Rana Badreddine
email: rana.badreddine@universite-paris-saclay.fr

ON THE GLOBAL WELL-POSEDNESS OF THE CALOGERO-SUTHERLAND DERIVATIVE NONLINEAR SCHR ÖDINGER EQUATION

Keywords: Mathematics Subject Classification. 37K10, 35Q55 Calogero-Sutherland-Moser systems, Derivative nonlinear Schrödinger equation (DNLS), Global well-posedness, Explicit solution, Hardy space, Integrable systems, Lax operators, L 2 -critical, Relatively compact orbits

We consider the Calogero-Sutherland derivative nonlinear Schrödinger equation in the focusing (with sign +) and defocusing case (with sign -)

where Π is the Szegő projector Π n∈Z u(n) e inx = n≥0 u(n) e inx . Thanks to a Lax pair formulation, we derive the explicit solution to this equation. Furthermore, we prove the global well-posedness for this L 2 -critical equation in all the Hardy Sobolev spaces H s + (T) , s ≥ 0 , with small L 2 -initial data in the focusing case, and for arbitrarily L 2 -data in the defocusing case. In addition, we establish the relative compactness of the trajectories in all H s + (T) , s ≥ 0 .

Introduction

1.1. Main results 1.2. Other related equations 1.3. Outline of the paper Acknowledges. 2. The Lax pair structure 2.1. The explicit formula of the solution 2.2. Global well-posedness of (CS + ) in H s + (T) , s > 3 2 3. Extension of the flow of (CS + ) to L 2 + (T) 3.1. Uniqueness of the limit and weak convergence in L 2 + (T) 3.1.1. Spectral properties of L u 0 for u 0 ∈ L 2 + (T) 3.1.2. Characterization of the limit u(t) 3.2. Strong convergence in L 2 + (T) and conservation of the L 2 -mass 4. Proof of Corollary 1.3 and Theorem 1.4 5. The Calogero-Sutherland DNLS defocusing equation (CS -) 6. Final remarks and open problems References

Introduction

This paper aims to prove the global well-posedness for the Calogero-Sutherland derivative nonlinear Schrödinger equations on the torus x ∈ T := R/(2πZ) :

i∂ t u + ∂ 2 x u ± 2D + (|u| 2 )u = 0 , u(t = 0, x) = u 0 , x ∈ T , (CS)
for small L 2 -initial data u 0 in the focusing case (with sign +) , and for arbitrarily L 2 -initial data in the defocusing case (with sign -). The operator D + in the nonlinear term of (CS) denotes DΠ , where D = -i∂ x , and Π is the Szegő projector acting on L 2 (T) as

Π n∈Z u(n) e inx := n≥0 u(n) e inx , (1.1)
with value onto the Hardy space

L 2 + (T) := u ∈ L 2 (T) | u(n) = 0 , ∀n ∈ Z ≤-1 ≡ Π(L 2 (T)) .
(1.2)

We equip L 2 + (T) with the standard inner product of L 2 (T) , ⟨u | v⟩ = 2π 0 uv dx 2π . Our interest focuses on studying this equation with an unknown function u taken in the Hardy space of the torus, with a certain regularity. Thus, we denote by H s + (T) , the subspace of the Sobolev space H s (T) , defined as

H s + (T) := H s (T) ∩ L 2 + (T) , s ≥ 0 , (1.3)
and equipped with the Sobolev norm

∥u∥ H s = ∥⟨D⟩ s u∥ L 2 , ⟨D⟩ s = (1 + |D| 2 ) s/2 .
In Physics, this dynamical (CS)-equation is derived from the classical Calogero-Sutherland-Moser system (or Toda system) introduced in the end sixties-early seventies [START_REF] Calogero | Ground State of a One-Dimensional N-Body System[END_REF][START_REF] Calogero | Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials[END_REF][START_REF] Sutherland | Exact results for a quantum many-body problem in one dimension[END_REF][START_REF] Sutherland | Exact ground-state wave function for a one-dimensional plasma[END_REF]. This physical model corresponds to a N -body problem describing the pairwise interactions of N identical particles. Abanov-Bettelheim-Wiegmann show in [START_REF] Abanov | Integrable hydrodynamics of Calogero-Sutherland model: bidirectional Benjamin-Ono equation[END_REF] that taking the thermodynamic limit of such a model, and applying a change of variables leads to the (CS)-equation. In Mathematics, this equation has recently been studied on the real line (x ∈ R) by [START_REF] Gérard | The Calogero-Moser Derivative nonlinear Schrödinger equation[END_REF] , who referred to the equation as the Calogero-Moser derivative NLS equation. The transition of nomenclature to the Calogero-Sutherland DNLS equation in the periodic setting (x ∈ T) is connected to the physicist Sutherland, who has studied the Calogero-Sutherland-Moser system in the case where the N particles lie on the circle and interact with an inverse sin-square potential (trigonometric-type potential). Besides, one can obtain the (CS)-equation formally as a limit of the intermediate nonlinear Schrödinger equation introduced by Pelinovsky [START_REF]Pelinovsky Intermediate nonlinear Schrödinger equation for internal waves in a fluid of finite depth[END_REF] ,

i∂ t u = ∂ 2 x u + (i -T )∂ x (|u| 2 ) u , ( INS
)
where T is the integral operator T u(t, x) = 1 2δ p.v. It turns out that the Calogero-Sutherland DNLS equation is completely integrable. Thus, what does the word "integrability" mean? In line with the different perspectives developed by various schools, a number of definitions have been raised. If the word "integrable system" means for some researchers the existence of action-angle variables, a coordinate system in which the equation is completely solvable by quadratures, others would say that it refers to the existence of a Lax operator associated with the equation, and satisfying the isospectral property 1 . However, a common facet of all these definitions is the presence of infinitely independent integrals of motion, or what we can also call conservation laws. Naturally, this infinite number of conservation laws plays a crucial role in proving some global well-posedness results.

In our case, Gérard-Lenzmann derived in [GL22, Lemma 5.1] , for u sufficient regular, a Lax operator so that the focusing Calogero-Sutherland DNLS equation (CS) + enjoys a Lax pair formulation on the real line R . i.e., for any u ∈ H s + (R) with s sufficiently large, there exist two operators (L u , B u ) such that the Lax equation

dL u dt = [B u , L u ] , [B u , L u ] := B u L u -L u B u , (1.4)
is satisfied with

L u = D -T u T ū , B u = T u T ∂x ū -T ∂xu T ū + i(T u T ū) 2 .
(1.5)

The operator T u is the Toeplitz operator of symbol u , and is defined for any u ∈ L ∞ by

T u f = Π(uf ) , ∀f ∈ L 2 + , (1.6)
where Π is the Szegő projector given in (1.1) . In what follows, we check that this Lax equation holds true on the torus T by retrieving the same Lax operators (L u , B u ) as on the real line . And, as expected, through this Lax formalism, we derive infinite conservation laws ⟨(L u + λ) s 1 | 1⟩ , λ >> 0 , s ≥ 0 , in order to control the growth of the Sobolev norms ∥u(t)∥ Ḣs uniformly for all t ∈ R . 2

Observe, the Calogero-Sutherland DNLS equation is invariant under the scaling u(t, x) -→ λ 1/2 u(λ 2 t, λx) , λ ∈ R , (t, x) ∈ I × R .

(1.7)

1 See Remark 2.1.

2 In particular, one can see that the usual conservation laws : the average ⟨1 | u⟩ , and the L 2 norm ∥u∥ L 2 are conserved for s = 1 and 2 , since by definition of L u = D -T u T ū we have L u 1 = -⟨1 | u⟩ u .

This suggests the L 2 -criticality of (CS) on R as well as on T . In [GL22, Theorem 2.1] , the local well-posedness of the (CS) equation was achieved in H s + (R) for s > 1 2 by following the analysis of [START_REF] De Moura | Local well-posedness for the nonlocal nonlinear Schrödinger equation below the energy space[END_REF]. In particular, for s > 3 2 , Gérard-Lenzmann [GL22, Proposition 2.1] used iterative schemes of Kato's type and energy estimates to derive the local well-posedness in H s + (R) [START_REF] Saut | Sur quelques généralisations de l'équation de Korteweg-de Vries[END_REF] . On T , the same proof of iterative schemes holds, and we deduce the local well-posedness in H s + (T) for s > 3 2 . Therefore, we denote by S + (t) the flow of the focusing Calogero-Sutherland DNLS equation (CS) + and by S -(t) the flow of the defocusing equation (CS) -: for all s > 3 2 , t ∈ I max ,

S ± (t) : H s + (T) -→ H s + (T) u 0 -→ u(t) , (1.8)
where I max denotes the maximal interval of the existence of the solution.

1.1. Main results. Some notation. In the sequel, we denote for any nonnegative integer a , by N ≥a the subset of Z given by {k ∈ Z | k ≥ a} . Moreover, we denote by B L 2 + (r) the open ball of L 2 + (T) centered at the origin, with radius r > 0 .

The goal of the paper is to prove the global well-posedness of the L 2 -critical equation (CS) in all H s + (T) , s ≥ 0 . As a starting point, we state the results for the more challenging equation, the focusing Calogero-Sutherland DNLS equation

i∂ t u + ∂ 2 x u + 2D + (|u| 2 )u = 0 , (CS + )
then, we present the results for the defocusing case

3 i∂ t u + ∂ 2 x u -2D + (|u| 2 )u = 0 . (CS -)
Theorem 1.1. For all s > 3 2 , the Calogero-Sutherland DNLS focusing equation (CS + ) is globally well-posed in H s + (T) ∩ B L 2 + (1) . Moreover, the following a priori bound holds, sup

t∈R ∥u(t)∥ H s ≤ C , where C(∥u 0 ∥ H s ) is a positive constant.
Remark 1.1. The restriction of smallness on the L 2 -norm of the initial data, namely ∥u 0 ∥ L 2 < 1 , appears after applying a sharp inequality (Lemma 2.7) in order to control the growth of the Sobolev norms ∥u(t)∥ Ḣs , s ≥ 0 , by the conservation laws. This sharp inequality is an equality when we take for example u 0 (x) = 1 -|q| 2 1 -q e ix , q ∈ D , 3 We refer to the introduction of Weinstein [START_REF] Weinstein | Localized states and dynamics in the nonlinear Schrödinger/Gross-Pitaevskii equation[END_REF] for a mathematical and physical meaning of the terms focusing and defocusing for any dispersive equation.

which correspond to the profile of a (CS + )'s traveling wave of L 2 -norm ∥u 0 ∥ L 2 = 1 [START_REF] Badreddine | Traveling waves & finite gap potentials for the Calogero-Sutherland derivative nonlinear Schrödinger equation[END_REF]. More details for an eventual way to avoid this condition are presented in Section 6 , but so far it is still an open problem.

As a second step, we focus on the main point of this paper : how the flow S + (t) defined globally on H 2 + (T) for u 0 ∈ B L 2 + (1) , can be extended to less regularity spaces for instance L 2 + (T) ? Recall, as noted in (1.7) , the Calogero-Sutherland DNLS equation is L 2 -critical. Based on the previous Theorem, and under the notation u ε (t) = S + (t)u ε 0 , ε > 0 , we state the following result.

Theorem 1.2. Let u 0 ∈ B L 2 + (1)
. There exists a unique potential u ∈ C(R; L 2 + (T)) such that for any sequence

(u ε 0 ) ⊆ H 2 + (T) where ∥u ε 0 -u 0 ∥ L 2 -→ ε→0 0 , the following convergence holds : for all T > 0 , sup t∈[-T,T ] ∥u ε (t) -u(t)∥ L 2 → 0 , ε → 0 .
Moreover, the L 2 -norm of the limit potential u is conserved

∥u(t)∥ L 2 = ∥u 0 ∥ L 2 , ∀t ∈ R.
(1.9)

As a consequence, Theorem 1.2 leads to the global well-posedness of the (CS + ) problem in L 2 + (T) in the following sense : There exists a unique continuous extension of the flow defined on

H 2 + (T) , to L 2 + (T) , generating a unique continuous map u 0 ∈ B L 2 + (1) -→ u ∈ C(R, L 2 + (T)) .
The key ingredient of the proof is to obtain H 1 2 bounds (inequality (3.21)) on the eigenfunctions of the Lax operator L u ε , which also constitute an orthonormal basis of L 2 + (T) . Therefore, we deduce the strong convergence of these eigenfunctions in L 2 . Finally, using Parseval's identity, we infer (1.9) .

We also need to emphasize the important aspect of the uniqueness of the limit potential u(t) , obtained independently of the choice of the sequence (u ε 0 ) that approximates u 0 ∈ L 2 + (T) . For this purpose, we derive in Proposition 2.5 , an explicit formula of the solution of the focusing (CS + ) equation. Thus, for any initial data u 0 , the solution of the (CS + ) focusing equation is given by

u(t, z) = (Id -z e -it e -2itLu 0 S * ) -1 u 0 | 1 , z ∈ D := {|z| < 1} , (1.10)
where S * denotes the adjoint of the Shift operator S : h → zh in L 2 + (T) , and L u 0 is the Lax operator at t = 0 . We underline two important facts about (1.10) : I. First, this inversion dynamical formula defined inside the open unit disc consists an explicit solution for the nonlinear PDE (CS + ) . This is not the first time that an explicit solution occurs while dealing with nonlinear integrable PDEs. Indeed, Gérard-Grellier derived in [START_REF] Gérard | An explicit formula for the cubic Szegő equation[END_REF] an explicit solution for the Szegő equation, and recently Gérard also prove in [START_REF] Gérard | An explicit formula for the Benjamin-Ono equation[END_REF] that the Benjamin-Ono equation has an explicit solution on R and on T . The common point to all these dynamical explicit formulas is that they all rely closely on the structure of the Lax operators induced by these equations.

