ŵf (t) H 1,0 ξ + t -α ŵf (t) H 0,2n+1
ξ is continuous. Then we will conclude by a connectivity argument. Let's remark that, for 1 < t 1 < t 2 < T , with the T defined in the previous section,

ŵf (t 2 ) -ŵf (t 1 ) = t2 t1 B g (s, ξ)R(ĝ, ĝ, f )(s, ξ)ds.
By the Lemma 3.1 and the estimate (3.1), we have that

ŵf (t 2 ) L ∞ ξ -ŵf (t 1 ) L ∞ ξ ≤ ŵf (t 2 ) -ŵf (t 1 ) L ∞ ξ ≤ t2 t1 R(ĝ, ĝ, f )(s) L ∞ ξ ds t 2 3α-δ -t 1 3α-δ -→ t1→t2 0. (4.4)
For the L 2 part, we remark that

t -α 2 ŵf (t 2 ) -t -α 1 ŵf (t 1 ) ≤ t -α 2 -t -α 1 | ŵf (t 2 )| + t -α 1 | ŵf (t 2 ) -ŵf (t 1 )| .
With the Proposition 2.1, we control the ŵf (t 2 ) H 1,0 ξ and ŵf (t 2 ) H 0,2n+1 ξ terms. Therefore, using this time the Lemma 3.1 and the estimates (3.1), (3.3), (3.4), (3.7) and (3.9), we have

ŵf (t 2 ) -ŵf (t 1 ) H 1,0 ξ ≤ t2 t1 ( R(ĝ, ĝ, f )(s) H 1,0 ξ + R(ĝ, ĝ, f )(s) L ∞ ξ ∂ ξ B(s) L 2 ξ )ds |t 2 α -t 1 α | + t 2 4α-δ -t 1 4α-δ -→ t1→t2 0. ŵf (t 2 ) -ŵf (t 1 ) H 0,2n+1 ξ ≤ t2 t1 R(ĝ, ĝ, f )(s) H 0,2n+1 ξ ds |t 2 α -t 1 α | -→ t1→t2 0.
Thus, we obtain by the previous computation the continuity of the application

t → ŵf (t) L ∞ ξ + t -α ŵf (t) H 1,0 ξ + t -α ŵf (t) H 0,2n+1 ξ .
Finally, taking the supremum of this application on [1, T ], we have the continuity of the application T → ŵf XT . Thanks to this continuity, we define the space

A := {T ≥ 1, ŵf XT ≤ 2ε} ⊂ [1, +∞).
Now, we show that A = [1, +∞). Let's remark that,

• A is non empty. Indeed, by the hypothesis of the Theorem 1.1 and the embedding of

H 1 (R) in L ∞ (R), 1 ∈ A. • A is closed.
This is due to the large inequality in the definition of A and the continuity of the application

T → ŵf XT . • A is open.
Here we use the Proposition 2.1, if T ∈ A, then we have that :

ŵf XT ≤ ε + M ŵf XT ŵg 2 XT (1 + ŵf 4 XT )(1 + ŵg 4 XT ) ≤ ε + 8M ε 3 (1 + 16ε 4 ) 2 .
Therefore, by the choice of ε in (4.1), ŵf XT ≤ 3 2 ε, we can go further for T . Finally, A is non empty, open, closed and included in the interval [1, +∞), therefore A = [1, +∞) by connectivity. Thus, we have the global existence under the assumptions of the Proposition 2.1. The last thing to check is that these assumptions are satisfied under the ones of the Theorem 1.1. Fix ε > 0 and suppose that u

1 H 1,0 x + u 1 H 0,2n+1 x ≤ ε 2 . We want to control ŵ1 f (ξ) = f (1, ξ) = e iξ 2 û(1, ξ), we have ŵ1 f (ξ) H 1,0 ξ ≤ û(1) L 2 ξ + ξ û(1) L 2 ξ + ∂ ξ û(1) L 2 ξ ≤ ε 2 , ŵ1 f (ξ) H 0,2n+1 ξ = û(1) H 0,2n+1 ξ ≤ ε 2 . Therefore, ŵ1 f (ξ) H 1,0 ξ + ŵ1 f (ξ) H 0,2n+1 ξ ≤ ε.
The hypothesis of the Proposition 2.1 are checked, and we have the global solution for the Theorem 1.1.

4.3.

The decay estimate. This estimate comes from the Lemma 2.3. Indeed, let (u, v) be the pair of solutions of our problem (1.1), thus we have that for all β ∈ (0, 1 4 ),

u(t) L ∞ x = e it∂xx f (t) L ∞ x 1 t 1 2 f (t) L ∞ ξ + 1 t 1 2 +β f (t) H 1,0 ξ 1 t 1 2 ŵf L ∞ ξ + 1 t 1 2 +β f (t) H 1,0 ξ .
By the definition of the X T norm and the bound (3.1), we get

u(t) L ∞ x t -1 2 (1 + t α-β ) ŵf XT (1 + ŵg 2
XT ). Remembering that α < β, we finally use the control of the X T norm from the previous subsection to get the desired decay of the Theorem 1.1.

