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On the well-posedness of the intermediate nonlinear Schrödinger equation on the line

Andreia Chapouto, Justin Forlano, Thierry Laurens

TL;DR

The paper develops a Fourier-analytic framework to study the intermediate nonlinear Schrödinger equation (INLS) on the real line, linking finite-depth models to continuum Calogero-Moser (CCM) dynamics as depth parameter $h$ varies. It achieves local well-posedness for $H^s(b R)$ with $s> frac{1}{4}$, by employing a gauge transform and a refined nonlinear decomposition that mitigates derivative losses, and it proves global well-posedness for small $L^2$ data via a new Lax pair that provides a priori bounds. An infinite-depth limit is established, showing convergence of INLS solutions to CCM and of polynomial invariants as $h o ty$. The results extend well-posedness theory beyond the Hardy-space setting, accommodate perturbations with $eta, abla o ext{real}$, and connect integrable CCM structure to non-integrable INLS through robust Fourier-analytic methods. Overall, the work advances low-regularity well-posedness, global behavior under small data, and the rigorous link between INLS and CCM, with implications for derivative-type NLS models and their stability properties.

Abstract

We consider a family of intermediate nonlinear Schrödinger equations (INLS) on the real line, which includes the continuum Calogero-Moser models (CCM). We prove that INLS is locally well-posed in $H^{s}(\mathbb{R})$ for any $s>\frac 14$, which improves upon the previous best result of $s>\frac 12$ by de Moura-Pilod (2008). This result is also new in the special case of CCM, as the initial condition is not required to lie in any Hardy space. Our approach is based on a gauge transformation, exploiting the remarkable structure of the nonlinearity together with bilinear Strichartz estimates, which allows to recover some of the derivative loss. This turns out to be sufficient to establish our main results for CCM in the Hardy space. For INLS and CCM outside of the Hardy space, the main difficulty comes from the lack of the Hardy space assumption, which we overcome by implementing a refined decomposition of the solutions, which observes a nonlinear smoothing effect in part of the solution. We also discover a new Lax pair for INLS and use it to establish global well-posedness in $H^{s}(\mathbb{R})$ for any $s>\frac 14$ under the additional assumption of small $L^2$-norm.

On the well-posedness of the intermediate nonlinear Schrödinger equation on the line

TL;DR

The paper develops a Fourier-analytic framework to study the intermediate nonlinear Schrödinger equation (INLS) on the real line, linking finite-depth models to continuum Calogero-Moser (CCM) dynamics as depth parameter varies. It achieves local well-posedness for with , by employing a gauge transform and a refined nonlinear decomposition that mitigates derivative losses, and it proves global well-posedness for small data via a new Lax pair that provides a priori bounds. An infinite-depth limit is established, showing convergence of INLS solutions to CCM and of polynomial invariants as . The results extend well-posedness theory beyond the Hardy-space setting, accommodate perturbations with , and connect integrable CCM structure to non-integrable INLS through robust Fourier-analytic methods. Overall, the work advances low-regularity well-posedness, global behavior under small data, and the rigorous link between INLS and CCM, with implications for derivative-type NLS models and their stability properties.

Abstract

We consider a family of intermediate nonlinear Schrödinger equations (INLS) on the real line, which includes the continuum Calogero-Moser models (CCM). We prove that INLS is locally well-posed in for any , which improves upon the previous best result of by de Moura-Pilod (2008). This result is also new in the special case of CCM, as the initial condition is not required to lie in any Hardy space. Our approach is based on a gauge transformation, exploiting the remarkable structure of the nonlinearity together with bilinear Strichartz estimates, which allows to recover some of the derivative loss. This turns out to be sufficient to establish our main results for CCM in the Hardy space. For INLS and CCM outside of the Hardy space, the main difficulty comes from the lack of the Hardy space assumption, which we overcome by implementing a refined decomposition of the solutions, which observes a nonlinear smoothing effect in part of the solution. We also discover a new Lax pair for INLS and use it to establish global well-posedness in for any under the additional assumption of small -norm.

Paper Structure

This paper contains 16 sections, 24 theorems, 356 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Going beyond H½
  4. Preliminaries
  5. Notation
  6. Product estimates
  7. The operators Th and Gh
  8. The gauge transform and properties of solutions
  9. Fourier restriction norm spaces
  10. Gauge transformation
  11. CCM regularity properties: using the Hardy space assumption
  12. INLS regularity properties: a decomposition
  13. INLS regularity properties: more nonlinear estimates
  14. Trilinear estimates
  15. Well-posedness and the infinite depth limit
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $s>\frac{1}{4}$. Then, for any $0<h\leq \infty$ and $\gamma,\beta\in \mathbb{R}$, INLS is locally well-posed in $H^{s}(\mathbb{R})$. More precisely, for any $u_0\in H^s(\mathbb{R})$, there exist $0<\delta\ll 1$ and $T=T(\|u_0\|_{H^s})>0$ and a unique solution $u$ to INLS in the space satisfying $u(0)=u_0$ and where $F$ is a primitive of $|u|^2$ defined in F, the spaces $X^{s,b}_{T}$ are the

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1: Fractional Leibniz rule
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 39 more