II. Beyond the fact that we have an explicit solution, this formula stresses out that the dynamics of the (CS + ) equation are encoded by the Lax operator L u 0 , suggesting thus, that the so-called actions-angles variables must be related to the spectral elements of the Lax operators L u .

In view of Theorem 1.2 , we state the third result.

Corollary 1.3. For all 0 ≤ s ≤ 3 2 , the Calogero-Sutherland DNLS focusing equation

(CS + ) is globally well-posed in H s + (T) ∩ B L 2 + (1)
. Moreover, the following a-priori bound holds,

sup t∈R ∥u(t)∥ H s ≤ C , where C = C(∥u 0 ∥ H s ) > 0 is a positive constant.
Remark 1.2. There is a subtlety hidden in the words of "globally well-posed" in the last statement. In fact, it is important to distinguish here the two different aspects of global well-posedness. First, we have the classical definition of GWP used in Theorem 1.1 : for any u 0 ∈ H s + there exists a unique solution u defined on R with value in H s + , such that u depends continuously on the initial data u 0 as a map

u 0 ∈ H s + → u ∈ C(R, H s + )
. The second definition is the one described in Theorem 1.2 in the sense : we suppose that the equation is defined at least in the distribution sense, then we extend the flow defined on high regularity spaces to low regularity spaces through continuous extension. In this corollary, the global well-posedness is in the sense used in Theorem 1.2 . This will become clearer once the proof is established (see Section 4). We also expect that, following arguments in [START_REF] De Moura | Local well-posedness for the nonlocal nonlinear Schrödinger equation below the energy space[END_REF] , one can go down for the global well-posedness in the classical sense to H s + (T) with s > 1 2 .

Beyond the global well-posedness results on the Cauchy Problem of (CS + ) , we are interested in some qualitative properties about the flow S + (t) of this equation.

Theorem 1.4. Given an initial data u 0 ∈ B L 2 + (1) ∩ H s + (T) , s ≥ 0 , the orbit of the solution {S + (t)u 0 ; t ∈ R} is relatively compact in H s + (T) .
The defocusing equation (CS -). Moving now to the defocusing case of the Calogero-Sutherland DNLS equation, this latter equation enjoys also a Lax pair structure : for any u(t) ∈ H s + , with s large enough, there exist two operators

Lu = D + T u T u , Bu = -T u T ∂xu + T ∂xu T u + i(T u T u ) 2 , satisfying the Lax equation d Lu dt = [ Bu , Lu ] .
Therefore, using the same methods as on the focusing case, we prove that the conservation laws ⟨( Lu + λ) s 1 | 1⟩ , s ≥ 0 , λ > 0 , controls uniformly the growth of the Sobolev norms without requiring any additional condition on the initial data. As a consequence, we obtain similar results in the defocusing case as in the focusing case, regardless of how large the initial data is in L 2 . To summarize, we have the following.

Theorem 1.5. The Calogero-Sutherland DNLS defocusing equation (CS -) is globally wellposed in H s + (T) for any s ≥ 0 in the sense of Remark 1.2 . In addition, for all

u 0 ∈ H s + (T) , u(t, z) = (Id -z e -it e -2it Lu 0 S * ) -1 u 0 | 1 ,
is the solution to the (CS -)-defocusing equation. Furthermore, the trajectories

S -(t)u 0 ; t ∈ R are relatively compact in H s + (T) .
1.2. Other related equations. As explained in [START_REF] Gérard | The Calogero-Moser Derivative nonlinear Schrödinger equation[END_REF], the Calogero-Sutherland DNLS equation (CS) can be seen as mass critical version of the Benjamin-Ono equation. We refer to [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF][START_REF] Gérard | Sharp well-posedness results of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solution[END_REF] for a deep study of this latter equation on the torus. Of course, the Calogero-Sutherland DNLS equation (CS) is also considered as part of the nonlinear Schrödinger's family. Several authors have been interested in different types of NLS-equations over the years. Some of these equations are classified and presented in [START_REF] Bourgain | Global solutions of nonlinear Schrödinger equations[END_REF] . Maybe the most closely related to the (CS)-equation are :

(i) Cubic NLS equation.

i∂ t u + ∂ 2 x u ± |u| 2 u = 0 , (NLS-cubic)
which is considered as one of the simplest PDE enjoying complete integrable properties. Zakharov-Shabat have studied this equation in [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF] using inverse scattering method. Moreover, global well-posedness results in L 2 (T) are presented in Bourgain [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrödinger equations[END_REF] after he introduced the X s,b -spaces. His proof relies on establishing L 4 (T)-Strichartz estimates and using L 2 -conservation norm. Actually, this result of L 2 -well-posedness is known to be sharp, and it is illustrated by various types of ill-posedness results below the regularity L 2 (T) . Indeed, Burq-Gérard-Tzvetkov proved in [START_REF] Burq | An instability property of the nonlinear Schrödinger equation on S d[END_REF] that the flow map of (NLS-cubic) fails to be uniformly continuous for Sobolev regularity below L 2 . Christ-Colliander-Tao [START_REF] Christ | Instability of the periodic nonlinear Schrödinger equation[END_REF] and Molinet [START_REF] Molinet | On ill-posedness for the one-dimensional periodic cubic Schrödinger equation[END_REF] showed the discontinuity of the map solution in H s (T) for s < 0 .

For a deep study of (NLS-cubic) using integrable tools, Birkhoff normal form, and some applications, we refer to Kappeler-Lohrmann-Topalov-Zung [KL+17] , Grébert-Kappeler [START_REF] Grébert | The Defocusing NLS Equation and Its Normal Form[END_REF] and Kappeler-Schaad-Topalov [START_REF] Kappeler | Scattering-like phenomena of the periodic defocusing NLS equation[END_REF]. For a study on the line R, we cite [START_REF] Harrop-Griffiths | Sharp well-posedness for the cubic NLS and mKdV in H s (R)[END_REF]. More references are also provided in [START_REF] Oh | On the one-dimensional cubic nonlinear Schrödinger equation below L 2[END_REF] . (ii) DNLS equation.

i∂ t u + ∂ 2 x u + ±i∂ x |u| 2 u = 0 , (DNLS)
which is also an integrable equation enjoying infinite conservation laws [START_REF] Kaup | An exact solution for a derivative nonlinear Schrödinger[END_REF]. Using the I-method, Win proved in [START_REF] Win | Global well-posedness of the derivative nonlinear Schrödinger equations on T[END_REF] the global well-posed of (DNLS)-equation in H s (T) , s > 1 2 for small data in L 2 (T) . More recently, Klaus-Schippa [START_REF] Klaus | A priori estimates for the derivative nonlinear Schrödinger equation[END_REF] presented law regularity a priori estimates of ∥u∥ H s for 0 < s < 1 2 upon small L 2 -norm, where u ∈ C ∞ (R, S(T)) and S(T) denotes the Schwartz space. Actually, they proved the a priori estimates

sup t∈R ∥u(t)∥ B s r,2 ≲ ∥u(0)∥ B s r,2
in any Besov space B s r,2 , with r ∈ [1, ∞] and 0 < s < 1 2 . For a study on the line R, we cite [JL+20, BP22, BLP21, KNV21, HKV21, HKNV22] .

1.3. Outline of the paper. The paper is organized as follows.

In Section 2 , we discuss some properties about the Lax operators of the Calogero-Sutherland DNLS focusing equation (CS + ) . We derive the explicit formula of the solution of (CS + ) in the first subsection 2.1 . Then, we prove in the second subsection 2.2 , the global well-posedness of the (CS + ) problem in H s + (T) for any s > 3 2 . In Section 3 , we extend the flow S + (t) of (CS + ) continuously from H 2 + (T) to L 2 + (T) ≡ H 0 + (T) . To this end, we use an approximation method, and we characterize in the first subsection 3.1.1 the limit potential u(t) for all t ∈ R . Then, in the second subsection 3.2, we make sure that the lack of compactness in L 2 + (T) do not occur while passing to the limit from H 2 + (T) to L 2 + (T) . In the same subsection, we derive an orthonormal basis of L 2 + (T) where the coordinates of the solution u(t) have nice evolution in this basis. This evolution suggests that the so-called "Birkhoff coordinates" are the coordinates of u(t) in this basis. After that, we deal in Section 4 with the problem of global well-posedness of (CS + ) in H s + (T) for 0 < s ≤ 3 2 . Moreover, we address the property of relative compactness of the orbits of (CS + ) in H s + (T) , s ≥ 0 . Moving to Section 5 , we present the Lax pair for the defocusing Calogero-Sutherland DNLS equation (CS -) and we state the analogous results of (CS + ) in the case of (CS -) . Finally, in Section 6 , we discuss some remarks and open problems related to this equation.

Acknowledges.

The author would like to thank warmly her Ph.D. advisor Patrick Gérard for his rich discussions and comments on this paper. Additionally, she expresses her appreciation to the anonymous referee for the thorough review.

The Lax pair structure

As noted in the introduction, we first check that the Lax pair defined in (1.5) holds the same in the context of the torus T as on the real line R, even though on the real line R , a complex function f is decomposed as

f = Πf + Πf , Πf (ξ) = 1 ξ>0 f (ξ) , ξ ∈ R , while on the torus T , f = Πf + Πf -⟨f | 1⟩ , Π n∈Z f (n) e inx := n∈N ≥0 f (n) e inx .
Proposition 2.1 (The Lax pair). For any s > 3 2 , let u ∈ C([-T, T ], H s + (T)) be a solution of the focusing equation (CS + ) . Then, there exist two operators

L u = D -T u T u , B u = T u T ∂xu -T ∂xu T u + i(T u T u ) 2
satisfying the Lax equation

dL u dt = [B u , L u ] ,
where T u is the Toeplitz operator defined in (1.6) .

Proof.

Let u ∈ C([-T, T ], H s + (T)) , s > 3 2
, be a solution of (CS + ) equation. On the one hand, we have by definition of L u and for all h

∈ H 1 + (T) , dL u (h) dt = -T ∂tu T ū(h) -T u T ∂tu (h) = -T i∂ 2 x u+2u∂xΠ(|u| 2 ) T ūh -T u T -i∂ 2 x ū+2ū ∂xΠ(|u| 2 ) h .
Therefore, since u belongs to the Hardy space,

dL u (h) dt = i T u T ∂ 2 x ū -T ∂ 2 x u T ū (h) -2u ∂ x Π(|u| 2 ) • Π(ūh) + Π ∂ x Π(|u| 2 ) • ūh . (2.1)
On the other hand, expanding the commutator

[B u , L u ](h) = B u L u h -L u B u h , we obtain T u T ∂x ūDh -T u T ∂x ūT u T ūh -T ∂xu T ūDh + T ∂xu T ūT u T ūh + i(T u T ū) 2 Dh -D(T u T ∂x ūh) + T u T ūT u T ∂x ūh + D(T ∂xu T ūh) -T u T ūT ∂xu T ūh -iD((T u T ū) 2 h) ,
where by the Leibniz rule, D(T u h) = -iT ∂xu h + T u Dh , so that

D(T u T ∂x ū •) = T u T ∂x ūD -iT ∂xu T ∂x ū -iT u T ∂ 2 x ū , D(T ∂xu T ū •) = T ∂xu T ūD -iT ∂ 2 x u T ū -iT ∂xu T ∂x ū , D((T u T ū) 2 •) = -i[T ∂xu T ūT u T ū + T u T ∂x ūT u T ū + T u T ūT ∂xu T ū + T u T ūT u T ∂x ū] + (T u T ū) 2 D • . As a consequence, [B u , L u ](h) = i T u T ∂ 2 x ūh -iT ∂ 2 x u T ūh -2 (T u T ūT ∂xu T ū + T u T ∂x ūT u T ū) (h) = i T u T ∂ 2 x ū -T ∂ 2 x u T ū -2u • Π ∂ x |u| 2 • Π(ūh) . (2.2)
Comparing (2.1) and (2.2) , it appears that all that remains to be proved is

∂ x Π(|u| 2 ) • Π(ūh) + Π ∂ x Π(|u| 2 ) • ūh = Π ∂ x |u| 2 • Π(ūh) , h ∈ H 1 + (T) .
(2.3)

In fact, any complex function f ∈ L 2 (T) can be decomposed as

f = Πf + Π f -⟨f | 1⟩ .
In particular, for f = ūh , we have Π

(∂ x Π(|u| 2 ) • ūh) equal to Π ∂ x Π(|u| 2 ) • Π(ūh) + Π ∂ x Π(|u| 2 ) • Π(u h) -⟨ūh | 1⟩ Π(∂ x Π(|u| 2 )) ,
where the last two terms vanishes, since Π is an orthogonal projector into the Hardy space. Therefore, the left-hand side of (2.3) coincides with

Π ∂ x Π(|u| 2 ) • Π(ūh) + Π ∂ x Π(|u| 2 ) • Π(ūh) , which is equal to Π (∂ x |u| 2 • Π(ūh)) since ⟨∂ x (|u| 2 ) | 1⟩ = 0 . □ 2.1.
The explicit formula of the solution. Using this Lax pair structure, we derive in this subsection the explicit formula of the solution of the focusing Calogero-Sutherland DNLS equation (CS + ) . To this end, we also need the shift operator introduced in the following paragraph.

