Solitons, peakons and periodic cusp wave solutions for the Fornberg-Whitham equation

TL;DR

This paper uses bifurcation methods to find exact travelling wave solutions for the Fornberg-Whitham equation, including solitons, peakons, and periodic cusp waves, revealing their interrelations.

Contribution

It provides explicit and implicit solutions for various wave types of the Fornberg-Whitham equation, advancing understanding of their structures and limits.

Findings

Explicit expressions for peakons and periodic cusp wave solutions.

Solitons and periodic cusp wave solutions tend to peakons in certain limits.

Bifurcation method effectively derives exact solutions for the equation.

Abstract

In this paper, we employ the bifurcation method of dynamical systems to investigate the exact travelling wave solutions for the Fornberg-Whitham equation. The implicit expression for solitons is given. The explicit expressions for peakons and periodic cusp wave solutions are also obtained. Further, we show that the limits of soliton solutions and periodic cusp wave solutions are peakons.