Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation

2023/12/18 by Yvan Martel, Martel, Yvan · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2312.11016

openalex publication_date 2023/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the Schrödinger equation with a cubic-quintic, focusing-focusing nonlinearity in one space dimension, this article proves the local asymptotic completeness of the family of small standing solitary waves under even perturbations in the energy space. For this model, perturbative of the integrable cubic Schrödinger equation for small solutions, the linearized equation around a small solitary wave has an internal mode, whose contribution to the dynamics is handled by the Fermi golden rule.

Citations

• Dynamics of solitons in nearly integrable systems
• Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities

Cited by

• Non-existence of internal mode for small solitary waves of the 1D Zakharov system
• Monotonicity Conjectures and Sharp Stability for Solitons of the Cubic-Quintic NLS on R³

Related

• Asymptotic stability of solitons for near-cubic NLS equation with an internal mode
• Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schrödinger equation in the absence of internal modes
• Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation under even perturbations
• Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation
• Long-time dynamics of small solutions to 1d cubic nonlinear Schrödinger equations with a trapping potential