4.4.

The long-range scattering. All the ingredients we need to exhibit the long-range scattering are already in the equation (4.4),

ŵf (t 2 ) -ŵf (t 1 ) L ∞ ξ t 2 3α-δ -t 1 3α-δ .
As 3αδ < 0, the Cauchy criterion for the existence of a limit of a function allows us to construct the application We still have to deal with the H 0,n case. For that purpose, we'll use the following embedding

W ∞ f ∈ L ∞ ξ (R
f L 1 ξ f 1 2 L 2 ξ xf 1 2 L 2 ξ . (4.6)
Let's prove this inequality, using Cauchy-Schwarz. Let M > 0,

f L 1 x = |x|<M 1 |f | + |x|>M x -1 x |f | M 1 2 f L 2 x + M -1 2 xf L 2 x . Choosing M = xf 1 2 L 2 x f -1 2 L 2 x
, we get the inequality (4.6). Thereby, we have that .

ξ n R L 2 ξ R 1 2 L ∞ ξ ξ n R 1 2 L 2 ξ R 1 2 L ∞ ξ ξ 2n R 1 2 L 1 ξ R 1 2 L ∞ ξ R 1 2 H 0,
These inequalities are verified for all α small enough and all 0 < δ < 1 4 . The idea is to take the limits for α → 0 and δ → 1 4 . Let ν := 1 4 -δ +4α ∈ (0, 1 4 ), we obtain the desired estimates of long-range scattering. The last thing to check is that W ∞ f = W n f . This is a consequence of the Riesz-Fischer Theorem for the completeness of L p . Indeed, it's proved in this theorem that if ŵf (t n ) converges to W in L p , then there exists a subsequence ŵf (t ϕ(n) ) that converges to W almost everywhere. Taking p = 2 and p = ∞, we show that W ∞ f = W n f almost everywhere. Thus we can define where, using the bound (3.1),

W f := W ∞ f = W n f ∈ L ∞ ξ ∩ H 0,
A f (t) L ∞ x t -1 2 -δ f H 1,0 ξ t -1 2 +α-δ t -3 4 +ν .
Thus, we obtain the following asymptotic expansion for u, for a large time t: u(t, x) = 1 (2it)

1 2 exp(i x 2 4t )W f ( x 2t ) exp( -i 2 √ 2π t 1 1 τ |v(τ, x 2t )| 2 dτ ) + O(t -3 4 +ν ). (4.8)
This is an implicit formula. Indeed we have a v term in the RHS of the equality. To deal with this term, we follow the method of Hayashi and Naumkin in [START_REF] Hayashi | Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations[END_REF]. Let us define γ g (t) := By using this bound in the previous equation we obtain |γ g (t)γ g (s)| t 3α-δs 3α-δ + t 3α-δ ln(t)s 3α-δ ln(s) .

Therefore, by the Cauchy criterion, it exists Γ g ∈ L ∞ ξ (R) defined by Γ g : ξ → lim t→+∞ γ g (t, ξ) in L ∞ ξ . Let t → +∞ in the previous equation. We use that ln(t) t α to obtain |γ g (s) -Γ g | s 4α-δ .

(4.10)

Back to the definition of γ g in (4.9), we have

4. 1 . 1 .

 11 The B g1 -B g2 estimates. First of all, let us rewrite B g1 -B g2 . For a, b ∈ R, e iae ib = 2i sin a-b 2 e i( a+b

  As we have the relation |ξ 2n+1 ĝ1 | = |ξ 2n+1 ŵg1 |, we have the same bound once again. 4.1.4. Conclusion. Set ψ(t) := (Φ 1 ( ŵf1 , ŵg1 ) -Φ 1 ( ŵf2 , ŵg2 )) (t), we remind that

t 1 (

 1 | ŵg (τ )| 2 -| ŵg (t)| 2 ) 1 ≤ s ≤ t, we have γ g (t)γ g (s) = t s (| ŵg (τ )| 2 -| ŵg (t)| 2 ) dτ τ + (| ŵg (s)| 2 -| ŵg (t)| 2 ) ln(s).We use the bounds of the Lemma 3.1 and the equation (3.1) to get| ŵg (τ )| 2 -| ŵg (t)| 2 || ŵg (τ )| -| ŵg (t)|| τ 3α-δt 3α-δ .

  ) by W ∞ f : ξ → lim t→+∞ ŵf (t, ξ) in the L ∞ sense. Let t 1 → +∞ in the previous equation, we obtain ŵf (t) -W ∞

	f	L ∞ ξ	t 3α-δ .	(4.5)

.

