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Orbital stability of solitary waves for the Schr odinger-Boussinesq system

Yilong Ma, Yamin Xiao

TL;DR

This work investigates the orbital stability of solitary waves for a Schrödinger–Boussinesq system in one spatial dimension, focusing on the SB system with $p=2$ and positive parameters $\\alpha,\\beta,\\gamma$. The authors construct explicit traveling-wave solitary solutions with profiles that take a sech form for the optical field and related profiles for the other fields, and they establish their orbital stability using the Grillakis–Shatah–Strauss framework by conducting a detailed spectral analysis of the linearized operator and Hessian of the associated constrained energy functional. Under a concrete parameter regime, they show the Hessian has one positive and one negative eigenvalue and that the linearized operator has exactly one negative direction, ensuring stability of the solitary waves. Their results extend prior stability findings for related systems and contribute to the mathematical understanding of nonlinear wave interactions in optics modeled by coupled Schrödinger–Boussinesq equations.

Abstract

This paper studies the orbital stability of solitary waves for the following Schrödinger-Boussinesq system \begin{equation*} \begin{cases} { \begin{array}{ll} i\varepsilon_t+\varepsilon_{xx}=n\varepsilon+γ|\varepsilon|^2\varepsilon, \\ n_{tt}-n_{xx}+ αn_{xxxx}-β(n^2)_{xx}=|\varepsilon|^2_{xx}, \end{array} } (t,x)\in \mathbb{R}^2. \end{cases} \end{equation*} By applying the abstract results and detailed spectral analysis, we obtain the orbital stability of solitary waves. The result can be regarded as an extension of the results of \cite{ F-P,H,W}.

Orbital stability of solitary waves for the Schr odinger-Boussinesq system

TL;DR

This work investigates the orbital stability of solitary waves for a Schrödinger–Boussinesq system in one spatial dimension, focusing on the SB system with and positive parameters . The authors construct explicit traveling-wave solitary solutions with profiles that take a sech form for the optical field and related profiles for the other fields, and they establish their orbital stability using the Grillakis–Shatah–Strauss framework by conducting a detailed spectral analysis of the linearized operator and Hessian of the associated constrained energy functional. Under a concrete parameter regime, they show the Hessian has one positive and one negative eigenvalue and that the linearized operator has exactly one negative direction, ensuring stability of the solitary waves. Their results extend prior stability findings for related systems and contribute to the mathematical understanding of nonlinear wave interactions in optics modeled by coupled Schrödinger–Boussinesq equations.

Abstract

This paper studies the orbital stability of solitary waves for the following Schrödinger-Boussinesq system \begin{equation*} \begin{cases} { \begin{array}{ll} i\varepsilon_t+\varepsilon_{xx}=n\varepsilon+γ|\varepsilon|^2\varepsilon, \\ n_{tt}-n_{xx}+ αn_{xxxx}-β(n^2)_{xx}=|\varepsilon|^2_{xx}, \end{array} } (t,x)\in \mathbb{R}^2. \end{cases} \end{equation*} By applying the abstract results and detailed spectral analysis, we obtain the orbital stability of solitary waves. The result can be regarded as an extension of the results of \cite{ F-P,H,W}.
Paper Structure (3 sections, 7 theorems, 67 equations)

This paper contains 3 sections, 7 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. The existence of solitary waves
  3. Orbital stability of solitary waves

Key Result

Theorem 2.1

For any real constants $\omega,v$ satisfying there exist solitary waves of (1.1) in the form of (2.1), with $\hat{\varepsilon}_{\omega,v}$ and $n_{\omega,v}$ satisfying (2.7).

Theorems & Definitions (11)

  • Definition 1.1: G-S-S1
  • Theorem 2.1
  • Proposition 3.1
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • proof
  • Theorem 3.7
  • Theorem 3.8
  • ...and 1 more