Asymptotic stability manifolds for solitons in the generalized Good Boussinesq equation

TL;DR

This paper investigates the long-term stability of standing solitary waves in the generalized Good-Boussinesq equation, constructing a manifold of initial data leading to asymptotic stability in the energy space.

Contribution

It introduces a detailed analysis of the asymptotic stability of standing waves, constructing a manifold of initial conditions ensuring stability, extending previous stability results.

Findings

Constructed a manifold of initial data for stability

Proved asymptotic stability of standing waves in energy space

Extended understanding of wave stability near zero speed

Abstract

We consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space $H^1\times L^2$. This model has solitary waves with speeds $-1<c<1$. When $|c|$ approaches 1, Bona and Sachs showed orbital stability of such waves. It is well-known from a work of Liu that for small speeds solitary waves are unstable. In this paper we consider in more detail the long time behavior of zero speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel and Mu\~noz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space.