  ) in the last inequality. Finally we have ŵf (t 2 ) -ŵf (t 1 ) H 0,n As 4αδ < 0 (see Remark 3.2), the Cauchy criterion for the existence of a limit of a function allows us to construct the application W n f ∈ H 0,n ξ (R) by W n f : ξ → lim t→+∞ ŵf (t, ξ) in the H 0,n sense. Let t 1 → +∞ in the previous equation, we obtain

	ŵf (t) -W n f H 0,n ξ	t	4α-δ 2
						(4.7)
					2n+1	,
					ξ
	where we used (4.6ξ		t 2	4α-δ 2	-t 1	4α-δ 2

  n . 4.5. The asymptotic formula. From the previous part, we know that it exists an unique W ∞

	such that	f (t, .) exp	2	i √ 2π	1	t	1 s	|v(s, .)|	2 ds -W f L ∞ ξ	t -1 4 +ν .	ξ (R) f ∈ L ∞
	By the equation (2.8) we have
		u(t, x) = e it∂xx f (t, x) = (2it) -1 2 e i x 2 4t f (	x 2t	) + A

f (t, x),

f (s, ξ)

Work partially supported by the grant ANAÉ, ANR-13-BS01-0010-03. The author warmly thanks B. Grébert and L. Thomann for several useful discussions and their many suggestions.


Baptiste Mélès

Systems and Teaching in Stoic and Confucian Philosophies

published or not. The documents may come

Introduction *

Systems have a lot of virtues. Among their abilities, the theoretical ones are often underlined. First, the system, defined as a rational organization of philosophical discourse, gives consistency: it assures that there is no contradiction between the propositions constituting the theory. It also offers unity: every part of the theory is bound to a single principle. It eventually allows for completeness: the author as well as the reader are sure that no proposition is missing, since each one stays exactly at its place, leaving no lacuna. Systematicity thus guarantees scientificity, which is the highest theoretical virtue for which one can hope, even when systematicity is not claimed and remains implicit.

But one is led to examine whether systems may also have some teaching virtues, especially in moral philosophy -thus not only theoretical virtues, but also practical ones. Are systems appropriate ways of teaching the path to wisdom? To consider this question, two philosophical schools will be compared: stoicism in the Western tradition and confucianism in the ancient Chinese tradition.

1 Systematicity as the royal way to wisdom It has often been said that the Stoics invented systematicity. Without, of course, being the first philosophers to try and develop a consistent theory, they are said to have paid particular attention to dividing philosophy in a few domains -ethics, physics and logic -and showing the links between those fields. Long and Sedley thus write that "Of all ancient philosophies, Stoicism makes the greatest claim to being utterly systematic. Arguably, the Stoics invented the notion of philosophy as "system 1 " [...]." I do not intend to examine whether this precedence is historically valid, but I will try and show which systematicity is engaged in Stoic philosophy.

Indeed the risk is to interpret retrospectively the ancient notion of system, keeping in mind the later fortune of this concept, and especially Kant's or Hegel's views about, and use of, systems. That Stoics have promoted systems for their theoretical virtues, above all for consistency and completeness, is beyond doubt, and -rightly -often emphasized. Victor Goldschmidt has shown how the Stoic theory of time not only belongs to physics, but is also correlated to ethics and even logic. For instance, physics shows that only the present exists, and that past and future do not exist in a strong sense; ethics thus claim that I have to keep my mind on the present, preventing myself from any remorse or hope2 . Ethics plays in this insightful analysis the role of an object for the system -time indeed has an ethical meaning -, but the system as educational method does not explicitly seem to possess any ethical virtue: only theoretical ones.

Less often mentioned are the educational virtues systems had for the Stoics, however important it may have been for them. The goal of Stoic philosophy is not exclusively theoretical. Wisdom is aimed beyond science, as Seneca writes: "I will state the difference between wisdom and philosophy. Wisdom is the human mind's good brought to perfection. Philosophy is the love and pursuit of wisdom; it strives for the goal which wisdom has achieved 3 ." And the fact that philosophy intends to lead to wisdom is corroborated by the ternary division of virtue (ἁρετή, arétè), i.e. excellence, reported by Aetius: "The Stoics said that wisdom is scientific knowledge of the divine and the human, and that philosophy is the practice of expertise (τέκνη, techne) in utility. Virtue singly and at its highest are triple -the physical one, the ethical one, and the logical one. For this reason philosophy also has three parts -physics, ethics and logic 4 ." Philosophy leads to wisdom because they share the same structure. There is more: wisdom preserves, at the highest level, the content of philosophy.

Even the order of the parts of philosophy may have an educational meaning. While it is an order of importance, it also refers to an order of teaching, showing what the last goals of philosophical learning are. Several different orders have been proposed: some Stoics "assign the first place to logic, the second to physics and the third to ethics; these include Zeno in his book On Discourse, Chrysippus, Archedemus and Eudromus. Diogenes of Ptolemais starts with ethics, Apollodores puts ethics second, while Panaetius and Posidonius start with physics 5 ." This order may also appear as an order of exhibition, if we presume that the most important parts of philosophy are treated at the end of the curriculum, insofar as they require the knowledge of the other parts.

