Instability of nonlinear dispersive solitary waves

TL;DR

This paper investigates the linear instability of solitary waves in various dispersive long wave models, including nonlocal operators, providing new criteria for exponential growth solutions without relying on Evans function techniques.

Contribution

It introduces novel instability criteria for models with nonlocal dispersive terms, extending analysis methods to cases where traditional spectral techniques are ineffective.

Findings

Derived instability criteria for nonlocal dispersive wave models

Reduced linearized problems to analyze nonlocal operators

Extended techniques to periodic waves and water wave problems

Abstract

We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to properties of solitary waves. A continuation argument with a moving kernel formula are used to find the instability criteria. Recently, these techniques have also been extended to study instability of periodic waves and to the full water wave problem.