Some Preliminaries. We recall one of the most important operator on Hardy's space, the shift operator, defined on L 2 + (T) as the isometric map

S : h ∈ L 2 + (T) -→ e ix h ∈ L 2 + (T) . Its adjoint in L 2 + (T) is given by S * : h ∈ L 2 + (T) -→ S * h = T e -ix h = Π(e -ix h) ∈ L 2 + (T) .
In particular, we have

S * S = Id, SS * = Id -⟨ • | 1⟩1, (2.4)
leading to the fact that the shift map S is injective but not surjective. Pointing out that the Hardy space can be defined with different approaches, for instance,

H 2 (D) := u ∈ Hol(D) ; sup 0≤r<1 2π 0 |u(r e iθ )| 2 dθ 2π < ∞ , which is equivalent via the isometric isomorphism u(z) = k≥0 u(k)z k -→ u * (x) := k≥0 u(k) e ikx ,
to the Hardy space L 2 + (T) defined in (1.2) , then one could read the shift operator acting as multiplication by z . In what follows, we use indifferently u and the boundary function u * , by making a slight abuse of notation and denoting both by u . *** Coming back to the problem, we need some commutator identities to obtain the explicit formula. This is the purpose of the next Lemma.

Lemma 2.2. Let u ∈ H s + (T) , s > 3 2 , then [S * , L u ] = S * -⟨ • | u⟩ S * u , (2.5) [S * , B u ] = i S * L 2 u -(L u + Id) 2 S * .
Proof. The first identity is a direct consequence of proving

L u S = SL u + S -⟨ • | S * u⟩ u , (2.6)
and taking the adjoint of all these operators in L 2 + (T) . Recall L u = D -T u T ū . On the one hand, we have by the Leibniz rule D(Sh) = S(Id +D)h , for all h ∈ H 1 + (T) . On the other hand, observe for all f ∈ L 2 (T),

Π (Sf ) = SΠ(f ) + ⟨Sf | 1⟩ .
In particular, for f = hū , we infer

T ū(Sh) = ST ūh + ⟨Sh | u⟩.
(2.7)

Hence, taking into consideration that the operators S and T u commute, we deduce identity (2.6) . Now, to prove the second point of (2.5) we use the first point. Recall that

B u = T u T ∂x ū -T ∂xu T ū +i(T u T ū) 2 , and by (2.7) we have [T ū, S] = ⟨ • | S * u⟩, in other words, [S * , T u ] = ⟨ • | 1⟩ S * u.
Thus, after noting that S * and T ū commute, we deduce

[S * , T u T ∂x ū] = ⟨ • | ∂ x u⟩ S * u . [S * , T ∂xu T ū] = ⟨ • | u⟩ S * ∂ x u , [S * , (T u T ū) 2 ] = ⟨ • | T u T ūu⟩ S * u + T u T ū(⟨ • | u⟩ S * u) . As a result, [S * , B u ] = ⟨ • | ∂ x u⟩ S * u -⟨ • | u⟩ S * ∂ x u + i ⟨ • | T u T ūu⟩ S * u + iT u T ū (⟨ • | u⟩ S * u) .
Using the adjoint Leibniz rule S * D = (D + Id)S * and since

L u = D -T u T ū , we infer [S * , B u ] = -i ⟨ • | L u u⟩ S * u -iL u (⟨ • | u⟩ S * u) -i ⟨ • | u⟩ S * u = -i (⟨ • | u⟩ S * u)L u -i(L u + Id)(⟨ • | u⟩ S * u) .
We conclude by the first identity of (2.5) that -⟨

• | u⟩ S * u = S * L u -L u S * -S * and hence [S * , B u ] = i S * L 2 u -(L u + Id) 2 S * . □ Proposition 2.3. Let u(t) ∈ H s + (T) , s > 3 2 . The Lax operator (L u(t) , H 1 + (T)
) is a selfadjoint operator with a discrete spectrum bounded from below. Moreover, B u(t) is a skewsymmetric bounded operator on L 2 + (T) .

Proof. The proof is a direct consequence of Kato-Rellich's theorem. Indeed, the differential operator (D, H 1 + (T)) is a positive self-adjoint operator on the Hardy space L 2 + (T). In addition, T u T u is relatively bounded with respect to D, since for all h ∈ H 1 + (T) ,

∥T u T ūh∥ L 2 ≤ ∥u∥ 2 L ∞ ∥h∥ L 2 ≤ ε ∥Dh∥ L 2 (T) + ∥u∥ 2 L ∞ ∥h∥ L 2 (T) , 0 ≤ ε < 1.
Furthermore, the spectrum of L u is discrete since the resolvent of L u is compact by the Rellich-Kondrachov theorem. And it is bounded from below as the operator L u is a semi-bounded operator. Besides, one can easily observe by definition of B u = T u T ∂xu -T ∂xu T u + i(T u T u ) 2 , that this operator is a skew-symmetric operator. □

In view of the previous proposition, we denote by (λ n (u)) n≥0 the eigenvalues of L u ordered by increasing modulus, and taking into account their multiplicity

λ 0 (u) ≤ λ 1 (u) ≤ λ 2 (u) ≤ . . . ≤ λ n (u) ≤ . . .
Remark 2.1 (Isospectral property). As discovered in the modern theory of integrable systems [START_REF] Gardner | Method for solving the Korteweg-de Vries equation[END_REF] and reformulated by [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF], the eigenvalues of a Lax operator are integrals of motion of the associated equation. In fact, any Lax operator satisfies the isospectral property, namely, there exists a one-parameter family of unitary operators U (t) such that

U (t) -1 L u(t,•) U (t) is independent of t . That is, U (t) -1 L u(t) U (t) = L u 0 .
(2.8)

This implies, that the eigenvalues (λ n (u)) of L u are all conserved along the flow of (CS + ) . Or in other words, for all n ∈ N ≥0 , λ n (u(t)) = λ n (u 0 ) for all t ∈ R .

The following lemma provides a rewrite of the Calogero-Sutherland DNLS equation focusing on (CS + ) in terms of the Lax operators L u and B u . This will certainly be useful during the proof of the dynamical explicit formula.

Lemma 2.4. Given u ∈ C([-T, T ], H s + (T)) , s > 3 2 , a solution of (CS + ) equation, then

∂ t u = B u u -iL 2 u u .
Proof. By definition of

B u := T u T ∂xu -T ∂xu T u + i(T u T u ) 2 , ∂ t u -B u u = i∂ 2 x u + 2iD + (|u| 2 )u -T u T ∂xu u + T ∂xu T u u -i(T u T u ) 2 u = -i D 2 u -2u • DΠ(|u| 2 ) + u • Π(Du • u) -Du • Π(|u| 2 ) + (T u T u ) 2 u . Applying Leibniz's rule on D (u • Π(|u| 2 )) , we infer ∂ t u -B u u = -i D 2 u -D Π(|u| 2 ) • u + uΠ(Du • u) -uDΠ(|u| 2 ) + (T u T u ) 2 u .
Again, using Leibniz's rule on the term DΠ(|u| 2 ) ,

∂ t u -B u u = -i D 2 u -D(T u T u u) -uΠ(u • Du) + (T u T u ) 2 u = -iL 2 u u .
□ Following [START_REF] Gérard | An explicit formula for the Benjamin-Ono equation[END_REF] and [START_REF] Gérard | An explicit formula for the cubic Szegő equation[END_REF], we derive the explicit formula for the solution of the Calogero-Sutherland DNLS focusing equation.

Proposition 2.5 (The explicit formula). Given u 0 ∈ H s + (T) , s > 3 2 , the solution of the focusing Calogero-Sutherland DNLS equation (CS + ) is given by

u(t, z) = (Id -z e -it e -2itLu 0 S * ) -1 u 0 | 1 , ∀ z ∈ D . Proof. Since u(t, •) ∈ H s + (T) , s > 3 2 , for all t ∈ [-T, T ] , then for all z ∈ D u(t, z) = ∞ k=0 u(t, k)z k = ∞ k=0 u(t) | S k 1 z k = ∞ k=0 (S * ) k u(t) | 1 z k ,
where by the Neumann series of

∞ k=0 (zS * ) k = (Id -zS * ) -1 , we infer u(t, z) = (Id -zS * ) -1 u(t) | 1 , ∀ z ∈ D .
(2.9)

Now, consider a one-parameter family U (t) solution of the Cauchy problem

d dt U (t) = B u(t,•) U (t) U (0) = Id .
(2.10)

Observe that U (t) is a unitary operator since B u is skew-adjoint. Moreover, using the Lax pair structure of Proposition 2.1 , and (2.10) ,

d dt (U (t) * L u(t) U (t)) = 0 ,
and thus

U (t) * L u(t) U (t) = L u 0 . (2.11)
Hence, by applying U (t) * to both sides of the inner product (2.9) ,

u(t, z) = U (t) * (Id -zS * ) -1 u(t) | U (t) * 1 (2.12) = (Id -zU (t) * S * U (t)) -1 U (t) * u(t) | U (t) * 1 .
The aim is to express differently U (t) * S * U (t) , U (t) * u(t) and U (t) * 1 . Using (2.10) and since B u is a skew-adjoint operator (Proposition 2.3), we find

• d dt [U (t) * 1] = -U (t) * B u(t) 1 = -iU (t) * L 2 u(t) 1 • d dt [U (t) * u(t)] = -U (t) * B u(t) u(t) + U (t) * ∂ t u(t) = -iU (t) * L 2 u(t) u(t) by Lemma 2.4 • d dt [U (t) * S * U (t)] = -U (t) * B u(t) S * U (t) + U (t) * S * B u(t) U (t) = U (t) * [S * , B u(t) ] U (t)
, where the third point is equal to

d dt [U (t) * S * U (t)] = iU (t) * S * L 2 u(t) -(L u(t) + Id) 2 S * U (t) ,
by Lemma 2.2 . Therefore, applying the identity U (t) * L u(t) = L u 0 U (t) * of (2.11) , we deduce

• d dt [U (t) * 1] = -iL 2 u 0 [U (t) * 1] • d dt [U (t) * u(t)] = -iL 2 u 0 [U (t) * u(t)] • d dt [U (t) * S * U (t)] = i [U (t) * S * U (t)]L 2 u 0 -(L u 0 + Id) 2 [U (t) * S * U (t)] .
As a consequence,

U (t) * 1 = e -itL 2 u 0 1 , U (t) * u(t) = e -itL 2 u 0 u 0 , (2.13)
and U (t) * S * U (t) = e -it(Lu 0 +Id) 2 S * e itL 2 u 0 .

(2.14)

Combining (2.12) , (2.13) and (2.14), the claimed formula follows. □

Global well-posedness of (CS

+ ) in H s + (T) , s > 3 2 .
To prove the global wellposedness of (CS + ) , we need to derive some conservation laws and energy estimates.

Lemma 2.6 (Conservation laws). Let u ∈ C [-T, T ], H r + (T) , r > 3 2 , solution of (CS + ) . For all λ >> 0 , the family {H s (u) := ⟨(L u + λ) s u | u⟩ ; 0 ≤ s ≤ 2r} is conserved by the flow of (CS) .
Remark 2.2.

• Using complex interpolation method [Ta81, Chapter I. 4.], one can observe as demonstrate in Proposition 2.8 , that the

H s (u) ≤ C∥u∥ 2 H s 2 . • The condition r > 3
2 is to guarantee the existence of u(t) . It can be omitted once we prove in Section 4 that the flow

u 0 ∈ B L 2 + (1) ∩ H r + (T) → u(t) ∈ H r + (T) exists for all r ≥ 0. Proof. Given u ∈ C [-T, T ], H r + (T) , r > 3 2
, solution of (CS + ) , we consider the unitary operator U (t) defined in (2.10) . Then, by (2.13) , we know that U (t) * u(t) = e -itL 2 u 0 u 0 . And, since L u is a self-adjoint operator by Proposition 2.3 , we infer by (2.11) ,

U (t) * (L u(t) + λ) s U (t) = (L u 0 + λ) s .
Therefore, for all 0 ≤ s ≤ 2r ,

H s (u(t)) = U (t) * (L u(t) + λ) s u(t) | U (t) * u(t) = ⟨(L u 0 + λ) s U (t) * u(t) | U (t) * u(t)⟩ = ⟨(L u 0 + λ) s e -itL 2 u 0 u 0 | e -itL 2 u 0 u 0 ⟩ As a consequence, H s (u(t)) = H s (u 0 ) as (L u 0 + λ)
s and e -itL 2 u 0 commute. □ Remark 2.3. Using the identity U (t) * 1 = e -itL 2 u 0 1 of (2.13) and repeating the same proof of Lemma 2.6 , one can also deduce for λ >> 0 , that the quantities ⟨(L u + λ) q 1 | u⟩ and ⟨(L u + λ) p 1 | 1⟩ are conserved by the flow. Another way to show this, is to observe by definition of L u = D -T u T ū we have L u 1 = -⟨1 | u⟩ u and the average ⟨u | 1⟩ is conserved along the evolution, since

∂ t ⟨u | 1⟩ = i ∂ 2 x u | 1 + 2i DΠ(|u| 2 ) | u = 0 .
To prove the energy estimates and for future requests, we need the following lemma.