Emphasizing the theoretical virtues of systems in Stoic philosophyconsistency, unity and completeness -as it has often, and rightly, been done, should therefore not hide some of its practical virtues recognized and promoted by the Stoics. Systematicity in itself does not directly lead to wisdom, but is the best way to express philosophy, which in turn leads to wisdom, once excellence is reached. It is not enough to say that systematicity does not obstruct the quest for wisdom: it is, in fact, the royal way that leads to it.

2 Systematicity as obstructing the Way to wisdom A glimpse at ancient Chinese philosophical texts suffices to establish that few are systematic. Why is that?

For theoretical reasons, one might think, concerning ancient Chinese cosmology, as it was developped since the Yìj īng (易經), which describes the world in an ever-changing state6 . For theoretical reasons too, if all phenomena are relative, as in Zhuāngzi (莊子, Tchouang-Tseu), or if language is not trustworthy, as in Lǎozi (老子, Lao-Tseu) 7 . With regard to these ancient Chinese philosophies, the lack of systematicity is not surprising.

More unexpected is the fact that even some relatively systematic philosophies can refuse the systematic way of teaching. It is mainly the case of Confucius' thought 8 . The comparaison may be all the more instructive as Confucian philosophy has often been compared with Stoicism 9 .

Of course, the word "system" is nowhere to be found in Confucius' work, which is not expressed in a systematic way, consisting in a set of words or acts of the Master related by his disciples, similarly to Epictetus' Discourses; but this did not prevent Confucius from expressing himself about notions which constitute the concept of system in the Western tradition.

"The way that can be spoken of Is not the constant way; The name that can be named Is not the constant name." See Laozi, Tao Te Ching (道得經, Dàodéjīng), Penguin Classics, translated by D. C. Lau, London: Penguin Books, 2006, 1: 《道可道非常道,名可名非常名》.

8 Kǒngfūzǐ (孔夫子). 9 Both link cosmology and ethics, cosmological and social order. The notion of duty (κατήκοντα or officia in Stoic philosophy) is not universel, but depends from the social position. This last point is exactly what distinguishes Mencius' (孟子, Mèngzi) orthodox confucianism from Mòzi's (墨子) thought.

Consistency is the minimal condition for a theory to be a system; it can without doubt be expected from Confucius. But there is more: even unitythe fact that several theories are correlated to a single principle -is claimed. Confucius strongly asserts this last criterion of "systematicity:" "There is a single thread binding (貫, guàn) my way (道, dào) together10 ." Further he also claims, in almost the same words: "I have a single thread binding it all together11 ." Thus Confucius does not deny, nor minimize, the consistency and unity of his thought: he emphasizes it.

Concerning completeness, some passages in the Analects, especially the sixteenth chapter, consist in a set of lists: three kinds of beneficial friendship, of harmful friendship, of beneficial pleasures, of harmful pleasures, etc. But these lists have no regularity, no rule of construction; they are very rare in Confucius' work; and above all, their authenticity is very doubtful. Confucius thus emphasizes unity but avoids completeness.

Confucius' relation to the form of his moral philosophy is itself regulated by educational interest. The way in which thoughts are exhibited is not neutral with respect to the content of the lessons. Confucian philosophy may be a system, but is not exposed as such. Keeping silent about what makes from the system a totality allows for the revelation of disciples' qualities 12 . The good student is the one who can develop by himself what follows from the thoughts of his master: "Ah, dear Zígòng (子貢)! I eventually can talk to you about the Book of Odes (詩經, Sh īj īng)! Tell such a man something and he can see its relevance to what he has not been told 13 ." This criterion, which characterizes good students, also enables the master to identify bad ones: "When I have pointed out one corner of a square to anyone and he does not come back with the other three, I will not point it out to him a second time 14 ." Thus the real wise is not the one who has reached excellence in all the fields of philosophy, as in the Stoic notion of wisdom; he is the one who can see much further than what the master says. As Confucius compares two of his disciples, Yán Huí (顏回) 15 and Zígòng, the latter tells him: "How dare I compare myself with Hui? When he is told one thing he understands ten. When I am told one thing I understand only two 16 !" Refusing any systematic discourse, any claim to sufficiency, is therefore a way to let the disciples become intellectually autonomous. The most important quality is not to know, be it at the highest level of excellence, but to study (學, xué) [START_REF]To see the importance of "study" (學, xué), first caracter of the Analects (I, 1)[END_REF] . Completeness is educationally dangerous: it prevents the students, but also the master, from studying.

Conclusion

Thus a difference emerges between two educational strategies. The Stoics choose systems in order to promote an idea of excellence, which implies an excellence of knowledge. Confucius, who avoids systems, reveals and shapes the understanding abilities of his disciples.

This conflict of strategies finds an analogon in mathematics, the theoretical science par excellence. While Euclides develops his mathematics in a systematic way, and tends to generality, the Nine Chapters, a classical book of Chinese mathematics, only exhibits particular cases. Not by inability, as Karine Chemla has shown [START_REF] Dunod | Les Neuf Chapitres. Le classique mathématique de la Chine ancienne et ses commentaires[END_REF] , but by choice: the reader has to find truths by himself. This confrontation reveals that systematicity does not only raise theoretical problems, but also educational ones.

