Wave breaking for the nonlinear variational wave equation

TL;DR

The paper analyzes the nonlinear variational wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$ under the conservative-solution framework built with energy measures, focusing on when wave-breaking occurs. By tracking the Riemann-invariant-like quantities $R=u_t+c(u)u_x$ and $S=u_t-c(u)u_x$ along backward and forward characteristics, it derives explicit criteria that predict imminent or recent wave-breaking, expressed via initial data and energy bounds in the Lagrangian formulation, including time-interval bounds $[t_l,t_u]$. It demonstrates that not all traveling-wave solutions are conservative and that energy concentrations can, in some regimes (notably when $c'(u)=0$), behave like linear waves, with energy moving along characteristic directions without immediate dispersion. These results leverage a generalized method of characteristics and the coupled Eulerian–Lagrangian description with Radon measures to quantify the evolution of energy concentration, providing sharp conditions for blow-up and illustrating the nuanced behavior of conservative solutions beyond simpler unidirectional models. The findings highlight how energy can remain localized for small times and how the two-energy measures interact to enforce conservation while allowing singularities to form along characteristics.

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.

Figures (3)

• Figure 1: The "hut" traveling wave profile $w(\zeta)$.
• Figure 2: The set $M$ in Eulerian coordinates and the corresponding box $\Omega$ in Lagrangian coordinates. The thick curve in each plot shows the set on which the initial data is given. The dotted lines in the first picture show the position of the discrete part of $\mu(t)$ and $\nu(t)$ inside $M$, while the two stripes bounded by the dotted lines indicate the set where $x_\eta$ (vertically) and $x_\xi$ (horizontally) equal zero.
• Figure 3: The set in Eulerian coordinates, which is bounded by the forward characteristic through $(0, -\beta)$ and the backward characteristic through $(0, \alpha)$ for $t\geq 0$ and the corresponding area in Lagrangian coordinates. The thick curve in each plot shows the set on which the initial data is given. The grey area in the picture in Eulerian coordinates indicates the region, where the measure $\mu$ is purely absolutely continuous while the dotted lines show the position of the discrete part of $\mu(t)$ and $\nu(t)$, respectively. In the second plot the shaded area represents the area in Lagrangian coordinates, which contains all the points in the grey area in Eulerian coordinates as a proper subset.

Theorems & Definitions (13)

• Definition 2.1: Eulerian coordinates
• Lemma 2.2
• proof
• Theorem 3.1: Wave breaking along backward characteristics - Part 1
• proof
• Theorem 3.2: Wave breaking along backward characteristics - Part 2
• Theorem 3.3: Wave breaking along forward characteristics - Part 1
• Theorem 3.4: Wave breaking along forward characteristics - Part 2
• Remark 4.1
• Remark 4.2