Lemma 2.7. Let h ∈ H

1 2 + (T) , u ∈ L 2 + (T) , ∥T ūh∥ 2 L 2 (T) ≤ ⟨Dh | h⟩ + ∥h∥ 2 L 2 (T) ∥u∥ 2 L 2 (T) .
(2.15)

Proof. By Parseval's identity,

∥T ūh∥ 2 L 2 (T) = n≥0 T ūh(n) 2 ,
where

T ūh(n) = Π(hū)(n) = p≥0 h(n + p) u(p) .
(2.16)

Applying Cauchy-Schwarz's inequality, we infer

∥T ūh∥ 2 L 2 (T) ≤ ∥u∥ 2 L 2 (T) p≥0 n≥0 | h(n + p)| 2 . Set k = n + p, then p≥0 n≥0 | h(n + p)| 2 = k≥0 (k + 1)| h(k)| 2 = ⟨Dh | h⟩ + ∥h∥ 2 L 2 (T) . □ Remark 2.4.
(1) Recall that the embedding H 1 2 (T) → L ∞ (T) fails to be true. However, taking the potential u as an element of the Hardy space L 2 + (T) , improved the estimate from ∥T ūh∥ L 2 ≤ ∥h∥ L ∞ ∥u∥ L 2 to (2.15) .

(2) From (2.16), one could see that, for all h ∈ H 1 2 + (T) , the Hilbert-Schmidt norm of the antilinear operator u ∈ L 2 + (T) → T ūh is given by

∥Π( • h)∥ 2 HS = p≥0 n≥0 | h(n + p)| 2 = k≥0 (k + 1)| h(k)| 2 = ⟨Dh | h⟩ + ∥h∥ 2 L 2 (T) .
In particular, we have u → T ūh is a compact antilinear operator in L 2 + (T) .

(3) The inequality (2.15) of Lemma 2.7 is a sharp inequality since its proof relies on a simple application of the Cauchy-Schwarz inequality. In particular, when h = u , inequality (2.15) is an equality, if and only if

u(z) = c 1 -qz , |q| < 1 , c ∈ C
-which corresponds to the profile of a traveling wave of (CS

+ ) when c = 1 -|q| 2 [Ba23]
. Indeed, following arguments used in [GG08, Lemma 1], one observe that the Cauchy-Schwarz inequality applied to (2.16) is an equality, if and only if for all n ≥ 0 , there exists c n ∈ C such that

u(n + p) = c n u(p) , ∀p ≥ 0 .
(2.17)

Hence, if u(1) ̸ = 0 and u(0) ̸ = 0 ,

c n = u(n + 1) u(1) = u(n) u(0) , leading to, u(n) = u(1) u(0) n u(0) , ∀n ∈ N .
Therefore, the sequence ( u(n)) is a geometric progression with common ratio q := u(1) u(0) , where 0

< |q| < 1 since ∞ n=0 | u(n)| 2 < +∞ . Hence, u(z) = ∞ n=0 u(n)z n = u(0) 1 -qz .
Now, if u(1) = 0 or u(0) = 0 then by (2.17) we infer for p = 0 or 1 , u = u(0) .

We recall that B L 2 + (r) denotes the open ball of L 2 + (T) centered at the origin, with radius r > 0 . And we denote for any s ∈ R by ⌊s⌋ := max{k ∈ Z ; k < s} .

Proposition 2.8. Let u 0 ∈ B L 2 + (1) ∩ H r + (T) , r > 3 2 .
Then, for all λ >> 0 , there exists

C = C(∥u 0 ∥ H ⌊2r⌋ 2
, λ) > 0 independent of t , such that for every f ∈ H s + (T) ,

1 C ∥f ∥ H s ≤ ∥(L u(t) + λ) s f ∥ L 2 ≤ C∥f ∥ H s , ∀ 0 ≤ s ≤ ⌊2r⌋ + 1 2 .
(2.18)

Remark 2.5. The condition r > 3 2 is to guarantee the existence of u(t) . It can be replaced by r ≥ 0 once we prove in Section 4 that the flow

u 0 ∈ B L 2 + (1) ∩ H r + (T) → u(t) ∈ H r + (T) exists for all r ≥ 0.
Proof. The proof is done by induction on every interval of length 1/2 .

Step 1 : s ∈ [0, 1 2 ] . Let s = 1 2 , we have by definition of L u = D -T u T ū , ⟨(L u + λ)f | f ⟩ = ⟨Df | f ⟩ -∥T ūf ∥ 2 L 2 + λ∥f ∥ 2 L 2 . (2.19)
Applying the sharp inequality of Lemma 2.7 ,

∥(L u + λ) 1 2 f ∥ 2 L 2 ≥ (1 -∥u∥ 2 L 2 ) ⟨Df | f ⟩ + (λ -∥u∥ 2 L 2 )∥f ∥ 2 L 2 .
(2.20) Thus, we infer since ∥u∥

L 2 = ∥u 0 ∥ L 2 < 1 , ∥(L u(t) + λ) 1 2 f ∥ 2 L 2 ≥ 1 C 2 ∥f ∥ 2 H 1 2
, where C = C(∥u 0 ∥ L 2 , λ) > 0 is a positive constant independent of t . On the other hand, using the definition of L u , it is easy to see that

∥(L u(t) + λ) 1 2 f ∥ L 2 ≤ C∥f ∥ H 1 2 .
Therefore, by complex interpolation [Ta81, Chapter I. 4] , we deduce that inequality (2.18) holds true for all s ∈ [0, 1 2 ] .

Step 2 : Uniform bounds on ∥u(t)∥ L p , p ∈ [2, ∞) . Applying step 1, and using the conservation laws of Lemma 2.6 , we infer for f = u ,

1 C ∥u(t)∥ H s ≤ ∥(L u(t) + λ) s u(t)∥ L 2 = ∥(L u 0 + λ) s u 0 ∥ L 2 ≤ C∥u 0 ∥ H s , for all s ∈ [0, 1 2 ] . Therefore, sup t∈R ∥u(t)∥ H 1 2 ≲ ∥u 0 ∥ H 1 2
, and thus by Sobolev embedding sup

t∈R ∥u(t)∥ L p ≲ ∥u 0 ∥ H 1 2 , ∀p ∈ [2, ∞) .
Step 3 : s ∈ [ 1 2 , 1] . As in Step 1, the idea is to prove that it is true for s = 1 and then by complex interpolation, infer that it is true for all s ∈

[ 1 2 , 1] . Let s = 1 , by (2.20) of Step 1, (L u(t) + λ)f 2 L 2 (T) = (L u(t) + λ)f | L u(t) f + λ (L u(t) + λ)f | f ≥ ∥L u(t) f ∥ 2 L 2 + λ (1 -∥u∥ 2 L 2 )∥f ∥ 2 Ḣ 1 2 + (λ -∥u∥ 2 L 2 )∥f ∥ 2 L 2 + λ (1 -∥u∥ 2 L 2 )∥f ∥ 2 Ḣ 1 2 -∥u∥ 2 L 2 ∥f ∥ 2 L 2 = ∥Df ∥ 2 L 2 + ∥uT ūf ∥ 2 L 2 -2Re ⟨Df | uT ūf ⟩ + 2λ(1 -∥u∥ 2 L 2 )∥f ∥ 2 Ḣ 1 2 + λ(λ -2∥u∥ 2 L 2 )∥f ∥ 2 L 2 .
Using Young's inequality, we deduce

(L u(t) + λ)f 2 L 2 (T) ≥ (1 -ε)∥Df ∥ 2 L 2 + (1 -C ε )∥uT ūf ∥ 2 L 2 (2.21) + 2λ(1 -∥u∥ 2 L 2 )∥f ∥ 2 Ḣ 1 2 + λ(λ -2∥u∥ 2 L 2 )∥f ∥ 2 L 2 .
Now, applying Cauchy-Schwarz's inequality on ∥uT ūf ∥ L 2 and since ∥u∥ L 8 and ∥u∥ L 4 are uniformly bounded by Step 2., we infer

(L u(t) + λ)f 2 L 2 (T) ≥ 1 C 2 ∥f ∥ 2 H 1 , C = C(∥u 0 ∥ H 1 2 , λ) .
On the other hand, we have by definition of L u ,

(L u(t) + λ)f L 2 (T) ≤ C∥f ∥ H 1 .
Therefore, inequality (2.18) holds for s = 1 and thus by complex interpolation [START_REF] Taylor | Pseudodifferential operators[END_REF], it holds for all s ∈ [ 1 2 , 1] .

Step 4 : Uniform bounds on ∥u(t)∥ H 1 Since (2.18) holds for s = 1 then we infer by repeating the same proof of Step 2 that sup t∈R ∥u(t)∥ H 1 < ∞ .

Step 5 : s ∈ [1, 3 2 ] Again, we prove that inequality (2.18) is true for s = 3 2 , then by complex interpolation, we deduce it for all s ∈ [1, 3 2 ] . Let s = 3 2 . By applying inequality (2.21) of Step 3 ,

(L u(t) + λ) 3 f | f = (L u(t) + λ) 2 f | L u(t) f + λ (L u(t) + λ) 2 f | f ≥ (D -uT ū + λ) 2 f | (D -uT ū)f + λ (1 -ε)∥Df ∥ 2 L 2 + (1 -C ε )∥uT ūf ∥ 2 L 2 + 2λ(1 -∥u∥ 2 L 2 )∥f ∥ 2 Ḣ 1 2 + λ(λ -2∥u∥ 2 L 2 )∥f ∥ 2 L 2
Now, expanding the first inner product on the right-hand side, and using the Cauchy-Schwarz inequality, Young's inequality, and Sobolev embedding , we obtain

(L u(t) + λ) 3 f | f ≳ D 3 f | f + p 1 (∥u∥ 2 H 1 , λ) D 2 f | f + p 2 (∥u∥ 2 H 1 , λ) ⟨Df | f ⟩ + p 3 (∥u∥ 2 H 1 , λ)∥f ∥ 2 L 2
where for all j = 1, 2, 3, p j (∥u∥ 2 H 1 , λ) is a positive polynomial for λ >> 0 . Therefore, there exists

C = C(∥u 0 ∥ H 1 , λ) > 0 such that (L u(t) + λ) 3 2 f L 2 (T) ≥ 1 C ∥f ∥ H 3 2 .
Finally, by repeating the same previous procedure, we infer that inequality (2.18) holds true for all s ≥ 0 . □ Theorem. 1.1. For all r > 3 2 , the Calogero-Sutherland DNLS focusing equation (CS + ) is globally well-posed in H r + (T) ∩ B L 2 + (1) . Moreover, the following a-priori bound holds,

sup t∈R ∥u(t)∥ H r ≤ C , where C = C(∥u 0 ∥ H r ) > 0 is a positive constant.
Proof. Let u 0 ∈ H r + (T) . Recall by (1.8) , there exists a unique solution u ∈ C([-T, T ], H r + (T)) , s > 3 2 , satisfying u(0, •) = u 0 . In addition, in view of the previous proposition, we infer for

∥u 0 ∥ L 2 < 1 , 1 C ∥u(t)∥ H r ≤ ∥(L u(t) + λ) r u(t)∥ L 2 = ∥(L u 0 + λ) r u 0 ∥ L 2 ≤ C∥u 0 ∥ H r . □ 3. Extension of the flow of (CS + ) to L 2 + (T)
In this section, we establish our main result, which states that, for ∥u 0 ∥ L 2 < 1 , the flow of (CS + ) S + (t) :

H 2 + (T) -→ H 2 + (T) u 0 -→ u(t) , (3.1)
defined globally on H 2 + (T) via Theorem 1.1 , can be extended continuously to the critical regularity L 2 + (T) . For this purpose, consider any initial data u 0 ∈ L 2 + (T) , ∥u 0 ∥ L 2 < 1 . Then we approximate u 0 by a sequence (u ε 0 ) ⊆ H 2 + (T) . Thus, in view of Theorem 1.1 , the time evolution u ε (t) := S + (t)u ε 0 of u ε 0 is well-defined for all t ∈ R . Our goal is to prove that the sequence (u ε ) converge to a unique limit u in C(R, L 2 + (T)) . This limit potential shall be called "solution" to the Cauchy problem (CS + ) . It will be uniquely well-defined, regardless of the chosen approximate sequence (u ε 0 ) that approximate u 0 ∈ L 2 + (T) . Moreover, it will satisfies the conservation of the L 2 -norm (i.e. ∥u(t)∥ L 2 = ∥u 0 ∥ L 2 for all t ∈ R ).

u 0 u ε 0 u ε (t) u(t) ? L 2 + (T) :
H 2 + (T) :

Remark 3.1. Note that, due to the presence of the nonlinear term DΠ + (|u| 2 )u in the equation, it may seem intriguing to say that there exists a solution with L 2 -regularity. Nevertheless, the equation is still well-defined in the distribution sense since the product of two functions with nonnegative frequencies is well-defined and continuous. Indeed, let D ′ + (T) denotes the following distribution space

D ′ + (T) = {u = k≥0 u(k) e ikx ; ∃M ∈ N, | u(k)| ≲ (1 + |k| 2M ) 1 2 } ,
and consider two sequences of smooth functions in the Hardy space

f n , g n ∈ C ∞ + (T) such that we suppose f n -→ f and g n -→ g in D ′ + (T) . Then, f n g n -→ f g in D ′ + (T) and for all k ∈ N ≥0 , f g(k) = k ℓ=0 f (k -ℓ) g(ℓ) , as f n g n (k) = k ℓ=0 f n (k -ℓ) g n (ℓ) .
3.1. Uniqueness of the limit and weak convergence in L 2 + (T). By passing to the limit as ε → 0 , it is necessary to first prove that the limit potential u(t) is uniquely well-characterized for all t ∈ R , and is independent of the choice of the sequence (u ε 0 ) ⊆ H 2 + (T) that approximate u 0 ∈ L 2 + (T) . The key point is to use the explicit formula of the solution of the focusing Calogero-Sutherland DNLS equation. Thus, for all (u ε (t)) ⊆ H 2 + (T) , we have by Proposition 2.5 ,

u ε (t, z) = (Id -z e -it e -2itL u ε 0 S * ) -1 u ε 0 | 1 , ∀ z ∈ D . (3.2)
Our goal in this subsection is to pass to the limit in this formula. Therefore, we need first to give a meaning to the operator L u 0 when u 0 ∈ L 2 + (T) . To handle this, we recall in a few lines the work of Gérard-Lenzmann [GL22, Appendix A] who defined the operator L u with u ∈ L 2 + (R) via the standard theory of quadratic form. This new operator will coincide with the former Lax operator L u = D -T u T ū when u ∈ H 2 + (R) . The same proof presented in [GL22, Appendix A], works out on the torus T , and thus, one can define L u for u ∈ L 2 + (T) . We recall the main points of the proof :