A. A.Long and D. N. Sedley, The Hellenistic Philosophers, vol. 1, Cambridge: Cambridge University Press, 1987, p. 160.

Victor Goldschmidt, Le Système stoïcien et l'idée de temps, Collège philosophique, Paris:Vrin, 1953, ch.

3, iii, p. 73. 3 Seneca, Letters, 89,[4][5] op. cit., 26 G, p. 160. 4 Aetius I, Preface 2, SVF II, 35. See ibid., 26 A, p. 158. 5 Diogenes LaërtiusVII,[39][40][41] 26 B, p. 158.

Although we could have expected a systematic treatment from this book, which is structured by regular combinations of plain lines "-" (陰, yīn) and broken ones "--" (陽, yáng).

The Book of Way and Virtue begins with the words:

Confucius, The Analects ( 論語, Lúnyǔ), Penguin Classics, translated by D. C. Lau, London: Penguin Books, 1979, IV, 15: 《吾道一以貫之》.

Ibid., XV, 3: 《子一以貫之》.

A deep analysis of ancient Chinese teaching strategy, especially in Confucian philosophy, can be found in François Jullien, Le Détour et l'accès. Stratégies du sens en Chine, en Grèce, Paris: Grasset, 1995, ch. ix. In the same book, François Jullien highlights the theoretical reasons of this educational strategy and compares it, among other things, to military strategy.

 13 Confucius, op. cit., I, 15. See also III, 8 for an example of a good disciple, understanding and pursuing the master's thought, and therefore able to talk with him from the

Shījīng 14 Ibid

., VII, 8.15 Yán Huí, who died precociously, was Confucius' favorite disciple; see ibid., V, 25; VI, 2, 5 and 9; VII, 10;IX, 10, 19 and 20; XI, 2,[6][7][8][9][10] 22; XII, 1; and probably, following the tradition, VIII, 5.


Daniel Alexandre

Georges Lutfalla

Michel Gross
email: michel.gross@univ-montp2.fr

Holographic imaging of Zebrafish embryo blood flow with dually oriented illumination beams

Keywords:

An holographic laser Doppler scheme using two illumination beams is proposed to image blood vessel in fish embryo. The coincidence of the reconstructed images made for each illumination orientation leads to an accurate z sectioning.

Blood flow imaging techniques are widely used in biomedical studies, since they can assess physiological processes or can be used for early detection of disease. However, many blood flow studies require, for imaging purposes, the use of a contrast agent, making the blood flow characterization invasive [START_REF] Sakurada | Measurement of local cerebral blood flow with iodo [14c] antipyrine[END_REF]. Combination of laser Doppler holography [START_REF] Atlan | Frequency-domain wide-field laser doppler in vivo imaging[END_REF] with transmission microscopy has been proposed to image blood flow in fish embryos [START_REF] Verrier | Laser doppler holographic microscopy in transmission: application to fish embryo imaging[END_REF]. To enhance the slicing along z, we have used two illumination beams oriented in two different directions. We have then selected in the Fourier space the field corresponding to each of the illuminations, and we have reconstructed the blood vessels images. By adjusting the reconstruction distance so that the reconstructed images for each illumination coincide, a precise measurement of the vessel z position can be obtained. The set-up (see Fig. 1) is an upright microscope that has been modified for making heterodyne holography [START_REF] Le Clerc | Numerical heterodyne holography with two-dimensional photodetector arrays[END_REF]. The main laser (frequency ω I ) is split into two arms (illumination and reference) by a beam splitter. Two acousto optic modulators (AOM) at ω 1,2 ≃ 80MHz control the frequency ω LO = ω I + ω 1 -ω 2 of the reference arm (E LO ). The illumination arm (E) is split into two branches so as to illuminate the sample in two directions. The sample is a zebrafish embryo (48 hours-old) between slide and cover slip. It is imaged with a microscope objective MO (NA= 0.25, Fig. 2. Pupil images reconstructed with 4 phases holograms H = (I 0 -I 2 ) + j(I 1 -I 3 ), and with ω 1 -ω 2 = ω CCD /4. Sample is ground glass (a) or zebrafish sample (b,c). G=10). To achieve off-axis holography, the beam splitter that recombines the signal (E) and reference (E LO ) fields is angularly tilted: θ = 0. The CCD camera (1280 × 1024 pixels, ω CCD = 110 Hz, 10 bits) record the interference pattern

Material and method

I = |E + E LO | 2 .
Sequences of 300 frames are recorded: I 0 ... I 299 . Recorded data are cropped into a 1024 × 1024 calculation grid to perform holographic calculations like Fast Fourier Transforms (FFT).

