Wave breaking for the nonlinear variational wave equation

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.