Continuous and discontinuous piecewise linear solutions of the linearly forced inviscid Burgers equation

TL;DR

This paper investigates piecewise linear solutions to the linearly forced inviscid Burgers equation, analyzing their dynamics and collision behavior, including both continuous and shock wave solutions.

Contribution

It introduces a novel analysis of soliton-like solutions and their interactions in the forced Burgers equation, including conditions for collisions and shock formation.

Findings

Triple collisions are impossible for continuous solutions.

Triple collisions can occur in shock wave solutions.

Explicit ODEs govern soliton dynamics.

Abstract

We study a class of piecewise linear solutions to the inviscid Burgers equation driven by a linear forcing term. Inspired by the analogy with peakons, we think of these solutions as being made up of solitons situated at the breakpoints. We derive and solve ODEs governing the soliton dynamics, first for continuous solutions, and then for more general shock wave solutions with discontinuities. We show that triple collisions of solitons cannot take place for continuous solutions, but give an example of a triple collision in the presence of a shock.