Discontinuous traveling waves as weak solutions to the Fornberg-Whitham equation

TL;DR

This paper investigates discontinuous traveling wave solutions for the Fornberg-Whitham equation, demonstrating their existence through dynamical system analysis and patching disconnected orbits, expanding understanding of weak solutions with discontinuities.

Contribution

It introduces a method to establish the existence of discontinuous weak traveling wave solutions for the Fornberg-Whitham equation using dynamical systems techniques.

Findings

Discontinuous weak solutions exist for the Fornberg-Whitham equation.

Analysis of a planar dynamical system reveals conditions for wave discontinuities.

Solutions are constructed by patching disconnected orbits in the phase space.

Abstract

We analyze the weak solution concept for the Fornberg-Whitham equation in case of traveling waves with a piecewise smooth profile function. The existence of discontinuous weak traveling wave solutions is shown by means of analysis of a corresponding planar dynamical system and appropriate patching of disconnected orbits.