(i) For u ∈ L 2 + (T) and f, g ∈ H 1 2 + (T) , consider the quadratic form Q u (f, g) = D 1/2 f | D 1/2 g -⟨T ūf | T ūg⟩ .
(ii) Observe that, by decomposing u in high and low frequency

u(x) = u N (x) + R N (u, x) , u N (x) := N n≥0 u(n)e inx R N (u, x) := n≥N +1 u(n) e inx
, and using Lemma 2.7 , one can prove that for all η > 0 , ∃ N η := N η (u) ∈ N ≥0 uniform on every compact set of L 2 + (T) , such that

∥T ūf ∥ 2 L 2 (T) < 2η 2 ⟨Df | f ⟩ + ∥f ∥ 2 L 2 (T) + 2N 2 η ∥u∥ 2 L 2 ∥f ∥ 2 L 2 (T) . (3.3) Therefore, Q u (f, f ) ≥ (1 -2η 2 )∥f ∥ 2 Ḣ1/2 -2(N 2 η ∥u∥ 2 L 2 + η 2 ) ∥f ∥ 2 L 2 (T) .
(3.4) (iii) Now, fixing η small enough, there exists K := K(u) > 0 uniform on every compact of L 2 + (T) , such that the following positive definite quadratic form

Qu (f, g) := Q u (f, g) + K ⟨f | g⟩ , f, g ∈ H 1 2 + (T) ,
define a new inner product on H 1 2 + (T) . (iv) Using the theory of quadratic forms (see [START_REF] Reed | Methods of modern mathematical physics. I. Functional analysis[END_REF]), we introduce for u ∈ L 2 + (T) ,

Dom(L u ) = h ∈ H 1 2 + (T) ; ∃ C > 0 , | Qu (h, g)| ≤ C ∥g∥ L 2 (T) , ∀g ∈ H 1 2 + (T) , (3.5)
and for any f ∈ Dom(L u )

⟨L u (f ) | g⟩ = Q u (f, g) , ∀g ∈ H 1 2 + (T) , (3.6)
and one shows that this new operator L u is a self-adjoint operator with a dense domain in H 1 2 + (T) . *** 3.1.1. Spectral properties of L u 0 for u 0 ∈ L 2 + (T). Now that the operator L u 0 has been introduced for u 0 ∈ L 2 + (T) , one can examine some of its spectral properties. As noted above, it is a self-adjoint operator with compact resolvent then it has discrete spectrum. Moreover, its quadratic form Q u 0 is bounded from below. therefore,

σ(L u 0 ) := {λ 0 (u 0 ) ≤ . . . ≤ λ n (u 0 ) ≤ . . .} , λ 0 > -∞ . (3.7)
To characterize this spectrum, we use the following proposition.

Proposition 3.1. For every n ∈ N ≥0 , the map

u ∈ L 2 + (T) → λ n (u) is Lipschitz continuous on compact subsets of L 2 + (T) . Proof. Let u ∈ L 2 + (T) .
The key ingredient is to use the max-min principle,

λ n (u) = max F ⊆L 2 + dim F ≤n min Q u (h, h) ; h ∈ F ⊥ ∩ H 1 2 + (T) , ∥h∥ L 2 = 1 . For any v ∈ L 2 + (T) , h ∈ H 1 2 + (T) , ∥h∥ L 2 = 1 , |Q u (h, h) -Q v (h, h)| = ∥T v h∥ 2 L 2 (T) -∥T u h∥ 2 L 2 (T) ≤ T (u-v) h L 2 (T) (∥T v h∥ L 2 (T) + ∥T u h∥ L 2 (T) ) ≤ ∥u -v∥ L 2 (T) ∥u∥ L 2 (T) + ∥v∥ L 2 (T) (1 + ⟨Dh | h⟩)
thanks to inequality (2.15) . Thus,

Q v (h, h) ≤ Q u (h, h) + ∥u -v∥ L 2 (T) ∥u∥ L 2 (T) + ∥v∥ L 2 (T) (1 + ⟨Dh | h⟩) . (3.8)
In particular, considering any subspace F of L 2 + (T) of dimension n , and for any h ∈

F ⊥ ∩ H 1 2 + (T) ∩ n k=0 ker(L u -λ k (u) Id)
, the latter inequality holds. In addition, observe by definition of L u = D -T u T ū , and by applying inequality

(3.3) with η = 1 2 , ⟨Dh | h⟩ ≤ ⟨L u h | h⟩ + ∥T ūh∥ 2 L 2 ≤ ⟨L u h | h⟩ + 1 2 (⟨Dh | h⟩ + 1) + 2N 2 ∥u∥ 2 L 2 ,
where N ∈ N is uniform on every compact subset of

L 2 + (T) . That is, ⟨Dh | h⟩ ≤ 2λ n (u) + 1 + 4N 2 ∥u∥ 2 L 2 , (3.9) since Q u (h, h) = ⟨L u h | h⟩ ≤ λ n (u) when h ∈ n k=0 ker(L u -λ k (u) Id)
. Furthermore, applying once more the max-min principle,

λ n (u) ≤ max F ⊆L 2 + dim F ≤n min ⟨Dh | h⟩ ; h ∈ F ⊥ ∩ H 1 2 + (T) , ∥h∥ L 2 = 1 (3.10) = min ⟨Dh | h⟩ ; h ∈ 1, . . . , e i(n-1)x ⊥ ∩ H 1 2 + (T) , ∥h∥ L 2 = 1 = n
Hence, combining (3.8) , (3.9) and (3.10) , we find for all h ∈ F ⊥ ∩ H

1 2 + (T) ∩ n k=0 ker(L u - λ k (u) Id) , and since Q u (h, h) ≤ λ n (u) , Q v (h, h) ≤ λ n (u) + 2 ∥u -v∥ L 2 (T) ∥u∥ L 2 (T) + ∥v∥ L 2 (T) n + 1 + 2N 2 ∥u∥ 2 L 2 . Therefore, λ n (v) ≤ λ n (u) + 2 (n + 1 + 2N 2 ∥u∥ 2 L 2 ) ∥u -v∥ L 2 (T) ∥u∥ L 2 (T) + ∥v∥ L 2 (T) . □ Corollary 3.2 (Characterization of the spectrum of L u 0 ). Let (u ε 0 ) ⊆ H 2 + (T) such that u ε 0 → u 0 in L 2 + (T) . The spectrum of L u 0 is given by σ(L u 0 ) = lim ε→0 λ n (u ε 0 ) | λ n (u ε 0 ) ∈ σ(L u ε 0 ) , n ∈ N ≥0 . Proof.
In light of the previous proposition, the result follows directly.

□ Proposition 3.3. Let u 0 ∈ L 2 + (T) and (u ε 0 ) ⊆ H 2 + (T) such that u ε 0 → u 0 in L 2 + (T). Then L u ε 0 → L u 0 in the strong resolvent sense as ε → 0 .
Proof. For all ε > 0 , we denote by ϕ ε λ the vector

ϕ ε λ := L u ε 0 + λ -1 h , h ∈ L 2 + (T)
, where λ ≫ 0 . Observe that λ can be chosen uniformly with respect to ε. Indeed, by inequality (3.3) , and since u ε 0 → u 0 in L 2 + (T) , there exists N ∈ N ≥0 uniform for all ε , such that for ε small enough

(L u ε 0 + λ)g | g ≥ 1 2 ∥g∥ 2 Ḣ 1 2 + (λ - 1 2 -2N 2 ∥u 0 ∥ 2 L 2 )∥g∥ 2 L 2 . (3.11)
Then, to apply the Lax-Miligram theorem, we choose in view of the last inequality λ ≫ 0 , such that (L

u ε 0 + λ) • | • is coercive and so (L u ε 0 + λ) is invertible for all ε small. Our goal is to prove that ϕ ε λ converges in L 2 + (T). For g = ϕ ε λ in (3.11) , L u ε 0 ϕ ε λ | ϕ ε λ + λ∥ϕ ε λ ∥ 2 L 2 ≥ 1 2 ∥ϕ ε λ ∥ 2 Ḣ 1 2 + λ - 1 2 -2N 2 ∥u ε 0 ∥ L 2 ∥ϕ ε λ ∥ 2 L 2 ,
which leads, for λ ≫ 0 , to

L u ε 0 ϕ ε λ | ϕ ε λ + λ∥ϕ ε λ ∥ 2 L 2 ≳ ∥ϕ ε λ ∥ 2 H 1 2 . Hence, as L u ε 0 ϕ ε λ | ϕ ε λ + λ∥ϕ ε λ ∥ 2 L 2 = ⟨h | ϕ ε λ ⟩ , ⟨h | ϕ ε λ ⟩ ≳ ∥ϕ ε λ ∥ 2 H 1 2 .
(3.12)

Using the Cauchy-Schwarz's inequality, we deduce

∥ϕ ε λ ∥ 2 H 1 2 ≤ ∥h∥ L 2 ∥ϕ ε λ ∥ L 2 ,
where, in view of Corollary 3.2 and by (3.7) , we have for all ε > 0 ,

∥ϕ ε λ ∥ L 2 ≤ sup n 1 |λ n (u ε 0 ) + λ| ∥h∥ L 2 ≤ C(λ)∥h∥ L 2 . (3.13)
Therefore,

∥ϕ ε λ ∥ 2 H 1 2 ≤ C(λ)∥h∥ 2 L 2 , ∀ε > 0 .
Thus, there exists ϕ λ ∈ L 2 + such that up to a subsequence,

ϕ ε λ ⇀ ϕ λ in H 2 + (T) and ϕ ε λ → ϕ λ in L 2 + (T) .
It remains to show that ϕ λ = (L u 0 + λ) -1 h . Indeed, for any g ∈ H

1 2 + (T) , we have by definition of ϕ ε λ , L u ε 0 + λ ϕ ε λ | g = ⟨h | g⟩. Namely, ⟨h | g⟩ = ⟨D 1 2 ϕ ε λ | D 1 2 g⟩ -T ūε 0 ϕ ε λ | T ūε 0 g + λ ⟨ϕ ε λ | g⟩ . (3.14) Since        T ūε 0 g -→ T ū0 g T ūε 0 ϕ ε λ ⇀ T ū0 ϕ λ D 1 2 ϕ ε λ ⇀ D 1 2 ϕ λ in L 2 + (T)
as ε → 0 , then passing to the limit in (3.14), we infer for all g ∈ H 1 2

+ (T), ⟨h | g⟩ = ⟨D 1 2 ϕ λ | D 1 2 g⟩ -T ūε 0 g ϕ ε λ | T ūε 0 + λ ⟨ϕ λ | g⟩ =: ⟨(L u 0 + λ)ϕ λ | g⟩ .
That is, ϕ λ ∈ Dom(L u 0 ) and ϕ λ = (L u 0 +λ) -1 h . Therefore, L u ε 0 → L u 0 in the strong resolvent sense as ε → 0 . □ 3.1.2. Characterization of the limit u(t).

Proposition 3.4 (Uniqueness of the limit potential u(t)). Let u 0 ∈ L 2 + (T) . There exists a unique potential u(t)

∈ L 2 + (T) , u(t, z) = (Id -z e -it e -2itLu 0 S * ) -1 u 0 | 1 , ∀ z ∈ D , (3.15)
such that, for any sequence

(u ε 0 ) ⊆ H 2 + (T) with ∥u ε 0 -u 0 ∥ L 2 -→ ε→0 0 , we have u ε (t) ⇀ u(t) in L 2 + (T) , ∀t ∈ R .
Proof. By the conservation of the L 2 -norm (Lemma 2.6),

∥u ε (t)∥ L 2 = ∥u ε 0 ∥ L 2 ≲ ∥u 0 ∥ L 2 , ∀ε ≪ 1 . Then, ∀t ∈ R , ∃ u * t ∈ L 2 + (T) such that u ε (t) ⇀ u * t in L 2 + (T) , and
∥u * t ∥ L 2 ≲ ∥u 0 ∥ L 2 . (3.16) Let u(t, z) = (Id -z e -it e -2itLu 0 S * ) -1 u 0 | 1 , ∀ z ∈ D ,
and recall by Proposition 2.5 ,

u ε (t, z) = (Id -z e -it e -2itL u ε 0 S * ) -1 u ε 0 | 1 , ∀ z ∈ D . (3.17)
Our goal is to prove that for all t ∈ R , z ∈ D , one has u ε (t, z) -→ u(t, z) , and thus, by the uniqueness of the limit, one can conclude that u * t is a well-defined function on L 2 + (T), given as a holomorphic function on D by u(t, z) . Indeed, by Proposition 3.3 , L u ε 0 → L u 0 in the strong resolvent sense as u ε 0 → u 0 in L 2 + (T) . Thus, for any bounded continuous functions f , we have f (L u ε 0 ) → f (L u 0 ) in the strong operator topology [deO09, Proposition 10.1.9] . In particular for all t ∈ R , and for f (x) = e -2ixt , e -2itL u ε 0 -→ e -2itLu 0 strongly as ε → 0 . Therefore, passing to the limit in (3.17) , we deduce

u ε (t, z) -→ u(t, z) , ε → 0 , ∀ z ∈ D , t ∈ R . □ 3.2. Strong convergence in L 2 + (T)
and conservation of the L 2 -mass. Our aim in this subsection is to prove Theorem 1.2 . In light of the previous subsection, it remains to have

∥u(t)∥ L 2 = ∥u 0 ∥ L 2 , (3.18)
in order to guarantee the strong convergence of

u ε (t) → u(t) in L 2 + (T) , as when ε → 0 , ∥u ε (t)∥ L 2 = ∥u ε 0 ∥ L 2 -→ ∥u 0 ∥ L 2 .
The main idea to prove (3.18) is to use Parseval's identity on u(t) , where u(t) is written in a suitable evolving L 2 + -basis (f t n ) , and satisfying

| u(t) | f t n | = | u 0 | f 0 n | , ∀n ∈ N ≥0 . (3.19) in (1.5) is not well-defined when u ∈ C t [L 2 + (T)]
x . Therefore, we should find another way to circumvent this problem.