To select the signal corresponding to each illumination, we have reconstructed the image of the field E scattered by the sample in the MO back pupil plane. Reconstruction is made by the 1 FFT method with 4 phases holograms H = (I 0 -I 2 ) + j(I 1 -I 3 ), and with ω 1 -ω 2 = ω CCD /4. Images are displayed in intensity by averaging data over 80 frames. We have imaged a ground glass (Fig. 2 (a) ) and a zebrafish sample (Fig. 2 (b,c) ). The pupil image, which is located in the upper right hand size of the calculation grid, has been displayed in the center of Fig. 2 (a...c). The pupil image is especially bright with the ground glass. It corresponds to the bright circular zone of Fig. 2 (a), whose edges are sharp (white dashed line circle). We have select the useful holographic information by cropping the pupil zone (and by filling the rest of the calculation grid with zeros).

Since the zebrafish sample is quite transparent, the two illumination beams yield two bright spots corresponding to the white arrows of Fig. 2 (b). Since these spots are well separated in the Fourier space, we have cropped 2 circular zones of radius 240 pixels centered on each spot (dotted white circle of Fig. 2 (c) ) in order to perform reconstruction with one or other of the illumination beams (i.e with the purple and green zone of Fig. 2 (c)). phases holograms H = (I 0 -I 2 ) + j(I 1 -I 3 ), and with ω 1 -ω 2 = ω CCD /4). Reconstruction is made by the 2 FFT method from the FFT -1 of the Fourier space selected data [START_REF] Verpillat | Dark-field digital holographic microscopy for 3d-tracking of gold nanoparticles[END_REF]. The images of Fig. 3 We have then made the 2 beams reconstruction by using a different holographic modality. To image the moving red blood cells (and thus the blood vessel by averaging the reconstructed images over 80 frames), we have consider 2 phases holograms H = I 0 -I 1 with ω 1 -ω 2 = 0. By the way, the signal which is scattered at the illumination frequency ω I is not detected, while the signal of the moving cells, which is Doppler shifted is detected. The reconstructed images obtained by selecting the purple and green zones are displayed on Fig. 4 (a) and (b). They correspond to a single illumination beam, and are very similar. To better visualise the small differences of the two images, we have displayed them in purple and green on Fig. 4 (c). For the vessels that are located in the reconstruction plane z = 0, the purple and green images coincide , as seen in the zoom of Fig. 4 (c)). Figure 4 (d) show the purple and green images obtained for z ≃ 360µm. The coinciding vessels are now well separated, and the separation is much larger than the slight blur of the vessel seen on the zoom of Fig. 4 (d). The coincidence of the purple and green images yield thus a much better determination of the vessel z position than the blurring of the reconstructed images.

Results

Exploiting the z information of the coincidence of purple and green image obtained with double illumination beam open the way to 3D imaging of the structure of the embryos blood flow. Motions of the individual reed blood cell with possible coincidence of images can be also studied from sequence of reconstructed images similar to the one displayed on Fig. 4. Experiment is under progress.

All animal experiments that are described in the present study were conducted at the University Montpellier 2 according to European Union guidelines for the handling of laboratory animals (http://ec.europa.eu/environment/chemicals/lab_animals/home_en.htm) and were approved by the Direction Sanitaire et Vétérinaire de l'Hérault and Comité d'Ethique pour l'Expérimentation Animale under reference CEEA-LR-13007.

Fig. 1 .

 1 Fig. 1. Holographic microscopy setup: BS: beam splitter; M: mirror; ω 1 , ω 2 : Acousto optics modu- lators (Bragg cell) that shift the frequency of the local oscillator beam. MO: microscope objective; CCD: CCD camera; E and E LO : signal and reference complex fields.

Fig. 3 .

 3 Fig. 3. Zebrafish images reconstructed with 4 phases holograms H = (I 0 -I 2 ) + j(I 1 -I 3 ), and with ω 1 -ω 2 = ω CCD /4 with the purple (a) and green (b) zone of Fig.2 (c).

Figure 3 (

 3 Figure 3 (a) and (a) shows the zebrafish reconstructed images obtained at illumination frequency ω I i.e. with 4 phases holograms H = (I 0 -I 2 ) + j(I 1 -I 3 ), and with ω 1 -ω 2 = ω CCD /4). Reconstruction is made by the 2 FFT method from the FFT -1 of the Fourier space selected data[START_REF] Verpillat | Dark-field digital holographic microscopy for 3d-tracking of gold nanoparticles[END_REF]. The images of Fig.3(a) and (b) are obtained by selecting the purple (a) and green (b) zones of the Fourier space (i.e. with one or the other illumination beam). The images of Fig. 3 (a, b) are quite different, since the illumination directions are different.

  Figure 3 (a) and (a) shows the zebrafish reconstructed images obtained at illumination frequency ω I i.e. with 4 phases holograms H = (I 0 -I 2 ) + j(I 1 -I 3 ), and with ω 1 -ω 2 = ω CCD /4). Reconstruction is made by the 2 FFT method from the FFT -1 of the Fourier space selected data[START_REF] Verpillat | Dark-field digital holographic microscopy for 3d-tracking of gold nanoparticles[END_REF]. The images of Fig.3(a) and (b) are obtained by selecting the purple (a) and green (b) zones of the Fourier space (i.e. with one or the other illumination beam). The images of Fig. 3 (a, b) are quite different, since the illumination directions are different.

