Nonlinear geometric optics for multidimensional shocks II: Oscillatory exact shocks

Abstract

We construct geometric optics expansions of high order for oscillatory multidimensional shocks, and then show that the expansions are close to exact shock solutions for small wavelengths. Expansions are constructed both for ψ ε, the oscillatory function defining the shock surface S ε and for u ε ± the solutions on each side of S ε. The profile equations yield detailed information on the evolution of ( u ε ±, ψ ε), showing for example how new interior oscillations are produced by nonlinear interaction between u ε ± and S ε. In this second Note we reduce the shock problem to a forward boundary problem and give a summary of the proofs.