The following proposition aims to answer question I. and to characterize the eigenfunctions of L u 0 for u 0 ∈ L 2 + (T), by finding a uniform bound on the growth of the Sobolev norm ∥f ε,0 n ∥ H 1 2 . For the second question II., we avoid the problem of defining (f t n ) via Definition 3.5 by using the same strategy done in the previous subsection, that is, we characterize the limit f t n , for all t ∈ R . Therefore, we should derive an explicit formula of f ε,t n , for all ε > 0 in order to pass to the limit. Unfortunately, we won't directly obtain that the limit (f t n ) forms an orthonormal basis of L 2 + (T) . However, it shall be an orthonormal family in L 2 + (T) , which will be sufficient to conclude.

Proposition 3.7. Given u 0 ∈ L 2 + (T) there exists a sequence (f 0 n ) ⊆ Dom(L u 0 ), such that for any sequence (u ε 0 ) ⊆ H 2 + (T) , u ε 0 → u 0 in L 2 + (T) , we have up to a subsequence lim ε→0 ∥f ε,0 n -f 0 n ∥ L 2 = 0 , ∀n ∈ N ≥0 .
In addition, for all n ,

L u 0 f 0 n = λ n (u 0 )f 0 n . Proof. By definition of L u ε 0 = D -T u ε 0 T ūε 0 , and since L u ε 0 f ε,0 n = λ n (u ε 0 )f ε,0 n , it follows λ n (u ε 0 ) + ∥T ūε 0 f ε,0 n ∥ 2 L 2 = ∥f ε,0 n ∥ 2 Ḣ 1 2 + , ∀n ≥ 0 .
Note that as u ε 0 → u 0 in L 2 + (T) , and by applying inequality (3.3), we infer that ∃ N ≥ 0 independent of ε , such that

λ n (u ε 0 ) + 1 2 ∥f ε,0 n ∥ 2 Ḣ 1 2 + 2N 2 ∥u ε 0 ∥ L 2 + 1 2 > ∥f ε,0 n ∥ 2 Ḣ 1 2 . Hence, by Proposition 3.1 and since u ε 0 → u 0 in L 2 + (T) , ∥f ε,0 n ∥ 2 Ḣ1/2 ≲ λ n (u 0 ) + ∥u 0 ∥ 2 L 2 . (3.21)
Therefore, up to a subsequence, ∃ (f 0 n ) such that, as ε → 0 ,

f ε,0 n ⇀ f 0 n in H 1 2 + (T) and f ε,0 n → f 0 n in L 2 + (T) . (3.22)
At present, for the second part of the proof we show that the (f 0 n ) are eigenfunctions of L u 0 . Note that by Lemma 2.7 , one can directly check that (f 0 n ) ⊆ Dom(L u 0 ) where Dom(L u 0 ) was defined in (3.5). Besides, by definition of L u ε 0 , we have for all g ∈ H 1 2

+ (T) , ⟨D 1 2 f ε,0 n | D 1 2 g⟩ -T ūε 0 f ε,0 n | T ūε 0 g = λ n (u ε 0 ) f ε,0 n | g , (3.23)
where by Lemma 2.7 T ūε 0 g -→ T ū0 g in L 2 + (T) , by Proposition 3.1 λ n (u ε 0 ) → λ n (u 0 ) , and by (3.22) T ūε 0 f ε,0 n ⇀ T ū0 f 0 n . Hence, passing to the limit in (3.23) , we infer

L u 0 f 0 n | g = λ n (u 0 ) f 0 n | g , ∀g ∈ H 1 2 + (T) , leading to L u 0 f 0 n = λ n (u 0 )f 0
n for all n ≥ 0 , where (λ n (u 0 )) denotes all the spectrum of L u 0 by Corollary 3.2 .

□

In the sequel, thanks to Corollary 3.2 and Proposition 3.7 , we denote by (f 0 n ) the orthonormal basis of L 2 + (T) made up of the eigenfunctions of L u 0 obtained in the previous proposition. The following lemma aims to give an explicit formula to the (f ε,t n ) defined in Definition 3.5 in order to characterize at a second stage their limits when ε → 0.

Lemma 3.8 (The explicit formula of f ε,t n ). Under the same notation of Definition 3.5 , we have for all ε > 0 , t ∈ R ,

f ε,t n (z) = Id -z e -it(L u ε 0 +Id) 2 S * e itL 2 u ε 0 -1 f ε,0 n | e -itL 2 u ε 0 1 , ∀ z ∈ D . (3.24)
Proof. Like the proof of Proposition 2.5 , we have

f ε,t n (z) = (Id -zS * ) -1 f ε,t n | 1 , ∀ z ∈ D .
Using the unitary operator U (t) introduced in (2.10) , we deduce

f ε,t n (z) = U (t) * (Id -zS * ) -1 f ε,t n | U (t) * 1 (3.25) = (Id -zU (t) * S * U (t)) -1 U (t) * f ε,t n | U (t) * 1 = (Id -zU (t) * S * U (t)) -1 f ε,0 n | U (t) * 1
. By the formulae of (2.13) and (2.14) , the explicit formula of f ε,t n follows. □ Proposition 3.9. Let u 0 ∈ B L 2 + (1) . Under the same notation of Definition 3.5 , there exists an orthonormal family (f t n ) of L 2 + (T) , such that for any sequence

(u ε 0 ) ⊆ H 2 + (T) , u ε 0 → u 0 in L 2 + (T) , we have up to a subsequence , ∥f ε,t n -f t n ∥ L 2 -→ ε→0 0 .
Proof. This proof is similar to the one done in Proposition 3.7. However, it presents two main differences. We will discuss these later in the upcoming remark. Now, coming back to the proof, recall by Proposition 3.4 , there exists a unique u(t) ∈ L 2 + (T) such that for any

u ε 0 → u 0 in L 2 + (T) we have u ε (t) ⇀ u(t) in L 2 + (T) as ε → 0 . Therefore, by definition of L u ε (t) = D -T u ε (t) T ūε (t) , and since L u ε (t) f ε,t n = λ n (u ε 0 )f ε,t n by the second point of Remark 3.2 , λ n (u ε 0 ) + ∥T ūε (t) f ε,t n ∥ 2 L 2 = ∥f ε,t n ∥ 2 Ḣ 1 2 , ∀n ≥ 0 .
Thus, applying Lemma 2.7 ,

(1 -∥u ε (t)∥ 2 L 2 ) ∥f ε,t n ∥ Ḣ 1 2 ≤ ∥u ε (t)∥ 2 L 2 + λ n (u ε 0 ) , ∀n ≥ 0 .
Taking ε small enough to guarantee ∥u ε (t)∥ L 2 = ∥u ε 0 ∥ L 2 < 1 , we deduce by Proposition 3.1 , for ε small

∥f ε,t n ∥ 2 Ḣ1/2 ≲ λ n (u 0 ) + ∥u 0 ∥ 2 L 2 1 -∥u 0 ∥ 2 L 2 , ∀n ≥ 0 . (3.26)
Hence, up to a subsequence,

f ε n (t) ⇀ f * n, t in H 1 2 + (T) , f ε n (t) → f * n, t in L 2 + (T) .
It remains to show that f * n, t is uniquely characterized for all t. Using the explicit formula of Lemma 3.8 ,

f ε,t n (z) = Id -z e -it(L u ε 0 +Id) 2 S * e itL 2 u ε 0 -1 f ε,0 n | e -itL 2 u ε 0 1 , ∀ z ∈ D ,
and applying Proposition 3.3 , one can conclude that there exists

f t n (z) = Id -z e -it(Lu 0 +Id) 2 S * e itL 2 u 0 -1 f 0 n | e -itL 2 u 0 1 , ∀ z ∈ D , (3.27)
where (f 0 n ) denotes the eigenfunctions of L u 0 obtained in Proposition 3.7 . Therefore, the limit f * n,t = f t n for all t on D. Finally, observe that since the (f ε,t n ) is an orthonormal basis of L 2 + (T) and as f ε,t n → f t n in L 2 + (T) , then (f t n ) forms an orthonormal family in L 2 + (T) . □ Remark 3.3. There are two main differences between the proof of Proposition 3.7 and Proposition 3.9 :

(i) First, note that in the last proof, we cannot control the growth of the Sobolev norm ∥f ε,t n ∥ H 1 2 uniformly for all t by using the inequality (3.3) , since the integer N η in (3.3) is not uniform for all t ∈ R . As an alternative, we rely on Lemma 2.7 . Consequently, the condition of ∥u ε (t)∥ L 2 < 1 for ε small enough, is crucial here in order to conclude. (ii) Second in the previous proof, we had to give a meaning to the limit f * n,t by characterizing this limit for all t ∈ R .

A common feature about these two proofs is to obtain a uniform bounds on the growth of the Sobolev norm H 1 2 + (T) of the eigenfunctions f ε,0 n and f ε,t n to be able to conclude.

In view of Proposition 3.1 , Lemma 3.6 , Proposition 3.7 and Proposition 3.9 , we infer the following lemma.

Lemma 3.10. Let u 0 ∈ B L 2 + (1) . There exists an orthonormal family

(f t n ) of L 2 + (T) such that for all n ≥ 0 , u(t) | f t n = u 0 | f 0 n e -itλ 2 n (u 0 ) , ∀t ∈ R . (3.28)
We are, at this stage, in a position to prove Theorem 1.2 .

Theorem. 1.2. Let u 0 ∈ B L 2 + (1)
. There exists a unique potential u ∈ C(R, L 2 + (T)) such that, for any sequence

(u ε 0 ) ⊆ H 2 + (T) , ∥u ε 0 -u 0 ∥ L 2 -→ ε→0
0 , the following convergence holds : for all T > 0 ,

sup t∈[-T,T ] ∥u ε (t) -u(t)∥ L 2 → 0 , ε → 0 . In addition, u(t, z) = (Id -z e -it e -2itLu 0 S * ) -1 u 0 | 1 , ∀ z ∈ D . (3.29)
Moreover, the L 2 -norm of the limit potential u is conserved

∥u(t)∥ L 2 = ∥u 0 ∥ L 2 , ∀t ∈ R. Proof. Let (t ε ) ⊆ R such that t ε → t as ε → 0. Since ∥u ε 0 -u 0 ∥ L 2 → 0 , then for ε small enough ∥u ε (t ε )∥ L 2 = ∥u ε 0 ∥ L 2 ≲ ∥u 0 ∥ L 2 .
Hence, for any t ε → t , there exists u * t ∈ L 2 + (T) such that up to a subsequence,

u ε (t ε ) ⇀ u * t in L 2 + (T) and ∥u * t ∥ L 2 ≤ lim inf ε→0 ∥u ε (t ε )∥ L 2 = lim inf ε→0 ∥u ε 0 ∥ L 2 = ∥u 0 ∥ L 2 . (3.30)
Our goal is to show that u ε (t ε ) converges strongly in L 2 + (T) . As a first step, we stress that the weak limit potential u * t is uniquely characterized for all t , and is equal to a unique limit u(t) . For that, we repeat the same proof of Proposition 3.4 by exchanging t into t ε with t ε → t, and we obtain

u ε (t ε ) ⇀ u(t) in L 2 + (T) ,
∥u ε (t ε )∥ L 2 (T) → ∥u(t)∥ L 2 (T) , ε → 0 .
In fact, it is actually sufficient to prove that ∥u

(t)∥ L 2 (T) = ∥u 0 ∥ L 2 (T) since ∥u ε (t ε )∥ L 2 = ∥u ε 0 ∥ L 2 -→ ∥u 0 ∥ L 2 , as ε → 0 . (3.32)
Thanks to (3.31) , we already have ∥u(t)∥ L 2 ≤ ∥u 0 ∥ L 2 . Now, to prove ∥u(t)∥ L 2 ≥ ∥u 0 ∥ L 2 , we use Lemma 3.10 to infer the existence of an orthonormal family

(f t n ) of L 2 + (T) such that ∞ n=0 | u(t) | f t n | 2 = ∞ n=0 | u 0 | f 0 n | 2 = ∥u 0 ∥ 2 L 2 .
Hence, by Bessel's inequality

∥u(t)∥ L 2 ≥ ∥u 0 ∥ L 2 .
As a conclusion, we have proved for any t ε → t , u ε (t ε ) → u(t) in L 2 + (T) . This means, u ∈ C(R, L 2 + (T)) and for all T > 0 , sup

t∈[-T,T ] ∥u ε (t) -u(t)∥ L 2 → 0 , ε → 0 .