Fig. 4 .

 4 Fig. 4. Recontructed image of zebrafish blood vessels made with 2 phases holograms H = I 0 -I 1 ) and with ω 1 -ω 2 = 0. Reconstruction is made by selecting the purple (a) or the green (b) or both zone (c,d). In the last case the 2 images are displayed in purple and green. Reconstruction is made with z = 0 (a...c) and z ≃ 370µm. Images are obtained by averaging reconstructed signal energy over 80 frames.

Acknowledgments. We acknowledge OSEO-ISI Datadiag grant, ANR Blanc Simi 10 (n 11 BS10 015 02) grant and Labex Numev (convention ANR-10-LABX-20) grant for funding, and the European Community's Seventh Framework Programme (FP7-PEOPLE-2011-ITN) under Grant Agreement PITN-GA-2011-289209 for the Marie-Curie Initial Training Network FishForPharma for animals.


Michel Gross
email: michel.gross@univ-montp2.fr

Heterodyne Holography: An Optimal and Versatile 2D Detection Scheme

Keywords: Holography, Digital holography, 040.2840, Detectors-Heterodyne

Heterodyne holography is a variant of phase shifting holography in which reference and signal arms are controlled by acousto optic modulators. We will briefly describe the method and illustrate its advantages on experimental examples.

Introduction

Heterodyne holography is a variant of phase shifting holography [START_REF] Yamaguchi | Phase-shifting digital holography[END_REF] introduced about 15 years ago [START_REF] Le Clerc | Numerical heterodyne holography with two-dimensional photodetector arrays[END_REF] in which the frequency, phase and amplitude of both reference and signal beams are controlled by acousto optic modulators (AOM). By shifting the frequency ω LO of the local oscillator beam with respect to the frequency ω I of illumination, heterodyne holography is able to detect the light scattered by the object at a frequency ω close to ω LO , which can be different than the frequency ω I of illumination of the object. To fully exploit this capability without being affected by a spurious detection at the alias frequency ≃ 2ω I -ω LO heterodyne holography must be combined with off-axis holography. This has been done in 2003 [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasoundmodulated optical imaging[END_REF], and since. • To detect the signal at the illumination frequency ω I with 4 phases and n max =4 frames, we must have: H = (I 0 -I 2 ) + j(I 1 -I 3 ) and ω LO = ω I + ω CCD /4 (i.e. ω AOM2 -ω AOM1 = ω CCD /4).

Setup and method

• To make detection at ω = ω I with 2 phases and n max frames, we must have similarly: (c,d) Sum of the detected signal | H| 2 in region 1 (c) and 2 (d) versus ω AOM2 -ω AOM1 : measurement (points) and theory (solid grey line). See [START_REF] Atlan | Accurate phase-shifting digital interferometry[END_REF].

H = ∑ n max n=0 (-1) n I n and ω LO = ω + ω CCD /2 (i.e. ω AOM2 -ω AOM1 = ω -ω I + ω CCD /2).

Advantages and examples

Heterodyne holography exhibit many advantages. First, phase shifting is made with a very accurate phase [START_REF] Atlan | Accurate phase-shifting digital interferometry[END_REF]. The hologram can thus be calculated without having to take account of the phase defects of the experimental setup. The frequency response of the holographic detection can thus be calculated and agrees with calculation. As it can be seen in Fig. 2, heterodyne holography performs the detection at the two sideband frequencies ω LO ∓ ω CCD /4 that correspond to 4 phases demodulation. The off-axis configuration of Fig. 1 allows to separate these 2 sidebands in the Fourier space (peak 1 and 2 are separated). It is then possible to select one of the two sideband, avoiding thus frequency alias.

The second advantage is selectivity and sensitivity. In Fig. 2 (a), the useful signal is peak 1. It is extracted in the recorded data (I 0 ...I 3 ) by a double filtering procedure. The hologram H is calculated by H = (I 0 -I 2 ) + j(I 1 -I 3 ). This means, that the time frequency component ω LO -ω CCD /4 is selected in the data. Then one calculates H = FFT (H), and one select the spatial frequency components near peak 1. This double filtering procedure is very efficient to filter off the zero and twin images grating order aliases. The technical noise thus cancel, and detection can easily reach shot noise sensitivity [START_REF] Gross | Digital holography with ultimate sensitivity[END_REF]. Lesaffre et al. [START_REF] Lesaffre | Noise and signal scaling factors in digital holography in weak illumination: relationship with shot noise[END_REF] have shown that the total amount of noise (for all pixels of the reconstructed image) is A photo electrons, where A is the area in pixel Units of the Cuche et al. spatial filter [START_REF] Cuche | Spatial filtering for zero-order and twin-image elimination in digital off-axis holography[END_REF]. The noise do not thus depends on the number n max of frames I n used to calculate H, while the detected signal is proportional to n max . One must also mention that the noise floor related to shot noise can be used to calibrate the signal that is detected by holography [START_REF] Gross | Heterodyne detection of multiply scattered monochromatic light with a multipixel detector[END_REF][START_REF] Absil | Photothermal heterodyne holography of gold nanoparticles[END_REF].

