On the resonant reflection of weak, nonlinear sound waves off an entropy wave

摘要

We analyze the resonant reflection of very weak, nonlinear sound waves off a weak sawtooth entropy wave for spatially periodic solutions of the one-dimensional, nonisentropic gas dynamics equations. The case of an entropy wave with a sawtooth profile is of interest because the oscillations of the reflected sound waves are nondispersive with frequency independent of their wavenumber, leading to an unusual type of nonlinear dynamics. On an appropriate long time scale, we show that a complex amplitude function for the spatial profile of the sound waves satisfies a degenerate quasilinear Schrodinger equation. We present some numerical solutions of this equation that illustrate the generation of small spatial scales by a resonant four-wave cascade and front propagation in compactly supported solutions.