ABSTRACT
The topic of nonlinear geometric optics concerns the formal analysis of small amplitude, high frequency oscillatory waves in nonlinear problems, and the rigorous justification of the asymptotic properties of oscillations. Most of the literature deals with the case of smooth background states. For the problem of two weak shocks perturbed by small amplitude, high frequency oscillatory waves, we obtain that the leading profiles of oscillatory shocks are solutions to an integro-differential system with free boundaries, the leading terms of shock fronts do not oscillate, and oscillations only appear in the leading terms of shock speeds. The chapter formulates the problems and state the main results, and sketches the proof of the main results briefly, which gives the nonlinear geometric optics.
