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Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation

Yongming Li

TL;DR

The paper proves the full asymptotic stability of solitary waves for the 1D focusing cubic NLS under small weighted perturbations by merging space–time resonances with the distorted Fourier transform and modulation theory. It overcomes threshold resonances that slow decay via a null structure in the quadratic nonlinearities and a moving-center smoothing mechanism, enabling modified scattering of the radiation and convergence of modulation parameters. A two-tier bootstrap controls both the radiation profile in a weighted energy space and the soliton parameters, with a careful normal-form reduction turning quadratic effects cubic and non-resonant cubic interactions into higher-order terms. The results extend previous work by handling threshold resonances and removing symmetry restrictions, providing robust techniques for low-power NLS models with delicate long-time dynamics.

Abstract

We establish the full asymptotic stability of solitary wave solutions for the 1D focusing cubic Schrödinger equation on the line under small perturbations in weighted Sobolev spaces, building upon our results in [58]. The proof integrates the space-time resonances approach, based on the distorted Fourier transform, with modulation techniques to show modified scattering for the radiation term and convergence for the modulation parameters. A key challenge throughout the nonlinear analysis is the slow local decay of the radiation term, caused by threshold resonances in the linearized operator. The presence of favorable null structures in the quadratic nonlinearities mitigates this problem through the use of normal form transformations. Another essential step in the proof involves developing a variant of the local smoothing estimate that incorporates a moving center.

Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation

TL;DR

The paper proves the full asymptotic stability of solitary waves for the 1D focusing cubic NLS under small weighted perturbations by merging space–time resonances with the distorted Fourier transform and modulation theory. It overcomes threshold resonances that slow decay via a null structure in the quadratic nonlinearities and a moving-center smoothing mechanism, enabling modified scattering of the radiation and convergence of modulation parameters. A two-tier bootstrap controls both the radiation profile in a weighted energy space and the soliton parameters, with a careful normal-form reduction turning quadratic effects cubic and non-resonant cubic interactions into higher-order terms. The results extend previous work by handling threshold resonances and removing symmetry restrictions, providing robust techniques for low-power NLS models with delicate long-time dynamics.

Abstract

We establish the full asymptotic stability of solitary wave solutions for the 1D focusing cubic Schrödinger equation on the line under small perturbations in weighted Sobolev spaces, building upon our results in [58]. The proof integrates the space-time resonances approach, based on the distorted Fourier transform, with modulation techniques to show modified scattering for the radiation term and convergence for the modulation parameters. A key challenge throughout the nonlinear analysis is the slow local decay of the radiation term, caused by threshold resonances in the linearized operator. The presence of favorable null structures in the quadratic nonlinearities mitigates this problem through the use of normal form transformations. Another essential step in the proof involves developing a variant of the local smoothing estimate that incorporates a moving center.
Paper Structure (30 sections, 37 theorems, 456 equations)

This paper contains 30 sections, 37 theorems, 456 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Main challenges and outline of the proof
  4. Related literature
  5. Notation
  6. Spectral Theory and Distorted Fourier Transform
  7. Spectrum of linearized operator
  8. Distorted Fourier Transform
  9. Mapping properties
  10. Linear Decay Estimates
  11. Dispersive decay
  12. Improved local decay
  13. Smoothing estimates with a moving center
  14. Low energy case. Proof of Lemma \ref{['lemma:dual_low_smoothing']}
  15. High energy case. Proof of Lemma \ref{['lemma:dual_high_smoothing']}
  16. ...and 15 more sections

Key Result

Theorem 1.1

For any $\omega_0 \in (0,\infty)$ there exist constants $0 < \varepsilon_0 \ll 1$, $0 < \delta \ll 1$, and $C_0 \geq 1$ with the following property: Let $(\gamma_0,p_0,\sigma_0) \in \mathbb R^3$ and let $u_0 \in H^1_x(\mathbb R) \cap L^{2,1}_x(\mathbb R)$ with Then the $H^1_x \cap L^{2,1}_x$--solution $\psi(t,x)$ to equ:cubic_NLS with initial condition exists globally in time and there exist con

Theorems & Definitions (78)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 68 more