Wave breaking for the nonlinear variational wave equation

TL;DR

This paper investigates wave breaking phenomena in the nonlinear variational wave equation, establishing criteria for wave breaking, analyzing the nature of conservative solutions, and exploring their local behavior and energy concentration properties.

Contribution

It introduces new criteria for wave breaking, distinguishes conservative solutions from traveling waves, and examines local linear behavior and energy concentration in solutions.

Findings

Wave breaking criteria are established.

Not all traveling waves are conservative solutions.

Conservative solutions can behave like linear wave solutions locally.

Abstract

Following conservative solutions of the nonlinear variational wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$ along forward and backward characteristics, we identify criteria, which guarantee that wave breaking either occurs in the nearby future or occurred recently. Thereafter, we apply the established criteria to show that not every traveling wave solution is a conservative solution. Furthermore, we show that conservative solutions can locally behave like solutions to the linear wave equation and hence energy that concentrates on sets of measure zero might remain concentrated instead of spreading out immediately.