Orbital Stability of Smooth Traveling Solitary Waves to the Fornberg-Whitham Equation

TL;DR

This paper investigates the orbital stability of smooth solitary wave solutions to the nonlocal Fornberg-Whitham equation, revealing stability properties using a variational approach.

Contribution

It demonstrates the orbital stability of certain smooth solitary waves in the FW equation, a non-integrable shallow water model, using variational methods.

Findings

Some smooth solitary waves are orbitally stable.

The stability is established through a variational framework.

The FW equation supports stable peaked traveling waves.

Abstract

The Fornberg-Whitham (FW) equation was introduced by Fornberg and Whitham [Fornberg and Whitham, Phil. Trans. R. Soc. Lond. A (1978)] as a nonlocal model for unidirectional shallow water waves capable of capturing wave steepening and breaking. Despite its similarities with integrable shallow-water equations, the FW equation is not completely integrable. Nevertheless, the FW equation is part of the family of peakon-type models as it supports peaked traveling wave solutions. In this paper, we consider smooth solitary wave solutions to the FW equation. We use a variational approach to show that some are orbitally stable.