# Stability properties of solitary waves for fractional KdV and BBM equations

## TL;DR

This paper investigates the stability and instability of solitary wave solutions in fractional KdV and BBM equations with low dispersion, providing new criteria and conditions for their linear and nonlinear stability or instability.

## Contribution

It introduces new stability criteria for solitary waves in fractional dispersive models using analytical and variational methods, especially under low dispersion conditions.

## Key findings

- Established conditions for exponential growth solutions indicating linear instability.
- Proved nonlinear stability and linear instability for specific parameter regimes.
- Analyzed stability of blow-up solutions for the critical fractional KdV equation.

## Abstract

This paper sheds new light on the stability properties of solitary wave solutions associated with models of Korteweg-de Vries and Benjamin\&Bona\&Mahoney type, when the dispersion is very lower. Via an approach of compactness, analyticity and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so a criterium of linear instability of solitary waves is obtained for both models. Moreover, the nonlinear stability and linear instability of the ground states solutions for both models is obtained for some specific regimen of parameters. Via a Lyapunov strategy and a variational analysis we obtain the stability of the blow-up of solitary waves for the critical fractional KdV equation. The arguments presented in this investigation has prospects for the study of the instability of traveling waves solutions of other nonlinear evolution equations.