□

In view of the last Theorem, we denote through on, u(t) the solution of (CS + ) in L 2 + (T) starting from an initial datum u 0 that lies inside the open ball B L 2 + (1) of L 2 + (T) .

Corollary 3.11. The spectrum σ(L u(t) ) is invariant under the flow of (CS + ) .

Proof.

Let (u ε 0 ) ⊆ H 2 + (T) such that ∥u ε 0 -u 0 ∥ L 2 → 0 as ε → 0 . Since, for all n ∈ N ≥0 , λ n (u ε 0 ) = λ n (u ε (t))
, ∀t ∈ R , then by passing to the limit, we infer by Proposition 3.1 and Theorem 1.2 that the spectrum of L u(t) is conserved in time. □ Proof. For s = 0 , we infer by Theorem 1.2 the global well-posedness of the problem in L 2 + (T) in the sense of continuous extension of the flow from H 2 + (T) to L 2 + (T) . For 0 < s ≤ 3 2 , let u 0 ∈ H s + (T) ∩ B L 2 + (1) and consider (u ε 0 ) ⊆ H 2 + (T) such that u ε 0 → u 0 in H s + (T) . Then applying Proposition 2.8 , and since u ε 0 → u 0 in H s + (T) , there exists C > 0 uniform with respect to all small ε , such that

1 C ∥u ε (t ε )∥ H s ≤ ∥(L u ε (t ε ) + λ) s u ε (t ε )∥ L 2 = ∥(L u ε 0 + λ) s u ε 0 ∥ L 2 ≤ C∥u ε 0 ∥ H s ,
thanks to Lemma 2.6 . Note that λ is also uniform with respect to all ε small for the same reasons presented in the proof of Proposition 3.3 . Therefore, for all ε > 0 small,

∥u ε (t ε )∥ H s ≤ C∥u 0 ∥ H s . ( 4
∥u ε (t ε ) -u(t)∥ 2 H s ≲ ∥(L u ε (t ε ) + λ) s (u ε (t ε ) -u(t))∥ 2 L 2 (4.3) = ∥(L u ε (t ε ) + λ) s u ε (t ε )∥ 2 L 2 + ∥(L u ε (t ε ) + λ) s u(t)∥ 2 L 2 -2Re (L u ε (t ε ) + λ) s u ε (t ε ) | (L u ε (t ε ) + λ) s u(t)
Recall that u ε (t ε ) → u(t) in L 2 + (T) which leads by Proposition 3.3 to L u ε (t ε ) → L u(t) in the strong resolvent sense. Thus, by functional calculus, (see the following lemma-Lemma 4.1) we infer

(L u ε (t ε ) + λ) s u(t) → (L u(t) + λ) s u(t) in L 2 + (T) , (L u ε (t ε ) + λ) s u ε (t ε ) ⇀ (L u(t) + λ) s u(t) in L 2 + (T) , as ε → 0 . In addition, for ε > 0 , recall by Lemma 2.6 , ∥(L u ε (t ε ) + λ) s u ε (t ε )∥ 2 L 2 = ∥(L u ε 0 + λ) s u ε 0 ∥ 2 L 2 . (4.4)
Therefore, passing to the limit in (4.3) , and since u ε 0 → u 0 in H s + (T) combined with Lemma 4.1 and Proposition 2.8 , we deduce for all ε small,

∥u ε (t ε ) -u(t)∥ 2 H s ≲ ∥(L u 0 + λ) s u 0 ∥ 2 L 2 -∥(L u(t) + λ) s u(t)∥ 2 L 2 .
At this stage, it remains to show that the right-hand side of the previous inequality is vanishing. Indeed, by Corollary 3.11 ,

(L u(t) + λ) 2s u(t) | u(t) = n≥0 (λ n (u(t)) + λ) 2s | u(t) | f t n | 2 = n≥0 (λ n (u 0 ) + λ) 2s | u 0 | f 0 n | 2 = (L u 0 + λ) 2s u 0 | u 0 ,
where (f t n ) are the orthonormal basis obtained in Corollary 3.12 since u ε (t

ε ) → u(t) in L 2 + (T) . As a result, ∥(L u 0 + λ) s u 0 ∥ 2 L 2 = ∥(L u(t) + λ) s u(t)∥ 2 L 2 , and as ε → 0 , ∥u ε (t ε ) -u(t)∥ 2 H s -→ 0 . Hence, u ∈ C(R, H s + (T)
) such that (4.2) is satisfied, and for all T > 0 , sup

t∈[-T,T ] ∥u ε (t) -u(t)∥ H s → 0 .

□

To conclude the proof of Corollary 1.3, we need to prove the following functional analysis result.

Lemma 4.1. Let (A ε ) be a sequence of positive self-adjoint operators in L 2 . Suppose that A ε → A in the strong resolvent sense as ε → 0 , and for all s ≥ 0 ,

Dom(A s ε ) = Dom(A s ) = H s , ε > 0 .
Moreover, assume that for all u ∈ H s , the (A s ε u) are uniformly bounded with respect to ε > 0 in the following sense

∥A s ε u∥ ≤ C∥u∥ H s . Then, for all s ≥ 0 , A s ε u -→ A s u in L 2 , ε → 0 . (4.5) Proof. For all R > 0, let χ R ∈ C ∞ (R + ) such that χ R ≡ 1 on [0, R] and supp(χ R ) ⊆ [0, 2R] .
Note that, for all s ≥ 0 , the subset {χ R (A) u ; u ∈ H s , R > 0} is dense in H s . Then, since the (A s ε u) are uniformly bounded with respect to ε , it is sufficient to prove for all R > 0 ,

A s ε χ R (A)u -→ A s χ R (A)u , ε → 0 ,
to obtain (4.5) . Toward this end, let R > 0, and write for any s ≥ 0 , for all ε > 0 ,

A s ε χ R (A)u = A s ε χ R(A ε )χ R (A)u + A s ε (1 -χ R(A ε ))χ R (A)u , (4.6) where χ R ∈ C ∞ (R + ) such that χ R ≡ 1 on [0, R] and supp(χ R) ⊆ [0, 2 R] , R ≥ 2R . Notice that, ∥A s ε (1 -χ R(A ε ))χ R (A)u∥ = ∥A -s ε (1 -χ R(A ε ))A 2s ε χ R (A)u∥ ≤ C (2 R) s ∥χ R (A)u∥ H 2s ≤ C (2 R) s ∥A 2s χ R (A)u∥ + ∥χ R (A)u∥ ≤ C (2 R) s (1 + (2R) s ) ∥u∥ .
Therefore, for all η > 0 , there exists R >> 0 , such that for all ε > 0 ,

∥A s ε (1 -χ R(A ε ))χ R (A)u∥ < η , and
so, by (4.6) ,

∥A s ε χ R (A)u -A s ε χ R(A ε )χ R (A)u∥ < η , ∀ε > 0 . (4.7)
Besides, recall that A ε → A in the strong resolvent sense as ε → 0 . Hence, by [deO09, Proposition 10.1.9] , f (A ε ) → f (A) as ε → 0 , in the operator norm for all continuous bounded f . In particular, for f (x) = x s χ R(x) , we have

A s ε χ R(A ε ) χ R (A)u -→ A s χ R (A)u , ε → 0 . (4.8)
Thus, combining (4.7) and (4.8) , we infer for all η > 0 , there exists ε 0 > 0 such that ∀ε ≤ ε 0 ,

∥A s ε χ R (A)u -A s χ R (A)u∥ < η .

□

Beyond the global well-posedness results of the (CS + ) Cauchy's Problem, we are interested in some qualitative properties about the flow S + (t) of this equation. Therefore, we prove that all weak limit points of the orbit are actually strong limit points.

Theorem. 1.4 . Given an initial data u 0 ∈ B L 2 + (1) ∩ H s + (T) , s ≥ 0 , the orbit of the solution {S + (t)u 0 ; t ∈ R} is relatively compact in H s + (T) . Proof. Let (t n ) ⊆ R such that t n → ∞ .
Step 1 : s = 0 . By Theorem 1.2 ,

∥u(t n )∥ L 2 = ∥u 0 ∥ L 2 . (4.9) Then, ∃ ũ ∈ L 2 + (T) such that, up to a subsequence, u(t n ) ⇀ ũ in L 2 + (T) and ∥ũ∥ L 2 ≤ ∥u 0 ∥ L 2 .
In order to obtain the strong convergence u(t n ) → ũ in L 2 + (T) , all it remains is to show that ∥u(t n )∥ L 2 → ∥ũ∥ L 2 , or by (4.9) ,

∥u 0 ∥ L 2 = ∥ũ∥ L 2 .
Observe that we already have ∥u 0 ∥ L 2 ≥ ∥ũ∥ L 2 . Now, to prove ∥u 0 ∥ L 2 ≤ ∥ũ∥ L 2 , recall by Corollary 3.12 ,

u(t n ) | f tn m = u 0 | f 0 m e -itn λ 2 m (u 0 ) , (4.10)
where (f tn m ) is the orthonormal basis of L 2 + (T) constituted of the eigenfunctions of L u(t n ) . The idea is to pass to the limit as t n → ∞ in the above identity and conclude by using Bessel's identity. First, we have u(t n ) ⇀ ũ in L 2 + (T) . Second, notice that the (f tn m ) converges strongly in L 2 + (T) as t n → ∞ to an orthonormal family denoted by (g m ) . Indeed, by definition of

L u(tn) , λ m (u 0 ) + ∥T u(tn) f tn m ∥ 2 L 2 = ∥f tn m ∥ 2 Ḣ 1 2 , leading to ∥f tn m ∥ H 1 2 ≲ λm(u 0 )+∥u 0 ∥ 2 L 2 1-∥u 0 ∥ 2 L 2
for all m , thanks to Lemma 2.7 . Hence, by Rellich-Kondrachov's Theorem , f m (t n ) → g m in L 2 + (T) as t n → ∞ . Third, using Cantor diagonalization procedure, one can extract a subsequence t n → θ m mod( 2π λ 2 m ) , as the circle is compact. Hence, by passing to the limit in (4.10), we obtain

⟨ũ | g m ⟩ = u 0 | f 0 m e -iλ 2 m (u 0 )θm .
As a result, using Bessel's inequality, we conclude

∥ũ∥ 2 L 2 ≥ ∞ n=0 |⟨ũ | g n ⟩| 2 = ∞ n=0 | u 0 | f 0 m | 2 = ∥u 0 ∥ 2 L 2 . Consequently, ∥u(t n )∥ L 2 → ∥ũ∥ L 2 and thus u(t n ) → ũ in L 2 + (T) .
Step 2 : s > 0 . By inequality (4.2) of the proof of Corollary 1.3 , we have ∥u(

t n )∥ H s ≲ ∥u 0 ∥ H s leading to u(t n ) ⇀ ũ in H s + (T) and ∥ũ∥ H s ≲ ∥u 0 ∥ H s .
In particular, u(t n ) → ũ in L 2 + (T) . Then, in view of Remark 2.5 ,

∥u(t n ) -ũ∥ 2 H s ≲ ∥(L u(tn) + λ Id) s (u(t n ) -ũ)∥ 2 L 2 (4.11) = ∥(L u(tn) + λ Id) s u(t n )∥ 2 L 2 + ∥(L u(tn) + λ Id) s ũ∥ 2 L 2 -2Re (L u(tn) + λ) s u(t n ) | (L u(tn) + λ) s ũ ,
where by the second point of Remark 2.2 ,

∥(L u(tn) + λ Id) s u(t n )∥ 2 L 2 = ∥(L u 0 + λ) s u 0 ∥ 2 L 2 .
Besides, as u(t n ) → ũ in L 2 + (T) , then L u(tn) → L ũ in the strong resolvent sense thanks to Proposition 3.3 . Hence, by functional calculus (see Lemma 4.1), we infer

(L u(tn) + λ) s ũ → (L ũ + λ) s ũ in L 2 + (T) , (L u(tn) + λ) s u(t n ) ⇀ (L ũ + λ) s ũ in L 2 + (T) ,
as n → ∞ . Therefore, by passing to the limit in (4.11), we deduce

lim n→+∞ ∥u(t n ) -ũ∥ 2 H s ≲ ∥(L u 0 + λ) s u 0 ∥ 2 L 2 -∥(L ũ + λ) s ũ∥ 2 L 2 ,
where the right-hand side vanishes. Indeed, by Corollary 3.11 and Proposition 3.1 , λ n (ũ) = λ n (u 0 ) . Hence,

(L ũ + λ) 2s ũ | ũ = n≥0 (λ n (u 0 ) + λ) 2s |⟨ũ | g n ⟩| 2 = n≥0 (λ n (u 0 ) + λ) 2s | u 0 | f 0 n | 2 = (L u 0 + λ) 2s u 0 | u 0 ,
where (g n ) is the orthonormal family of L 2 + (T) found in Step 1. Nevertheless, since u(t n ) → ũ in L 2 + (T) , one could show as in Corollary 3.12 that this orthonormal family is indeed an orthonormal basis of L 2 + (T) by proving that the (g n ) constitutes all the eigenfunction of the self-adjoint operator L ũ . As a consequence,

∥(L u 0 + λ) s u 0 ∥ 2 L 2 = ∥(L ũ + λ) s ũ∥ 2 L 2 ,
and thus ∥u(t n ) -ũ∥ 2 H s → 0 , as n → ∞ . □

The Calogero-Sutherland DNLS defocusing equation (CS -)

In this section, we consider the defocusing equation of (CS)

i∂ t u + ∂ 2 x u -2D + (|u| 2 )u = 0 . (CS -)
Note that by adapting the argument of [GL22, Proposition 2.1] to the defocusing equation, one can infer the local well-posedness of the (CS -) problem in H s + (T) for s > 3 2 . And we expect that one can go down to s > 1 2 by following [START_REF] De Moura | Local well-posedness for the nonlocal nonlinear Schrödinger equation below the energy space[END_REF] .

