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Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations

Liming Ling, Huajie Su

TL;DR

This work proves nonlinear stability for vector solitons in coupled NLS and CmKdV systems by marrying integrability with Lyapunov methods. It builds a Lyapunov functional from an infinite hierarchy of conserved quantities and analyzes the second variation via squared eigenfunctions tied to the Lax pair and zero-curvature structure, enabling precise counts of negative directions and kernel dimensions. The results establish that CNLS N-solitons are orbitally stable in $H^{N}$ and CmKdV $(N_{1},N_{2})$-solitons are stable in $H^{2N_{1}+N_{2}}$, with a detailed spectral-theory backbone that also covers single solitons and breathers. A key novelty is deriving linearized dynamics for a broad class of mixed flows from stationary zero-curvature equations in Lie algebras, which both generalizes the stability framework and underpins the constructive approach via Darboux transformations and squared eigenfunctions.

Abstract

We prove that the $N$-solitons, including breathers and multi-hump solitons, of the coupled nonlinear Schrödinger (CNLS) equations are nonlinearly stable in the Sobolev space $H^{N}$. Moreover, $(N_{1},N_{2})$-solitons of the coupled modified Korteweg--de Vries (CmKdV) equations are shown to be nonlinearly stable in the Sobolev space $H^{2N_{1}+N_{2}}$. The number of negative eigenvalues of the second variation of the Lyapunov functional is $N$ for $N$-solitons of the CNLS equations, and $N_{1}+\lfloor (N_{2}+1)/2 \rfloor$ for $(N_{1},N_{2})$-solitons of the CmKdV equations, which is obtained by exploiting integrable properties. The stability of solitons for the classical NLS and mKdV equations also follows from the same method. In addition, we show that solutions to the linearized spectral problem of the mixed flow equation can be constructed from solutions of the stationary zero curvature equations in a large class of Lie algebras.

Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations

TL;DR

This work proves nonlinear stability for vector solitons in coupled NLS and CmKdV systems by marrying integrability with Lyapunov methods. It builds a Lyapunov functional from an infinite hierarchy of conserved quantities and analyzes the second variation via squared eigenfunctions tied to the Lax pair and zero-curvature structure, enabling precise counts of negative directions and kernel dimensions. The results establish that CNLS N-solitons are orbitally stable in and CmKdV -solitons are stable in , with a detailed spectral-theory backbone that also covers single solitons and breathers. A key novelty is deriving linearized dynamics for a broad class of mixed flows from stationary zero-curvature equations in Lie algebras, which both generalizes the stability framework and underpins the constructive approach via Darboux transformations and squared eigenfunctions.

Abstract

We prove that the -solitons, including breathers and multi-hump solitons, of the coupled nonlinear Schrödinger (CNLS) equations are nonlinearly stable in the Sobolev space . Moreover, -solitons of the coupled modified Korteweg--de Vries (CmKdV) equations are shown to be nonlinearly stable in the Sobolev space . The number of negative eigenvalues of the second variation of the Lyapunov functional is for -solitons of the CNLS equations, and for -solitons of the CmKdV equations, which is obtained by exploiting integrable properties. The stability of solitons for the classical NLS and mKdV equations also follows from the same method. In addition, we show that solutions to the linearized spectral problem of the mixed flow equation can be constructed from solutions of the stationary zero curvature equations in a large class of Lie algebras.
Paper Structure (23 sections, 24 theorems, 478 equations)

This paper contains 23 sections, 24 theorems, 478 equations.

Table of Contents

  1. Introduction
  2. Review of stability results for integrable equations
  3. Main results
  4. Main steps of the proof
  5. Outline
  6. Linearized operator and $\mathbf{L}$ matrix
  7. The variation of $\mathbf{L}$ matrix
  8. Steady-state solutions to flow equations
  9. Examples for $\mathrm{gl}(3,\mathbb{C})$
  10. $N$-soliton solutions for CNLS equations
  11. Darboux transformation for CNLS equations
  12. The variational characterization for $N$-soliton solutions
  13. The squared eigenfunction matrices for CNLS equations
  14. Spectral analysis and nonlinear stability of CNLS solitons
  15. The integral between squared eigenfunctions and adjoint squared eigenfunctions
  16. ...and 8 more sections

Key Result

Theorem 1

The $N$-soliton solutions CNLS-Nsoliton for CNLS equations are nonlinearly stable in the Sobolev space $H^{N}$, and the $(N_{1},N_{2})$-soliton solutions CmKdV-Nsoliton for CmKdV equations are nonlinearly stable in $H^{2N_{1}+N_{2}}$. Denote $\tilde{N}=N$ and $\mathbf{q}_{sol}=\mathbf{q}^{[N]}$ for for some soliton solution with spectral parameters $\mathbf{\Lambda}$ and scattering parameters $\m

Theorems & Definitions (54)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Corollary 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • Lemma 1
  • proof
  • Remark 3
  • ...and 44 more