Wave breaking for nonlinear wave equations

Nonlinear wave equations are a central theme of research both in mathematics and physics One of the fascinating phenomena in that context, which can be described mathematically, is wave breaking. Hence the question arises how solutions can be continued beyond wave breaking as there are always two possibilities. On the one hand conservative solutions, where the energy is preserved and on the other hand dissipative solutions, where the energy vanishes after wave breaking occurs. One famous example of a wave equation enjoying wave breaking is the Camassa-Holm (CH) equation, which serves as a modell for shallow water waves and is rich of mathematical structure. A special group of solutions are the so-called multipeakons, which serve as an illustrating example of how solutions can be continued beyond wave breaking. A generalisation of the CH equation is the so-called 2-component Camassa-Holm (2CH) system. We are interested in the question for which parameters wave breaking occurs and how solutions can be described. Another example of an equation, which enjoys wave breaking is given by the Hunter-Saxton (HS) equation, which serves as a modell for liquid crystals. Here some conservative and dissipative solutions can be computed explicitely. Also this equation can be generalised to the so-called generalised Hunter-Saxton (gHS) system, for which the question of when wave breaking can occur has already been considered and therefore we will try to describe conservative and dissipative solutions in the cases where wave breaking occurs. The third and last example of an equation enjoying wave breaking, we want to consider, is given by the nonlinear variational wave (NVW) equation, for which in contrast to the CH and HS equation no explicit solution is known. Therefore only numerical methods can be used to illustrate that wave breaking occurs. So far global conservative solutions have been described and hence we will try to construct a Lipschitz metric for the NVW equation. Another related question is how one can describe dissipative solutions.