The third advantage is the ability of detecting a signal at a frequency different from that illumination. This point splits in two sub points. In some cases, the object moves and the scattered light exhibit a broads continuous frequency spectrum, which can be analyzed by sweeping ω AOM1 or ω AOM2 (in order to sweep the detection frequency ω). The method is called Laser Doppler holography [START_REF] Atlan | Laser doppler imaging, revisited[END_REF] and has been applied to image blood flow in mouse crania [START_REF] Atlan | Frequency-domain wide-field laser doppler in vivo imaging[END_REF][START_REF] Atlan | Cortical blood flow assessment with frequency-domain laser doppler microscopy[END_REF], rat eye [START_REF] Simonutti | Holographic laser doppler ophthalmoscopy[END_REF] and fish embryo [START_REF] Verrier | Laser doppler holographic microscopy in transmission: application to fish embryo imaging[END_REF]. The method has been also used to analyse the broadening of the light that travel through a women breast in strong multiple scattering condition [START_REF] Gross | Heterodyne detection of multiply scattered monochromatic light with a multipixel detector[END_REF]. In that case, the signal is very week and corresponds to less than 0.1 photo electron per pixel at maximum. The object that is imaged to detect the signal is not the breast, but a rectangular slit (of 5 mm width) that is back illuminated by the light that travel through the breast. The measurement is made by comparing the signal detected in and outside the slot.

In some other cases, the objet vibrates at a fixed frequency ω A , and the scattered light exhibits frequency sidebands that can be detected selectively by heterodyne holography. The method is then called Sideband Digital holography [START_REF] Joud | Imaging a vibrating object by sideband digital holography[END_REF]. It is more efficient than time averaged holography [START_REF] Picart | Time-averaged digital holography[END_REF] that makes the holographic detection at the carrier frequency( ω = ω I ), especially for studying vibrations of large [17] and small amplitude [START_REF] Psota | Measurement of piezoelectric transformer vibrations by digital holography[END_REF][START_REF] Verrier | Absolute measurement of small-amplitude vibrations by time-averaged heterodyne holography with a dual local oscillator[END_REF]. We must also mention that the RF signals that drive the AOMs can be used to switch on an off both illumination and detection in order to combine sideband holography with strobe illumination and detection. This stroboscopic sideband holography is sensitive to the phase of the vibration motion [START_REF] Verpillat | Imaging velocities of a vibrating object by stroboscopic sideband holography[END_REF]. Vibration of extremely low amplitude can be also imaged in amplitude and phase by displaying the carrier versus first sideband signals correlation [START_REF] Verrier | Vibration of low amplitude imaged in amplitude and phase by sideband versus carrier correlation digital holography[END_REF].

Fig. 1 .

 1 Fig. 1. Typical heterodyne holography setup in transmission geometry. L: main laser; BS: beam splitter; AOM1, AOM2: acousto-optic modulators; BE: beam expander; M, mirror; A1, A2: light attenuators; CCD: camera; E L , E I , E LO , and E: laser, illumination, reference (i.e. local oscillator LO) and object fields; ω AOM1/2 : driving frequencies (≃ 80 MHz) of the acousto optics modulators AOM1 and AOM2; θ : off-axis angular tilt. where

Fig. 2 .

 2 Fig. 2. (a,b) Fourier space holograms | H| 2 recorded with ω AOM2 -ω AOM1 = +ω CCD /4 (a) and -ω CCD /4 (b).(c,d) Sum of the detected signal | H| 2 in region 1 (c) and 2 (d) versus ω AOM2 -ω AOM1 : measurement (points) and theory (solid grey line). See[START_REF] Atlan | Accurate phase-shifting digital interferometry[END_REF].

  Figure 2 (a) and (b) shows the Fourier hologram H = FFT(H) of a USAF target obtained with H = (I 0 -I 2 ) + j(I 1 -I 3 ) and ω AOM2 -ω AOM1 = +ω CCD /4 (a) and -ω CCD /4 (b). Peak 1 corresponds to the transmitted ballistic light detected by 4 phase detection at ω LO -ω CCD /4, while peak 2 is the spurious alias at ω LO + ω CCD /4. We have plotted on Fig. 2 (c) and (d) the frequency response of the holographic detection made on peak 1 (c), and peak 2 (d). Experiment (points) agrees with theory (solid grey line).

Acknowledgments. We acknowledge ANR Blanc Simi 10 (n • 11 BS10 015 02), ANR Blanc Simi 4 (n • 11-BS04 017 04) and Labex Numev (convention ANR-10-LABX-20) grants for funding.