The existence and decay of solitary waves for the Fornberg-Whitham equation

TL;DR

This paper investigates solitary wave solutions of the Fornberg-Whitham equation, establishing their existence via minimization, proving their orbital stability, and demonstrating exponential decay for wave speeds exceeding 1.

Contribution

It introduces a novel approach using minimization and concentration-compactness to prove the existence and stability of solitary waves for the Fornberg-Whitham equation.

Findings

Existence of solitary wave solutions via minimization

Orbital stability of the solutions

Exponential decay when wave speed > 1

Abstract

In this paper, we consider the Fornberg-Whitham equation and a family of solitary wave solutions is found by using minimization principle, where a related penalization function and the concentration-compactness lemma play a key role in our proof. Besides, we also prove that the family of solitary solutions is orbital stable and decay exponentially when speed wave c is bigger than 1.