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Asymptotic stability of travelling waves for general nonlinear Schrödinger equations with non-zero condition at infinity

Jordan Berthoumieu

TL;DR

This work extends the asymptotic stability theory to general nonlinear Schrödinger equations with nonzero boundary conditions by analyzing travelling waves near the sonic speed $c_s=\\sqrt{-2f'(1)}$. The authors combine a hydrodynamic formulation, modulation theory, a Liouville-type rigidity result, and monotonicity-smoothing arguments to show that solutions starting near a travelling wave with speed close to $c_s$ converge to a (potentially different) travelling wave, up to translations and phase shifts, with precise modulation parameters tracked over time. Key contributions include the construction of a limit profile in hydrodynamic variables, a rigorous Liouville rigidity under algebraic decay, and the transfer of stability from the hydrodynamic to the classical variables, all under carefully crafted smoothness and growth hypotheses on $f$. The results provide a robust framework for understanding long-time dynamics of dark solitons and related structures in nonzero background settings, with potential implications for Bose–Einstein condensates and nonlinear optics. Overall, the paper advances the theory of asymptotic stability for nonzero-background NLS by marrying modulation analysis with virial-type rigidity and smoothing techniques.

Abstract

In previous works [4, 5], existence and uniqueness of travelling waves for the nonlinear Schrödinger equations have been shown for speeds close to the speed of sound. Furthermore, it has been proved that a chain of dark solitons of well-ordered speeds near the sound speed, taken initially apart from each other, is orbitally stable. In this article, we complete this study by proving the asymptotic stability of these travelling waves, namely that a solution initially close to a travelling wave eventually converges towards a travelling wave of close speed. This relies on the methods used by F. Béthuel, P. Gravejat and D. Smets in [6] and first introduced by Y. Martel and F. Merle in [22].

Asymptotic stability of travelling waves for general nonlinear Schrödinger equations with non-zero condition at infinity

TL;DR

This work extends the asymptotic stability theory to general nonlinear Schrödinger equations with nonzero boundary conditions by analyzing travelling waves near the sonic speed . The authors combine a hydrodynamic formulation, modulation theory, a Liouville-type rigidity result, and monotonicity-smoothing arguments to show that solutions starting near a travelling wave with speed close to converge to a (potentially different) travelling wave, up to translations and phase shifts, with precise modulation parameters tracked over time. Key contributions include the construction of a limit profile in hydrodynamic variables, a rigorous Liouville rigidity under algebraic decay, and the transfer of stability from the hydrodynamic to the classical variables, all under carefully crafted smoothness and growth hypotheses on . The results provide a robust framework for understanding long-time dynamics of dark solitons and related structures in nonzero background settings, with potential implications for Bose–Einstein condensates and nonlinear optics. Overall, the paper advances the theory of asymptotic stability for nonzero-background NLS by marrying modulation analysis with virial-type rigidity and smoothing techniques.

Abstract

In previous works [4, 5], existence and uniqueness of travelling waves for the nonlinear Schrödinger equations have been shown for speeds close to the speed of sound. Furthermore, it has been proved that a chain of dark solitons of well-ordered speeds near the sound speed, taken initially apart from each other, is orbitally stable. In this article, we complete this study by proving the asymptotic stability of these travelling waves, namely that a solution initially close to a travelling wave eventually converges towards a travelling wave of close speed. This relies on the methods used by F. Béthuel, P. Gravejat and D. Smets in [6] and first introduced by Y. Martel and F. Merle in [22].

Paper Structure

This paper contains 16 sections, 26 theorems, 316 equations.

Table of Contents

  1. Introduction
  2. The hydrodynamic framework
  3. Orbital stability of a travelling wave
  4. Sketch of the proof
  5. Construction of the profile in hydrodynamic variables
  6. Liouville property around a travelling wave
  7. Algebraic decay of the solution $\widetilde{Q}$
  8. Proof of the asymptotic stability in the classical variables
  9. Monotonicity and smoothing properties of \ref{['NLS']}
  10. Proof of the Liouville property
  11. Equivalence of the topologies in a neighborhood of the travelling wave
  12. Weak-continuity of the classical and hydrodynamical flow
  13. Estimates on the linearized operator $\mathcal{H}_c$
  14. Useful estimates
  15. Improvement of the orbital stability result
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $u_0\in \mathcal{X}(\mathbb{R})$. Take $f$ in $C^2(\mathbb{R})$ satisfying hypothèse de croissance sur F minorant intermediaire below. In addition, assume that there exist $\alpha_1\geq 1$ and $C_0 >0$ such that for all $\rho\geq 1$, If $\alpha_1 > \frac{3}{2}$, assume moreover that there exists $\alpha_2\in [ \alpha_1-\frac{1}{2},\alpha_1]$ such that for $\rho\geq 2$, $C_0 \rho^{\alpha_2} \l

Theorems & Definitions (64)

  • Theorem 1.1: Gallo1
  • Theorem 1.2: Bert2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9: Bert2
  • Remark 1.10
  • ...and 54 more