Caustics of nonlinear waves
Geometrical optics expansions for solutions of first order hyperbolic conservation laws are derived. The emphasis lies on the derivation of expansions at caustics of the solution. It is shown that for moderately large amplitudes in the caustic region the linear theory for caustics suffices to approximate the solution accurately, whereas away from the caustic for such amplitudes a nonlinear geometric optic expansion is needed. This is due to the fact that the caustic region is small, which prevents disturbances generated by nonlinear terms to accumulate. Moreover, for somewhat stronger waves, for which linear theory does not suffice at caustics, a weakly nonlinear caustic theory is developed. The expansions in this paper are derived and presented carefully. Estimates for the difference between the exact solution and the approximate solution are not given.
scientific article; zbMATH DE number 4052148
scientific article