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

2025/10/14 by Liming Ling, Ling, Liming, Huajie Su +1 · 1 voice · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Numerical methods for differential equations #math-ph #math.AP #nlin.SI

paper · pdf · doi:10.48550/arxiv.2510.12129

openalex publication_date 2025/10/14 · arxiv published 2025/10/14 · arxiv updated 2025/10/14 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

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, (N₁,N₂)-solitons of the coupled modified Korteweg--de Vries (CmKdV) equations are shown to be nonlinearly stable in the Sobolev space H^2N₁+N₂. The number of negative eigenvalues of the second variation of the Lyapunov functional is N for N-solitons of the CNLS equations, and N₁+\lfloor (N₂+1)/2 \rfloor for (N₁,N₂)-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.

Cited by

• Numerical inverse scattering transform for the coupled modified Korteweg-de Vries equation

Discussions

• Liming Ling, Huajie Su: Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations https://arxiv.org/abs/2510.12129 https://arxiv.org/pdf/2510.12129 https://arxiv.org/html/ [bsky, 0 points, 0 comments]

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