On asymptotic stability of stable Good Boussinesq solitary waves

TL;DR

This paper proves the asymptotic stability of stable solitary waves in the generalized Good-Boussinesq model for a broad class of initial data and wave speeds, using novel virial estimates and linear operator analysis.

Contribution

It introduces new virial estimates and linear operator analysis to establish asymptotic stability of stable solitary waves in the GB system for general initial data.

Findings

Asymptotic stability holds for all initial data in the energy space for certain wave speeds.

New virial estimates with mixed variables are developed for the GB system.

Analysis of the linear matrix operator under mixed orthogonality conditions is provided.

Abstract

We consider the generalized Good-Boussinesq (GB) model in one dimension, with subcritical power nonlinearity $1<p<5$ and data in the energy space $H^1\times L^2$. This model has solitary waves with speeds $c\in (-1,1)$. If $c^2>\frac{p-1}{4}$, Bona and Sachs showed the orbital stability of such waves. Previously, one of us proved that unstable GB standing waves can be perturbed with particular odd-even data in a suitable submanifold of the energy space, leading to the asymptotic stability property if $p\ge 2$. In this paper we prove that stable GB solitary waves are asymptotically stable in the case of general initial data placed in the energy space for any $p\ge 2$ and speeds $|c|>c_+(p)\geq \sqrt{\frac{p-1}{4}}$. The proof involves the introduction of a new set of virial estimates specifically adapted to the GB system in a moving setting. In particular, a new virial estimate with mixed variables is considered to treat arbitrary scaling and shift modulations. Another new ingredient is the understanding the corresponding linear matrix operator under mixed orthogonality conditions, a feature absent in our previous works.