Asymptotic stability of fast solitary waves to the Benjamin Equation

TL;DR

This paper proves the asymptotic stability of high-speed solitary waves in the Benjamin equation by leveraging the KdV limit and a Liouville property, addressing challenges from non-local operators and energy non-positivity.

Contribution

It establishes the asymptotic stability of solitary waves for the Benjamin equation, extending stability analysis to non-local and physically relevant cases.

Findings

Proves asymptotic stability of high-speed solitary waves.

Uses KdV limit and Liouville property for analysis.

Addresses challenges from Hilbert transform and energy issues.

Abstract

We prove the asymptotic stability of the high speed solitary waves to the Benjamin equation. This is done by establishing a Liouville property for the nonlinear evolution of the Benjamin equation around these solitary waves. To do this, inspired by Kenig-Martel-Robbiano 2011, we make use of the KdV limit of the Benjamin equation together with known rigidity property of the KdV flow. The main difficulties are linked to the presence of the Hilbert transform, that is a non-local operator, as well as the non-positivity of the quadratic part of the energy in the case $ \gamma<0$ which is the physical case.