Travelling Waves in Systems of Hyperbolic Balance Laws

Summary

We discuss several results on the existence of continuous travelling wave solutions in systems of conservation laws with nonlinear source terms.

In the first part we show how waves with oscillatory tails can emerge from the combination of a strictly hyperbolic system of conservation laws and a source term possessing a stable line of equilibria. Two-dimensional manifolds of equilibria can lead to Takens-Bogdanov bifurcations without parameters. In this case there exist several families of small heteroclinic waves connecting different parts of the equilibrium manifold.

The second part is concerned with large heteroclinic waves for which the wave speed is characteristic at some point of the profile. This situation has been observed numerically for shock profiles in extended thermodynamics. We discuss the desingularization of the resulting quasilinear implicit differential-algebraic equations and possible bifurcations. The results are illustrated using the p-system with source and the 14-moment system of extended thermodynamics.

Our viewpoint is from dynamical systems and bifurcation theory. Local normal forms at singularities are used and the dynamics is described with the help of blowup transformations and invariant manifolds.