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Asymptotic Stability of multi-solitons for $1$d Supercritical NLS

Gong Chen, Abdon Moutinho

TL;DR

This work proves the asymptotic stability of multi-solitons for the 1D $L^{2}$-supercritical NLS $i\partial_t\psi+\partial_x^2\psi+|\psi|^{2k}\psi=0$ with $k>2$ on a finite-codimension center-stable manifold, extending Krieger–Schlag’s single-soliton results to multi-solitons under a well-separated, nondegenerate spectral framework. The authors develop and leverage a linear theory for one-dimensional matrix charge transfer models to obtain dispersive and weighted estimates for the linearized flow around moving solitons, then implement a finite-time iterative contraction scheme that terminates unstable modes and passes to a limit. They construct a modulated multi-soliton decomposition plus a radiative component, proving precise decay and scattering properties for the remainder and establishing a center-stable manifold of codimension $m$ (and, via a more detailed analysis, a codimension-$m$ manifold on the soliton family). The results hinge on a careful spectral analysis of the linearized operators, sharp dispersive estimates for multi-potential configurations, and a robust nonlinear control strategy that accommodates slow soliton-soliton interactions in one dimension. This advances understanding of soliton resolution in 1D non-integrable settings and demonstrates asymptotic stability for more complex multi-soliton configurations than previously known.

Abstract

Consider the one-dimensional $L^2$ supercritical nonlinear Schrödinger equation \begin{equation} i\partial_{t}ψ+\partial^{2}_{x}ψ+\vert ψ\vert^{2k}ψ=0 \text{, $k>2$}. \end{equation} It is well known that solitary waves for this equation are unstable. In the pioneering work of Krieger and Schlag \cite{KriegerSchlag}, the asymptotic stability of a solitary wave was established on a codimension-one center-stable manifold. In the present paper, using linear estimates developed for one-dimensional matrix charge transfer models in our previous work, \cite{dispanalysis1}, we prove asymptotic stability of multi-solitons on a finite-codimension manifold for $k>\frac{11}{4}.$

Asymptotic Stability of multi-solitons for $1$d Supercritical NLS

TL;DR

This work proves the asymptotic stability of multi-solitons for the 1D -supercritical NLS with on a finite-codimension center-stable manifold, extending Krieger–Schlag’s single-soliton results to multi-solitons under a well-separated, nondegenerate spectral framework. The authors develop and leverage a linear theory for one-dimensional matrix charge transfer models to obtain dispersive and weighted estimates for the linearized flow around moving solitons, then implement a finite-time iterative contraction scheme that terminates unstable modes and passes to a limit. They construct a modulated multi-soliton decomposition plus a radiative component, proving precise decay and scattering properties for the remainder and establishing a center-stable manifold of codimension (and, via a more detailed analysis, a codimension- manifold on the soliton family). The results hinge on a careful spectral analysis of the linearized operators, sharp dispersive estimates for multi-potential configurations, and a robust nonlinear control strategy that accommodates slow soliton-soliton interactions in one dimension. This advances understanding of soliton resolution in 1D non-integrable settings and demonstrates asymptotic stability for more complex multi-soliton configurations than previously known.

Abstract

Consider the one-dimensional supercritical nonlinear Schrödinger equation \begin{equation} i\partial_{t}ψ+\partial^{2}_{x}ψ+\vert ψ\vert^{2k}ψ=0 \text{, }. \end{equation} It is well known that solitary waves for this equation are unstable. In the pioneering work of Krieger and Schlag \cite{KriegerSchlag}, the asymptotic stability of a solitary wave was established on a codimension-one center-stable manifold. In the present paper, using linear estimates developed for one-dimensional matrix charge transfer models in our previous work, \cite{dispanalysis1}, we prove asymptotic stability of multi-solitons on a finite-codimension manifold for

Paper Structure

This paper contains 64 sections, 31 theorems, 473 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Some notations
  5. Asymptotic stability
  6. Center-stable manifolds
  7. Organization and notations
  8. Organization
  9. Notations
  10. Acknowledgement
  11. Preliminaries and main ideas
  12. Spectral theory
  13. Linear theory for one-dimensional charge transfer models
  14. Some technical preparations
  15. Main ideas for the proof of Theorem \ref{['Tsigma(t)']}
  16. ...and 49 more sections

Key Result

Theorem 1.1

Assume that hypotheses $\mathrm{(H1)}$ and $\mathrm{(H2)}$ hold. Let $\delta_0\in (0,1)$ be a small constant only depending on the prescribed constants Consider the linear stable space: and a small ball inside it If $\delta<\delta_0,$ then there exists a Lipschitz mapStarting from Section sec:prelim, without additional confusion, we will drop the dependence on $\sigma$ in the subscript in $g_\si

Theorems & Definitions (73)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • proof : Proof of Theorem \ref{['asy']} using Theorem \ref{['Tsigma(t)']}
  • Theorem 1.5
  • proof
  • Theorem 1.6
  • proof
  • Remark 2.1
  • ...and 63 more