Below are a series of lemmas, propositions, and theorems that can be proved similarly to their analogs in the focusing case. Again, the integrable methods are the main ingredients to conclude. The first proposition is to announce that the defocusing equation of (CS -) enjoys also a Lax pair formalism. As a consequence, the quantities Hs (u) := ⟨( Lu + λ) s u | u⟩ , λ > 0 , are conserved by the flow S -(t) of (CS -) for all 0 ≤ s ≤ 2r .

Remark 5.1. Expanding the conservation laws Hk (u) for all k ∈ N ≥0 , we have

H0 (u) = ⟨u(t) | u(t)⟩ = ∥u(t)∥ 2 L 2 = ∥u 0 ∥ L 2 H1 (u) = ⟨ Lu(t) u(t) | u(t)⟩ = ∥u(t)∥ 2 Ḣ1/2 + ∥T ū(t) u(t)∥ 2 L 2 ≥ ∥u(t)∥ 2 Ḣ 1 2 H2 (u) = ∥ Lu(t) u(t)∥ 2 L 2 ≥ (1 -ε)∥Du(t)∥ 2 L 2 + (1 -C ε )∥T u(t) T ū(t) u(t)∥ 2 L 2 ≥ (1 -ε)∥u(t)∥ 2 Ḣ1 -C(∥u∥ H 1
2 ) , . . . thanks to Young's and Sobolev's inequalities. Unlike the focusing case, here we deduce the uniform control of the growth of Sobolev norms of the solution u by the conservation laws, without requiring any additional condition of smallness on the initial data u 0 . Therefore, Proposition 2.8 holds in the defocusing case for all u 0 ∈ H r + (T) , r > 3 2 .

As a result, we state the following theorem which is the analog of Theorem 1.1 but for equation (CS -).

Theorem 5.3. For all s > 3 2 , let u 0 ∈ H s + (T) . There exists a unique global solution u ∈ C(R, H s + (T)) of the defocusing equation (CS -), satisfying at t = 0, u(0, •) = u 0 . Furthermore, for all s > 3 2 , sup t∈R ∥u(t)∥ H s ≤ C , where C = C(∥u 0 ∥ H s ) > 0 is a positive constant.

As for the focusing case, the defocusing Calogero-Sutherland DNLS has an explicit solution.

Lemma 5.4. Let u 0 ∈ C(R, H s + (T)) , s > 3 2 then the solution of the defocusing Calogero-Sutherland DNLS equation (CS -) is given by u(t, z) = (Id -z e -it e -2it Lu 0 S * ) -1 u 0 | 1 .

In particular, using this explicit formula, we extend the flow S -(t) continuously from H 2 + (T) to H s + (T) , for 0 ≤ s ≤ 3 2 . Therefore, we have : Theorem. 1.5 . The Calogero-Sutherland DNLS defocusing equation (CS -) is globally well-posed in H s + (T) for any s ≥ 0 , in the sense of Remark 1.2 . In addition, for all u 0 ∈ H s + (T) , u(t, z) = (Id -z e -it e -2it Lu 0 S * ) -1 u 0 | 1 , is solution of the (CS -)-defocusing equation. Furthermore, the trajectories S -(t)u 0 ; t ∈ R are relatively compact in H s + (T) .

Final remarks and open problems

Let us briefly discuss here some remarks related to the previous sections.

1. One interesting feature about the focusing Calogero-Sutherland DNLS equation is that it admits a rich dynamic in comparison to the defocusing equation. For instance, as we shall see [START_REF] Badreddine | Traveling waves & finite gap potentials for the Calogero-Sutherland derivative nonlinear Schrödinger equation[END_REF] , the focusing equation has a wider collection of traveling wave solutions. is a key ingredient to answer this question. Indeed, writing for all t ∈ R , Σ t the operator S e 2itLu 0 e it , we have by (6.1)

u(t, z) = n≥0 ⟨u 0 | Σ n t 1⟩ z n . (6.2)
Observe that, if u 0 belongs to the space J generated by the orthonormal family {Σ n t 1 , n ≥ 0} , then using Parseval's identity on (6.2) , we infer ∥u(t)∥ L 2 (T) = ∥u 0 ∥ L 2 , t ∈ R , (6.3) leading to say that the set {u(t) , t ∈ R} is relatively compact in L 2 + (T) . Hence, the integer N η set out in inequality (3.3) is now independent of t , and thus applying inequality (3.3) to (2.19) , we obtain for all η > 0 , ∥(L u(t) + λ)

1 2 f ∥ 2 L 2 ≥ (1 -2η 2 )∥f ∥ 2 Ḣ 1 2 + (λ -2η 2 -2N 2 η ∥u 0 ∥ L 2 )∥f ∥ 2 L 2 ,
instead of having inequality (2.20) :

∥(L u(t) + λ)

1 2 f ∥ 2 L 2 ≥ (1 -∥u∥ 2 L 2 )∥f ∥ 2 Ḣ 1 2 + (λ -∥u∥ 2 L 2 )∥f ∥ 2 L 2 .
Therefore, using (3.3) , we control the growth of all the Sobolev norm ∥u(t)∥ H s for all s ≥ 0 , and we infer the global well-posedness of the focusing (CS + ) in all H s + (T) , s > 3 2 for arbitrary initial data. In addition, by the same manner, we deduce also H 1 2 -bounds on the eigenfunctions (f ε,t n ) -inequality (3.26)-implying that the flow S + (t) can be extended to L 2 + (T) , for arbitrary initial data. Besides, if u 0 does not belong to J , then we expect blow-up results in finite time T .

  coth π(x -y) 2δ u(t, y) dy , by taking δ → ∞ . The complex function u in (INS) represents the envelope of the fluid, and δ denotes its total depth. By passing to the limit δ → ∞ , one obtains the same equation as (INS) but with the Hilbert transform Hu(t, x) = 1 π p.v.

  of T[START_REF]Pelinovsky Intermediate nonlinear Schrödinger equation for internal waves in a fluid of finite depth[END_REF]. And since the Szegő projector Π T = 1 2 (Id +iH + ⟨• | 1⟩) , then the (CS)-equation can also be interpreted as a model describing the interfacial wave packets in a deep stratified fluid.

  Proposition 5.1 (Lax pair for (CS -)). Let u ∈ C([-T, T ] , H s + (T)) , s > 3 2 , be a solution of (CS -) . There exist two operatorsLu = D + T u T u , Bu = -T u T ∂xu + T ∂xu T u + i(T u T u ) 2satisfying the Lax equationd Lu dt = [ Bu , Lu ] .Lemma 5.2. Given u ∈ C([-T, T ], H r + (T)) , r > 3 2 , a solution of (CS -) equation, then∂ t u = Bu u -i L2 u u .

2.

  The problem of global well-posedness of the focusing Calogero-Sutherland DNLS equation (CS + ) without restriction on the initial data is wide open. Nevertheless, we expect that the explicit solution (equation (3.29)) u(t, z) = (Id -z e -it e -2itLu 0 S * ) -1 u 0 | 1 , (6.1)

  .1) Hence, ast ε → t , we have u ε (t ε ) ⇀ u(t) in H s + (T), where u is a characterized function for all t obtained as in Proposition 3.4 . In particular, we inferu ε (t ε ) → u(t) in L 2 + (T) with ∥u(t)∥ H s ≲ ∥u 0 ∥ H s ,

	(4.2)
	by (4.1) . As of now, to deduce the strong convergence in H s + (T) we use Proposition 2.8 .
	Thus, for all ε > 0 ,

Definition 3.5 (An orthonormal basis of L 2 + (T)). For all ε > 0 , let u ε ∈ C(R, H 2 + (T)) . We denote by (f ε,t n ) the evolving orthonormal basis of L 2 + (T) along the curve t → u ε (t) and satisfying the Cauchy problem

for all n , where (f ε,0 n ) is the orthonormal basis of L 2 + (T) constituted from the eigenfunctions of the self-adjoint Lax operator L u ε 0 , and B u ε (t) is the skew-adjoint operator defined in (1.5) . Remark 3.2.

(1) Since for all ε > 0 , B u ε (t) is a skew-adjoint bounded operator (cf. Proposition 2.3) then the orthogonality of the (

(2) By [Ku06, Lemma 4.1] , such orthonormal basis is formed by the eigenfunctions of the Lax operator L u ε (t) . Typically, we have for all ε > 0 , for all t ∈ R ,

With this choice of L 2 + (T)-basis, we have a nice description of the evolution of the coordinates of u ε (t) . This is the aim of the next Lemma. Lemma 3.6. For all ε > 0 , let u ε ∈ C(R, H 2 + (T)) solution of (CS + ) . Under the same notation of Definition 3.5 , we have for any n ∈ N ≥0 ,

Proof. By Lemma 2.4 , and since L u ε (t) and B u ε (t) are respectively self-adjoint and skewadjoint operators

which leads to the statement. □

Consequence. From the previous lemma, we infer for all ε > 0 , n ∈ N ≥0 ,

At this stage, we want to take ε → 0 in the latter identity in order to deduce (3.19) . However, one first might ask two questions :

I. Does the orthonormal basis (f ε,0 n ) constituted from the eigenfunctions of the self-adjoint Lax operator L u ε 0 remains an orthonormal basis of L 2 + (T) under the limit ε → 0 ? II. Suppose that the answer to the former question is affirmative, and denote by (f 0 n ) this orthonormal basis limit. Based on Definition 3.5 , could we construct a time-evolving orthonormal basis, coinciding at t = 0 with (f 0 n ) , and inducing a nice evolution as in Lemma 3.6 of the coordinates of u in this basis ? A priori, the operator B u defined Corollary 3.12. Let u 0 ∈ B L 2 + (1) . There exists an orthonormal basis (f t n ) of L 2 + (T) constituted from the eigenfunction of L u(t) , such that for all n ∈ N ≥0 ,

Proof. Taking into account Lemma 3.10 , we only need to prove that the orthonormal family (f t n ) found in Proposition 3.9 as

is actually an orthonormal basis of L 2 + (T) . On the one hand, using (3.34) and since

On the other hand, using Proposition 3.1 , we infer that taking ε → 0 in

+ (T) . As a result, the (f t n ) describes all the eigenfunctions of the self-adjoint operator L u(t) , thanks to Corollary 3.11 and Corollary 3.2 . Hence, they form an orthonormal basis of L 2 + (T) . □ Remark 3.4. The nice evolution in (3.33) of such coordinates suggests that the so-called "Birkhoff coordinates" of (CS + ) are the (⟨u(t) | f t n ⟩) . To be sure, we need to construct a one-by-one Birkhoff map u ←→ (⟨u(t) | f t n ⟩) , similar to the remarkable achievement for the Benjamin-Ono equation in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]. This construction can unlock several significant outcomes regarding the equation's dynamics. In particular, the global well-posedness of the focusing equation when ∥u 0 ∥ L 2 ≥ 1 .

Proof of Corollary 1.3 and Theorem 1.4

To summarize, we have proved the global well-posedness of (CS + )-equation in H s + (T) , s > 3 2 , and for s = 0 which correspond to H 0 + (T) ≡ L 2 + (T) . The following corollary aims to prove the global well-posedness for 0 < s ≤


Julia Wagner

K Guérin

Sandrine Berthon-Fabry

Comparative study of kapok-based self supported composites with TiO 2 or SiO 2 usable in Li-ion batteries

Comparative study of kapok-based self supported composites with TiO 2 or SiO 2 usable in Li-ion batteries

Julia Wagner* 1,2 , Katia Guérin 2,1 , Sandrine Berthon-Fabry Biobased and recycled materials are becoming more commonly used to create anodes for lithiumion batteries. [1], [2], [3] Carbon-based materials can be obtained by pyrolyzing organic materials while keeping their initial shape. [4] In this work, we synthesized self-supported carbon-metal oxide composites using pyrolyzed kapok fibers. We chose two well known anode materials: TiO 2 processing through insertion and SiO 2 processing through conversion-alloying process. The main idea is to compare the efficiency and durability of self-supported anodes upon electrochemical mechanisms.

The TiO 2 -based composite was synthesized by sol-gel. In order to reach the theoretical capacity of TiO 2 the physico-chemical properties of the composite were optimized. The lithium-ion-conducting anatase phase was obtained by modifying the acid used as a sol-gel catalyst and the pyrolysis temperature. The specific surface area and the material texture were varied by testing two drying processes (evaporative and CO 2 supercritical drying) and different thermal treatments. Finally, the optimal synthesis method was determined by galvanostatic cycling of the anodes.

The SiO 2 -based composite was obtained from the pyrolysis of a silica-resorcinol-formaldehyde gel. Following the same methodology as TiO 2 , different synthesis parameters were optimized. Specifically, in order to increase durability of the anode the effects of silica content, specific surface area and morphology were tuned by adjusting reagent and catalyst quantities. Finally, the optimal synthesis conditions were determined by galvanostatic cycling.

  2 adsorption isotherms of C-TiO 2 and C-SiO 2 composites, B, C: SEM micrographs of C-TiO 2 (B) and C-SiO 2 (C) composites

  1 1 MINES Paris, PSL University, PERSEE, CS 10207 rue Claude Daunesse 06904 Sophia Antipolis Cedex,

	France	2
	Clermont Université, ICCF, F-63000 Clermont-Ferrand, France
	(*) julia.wagner@minesparis.psl.eu