Nonlinear Asymptotics for Hyperbolic Internal Waves of Small Width

In the limit ε → 0, the solutions converge to piecewise smooth functions which are discontinuous accross a characteristic surface. Such solutions have source terms which are also piecewise smooth. Discontinuous sources are idealizations of smooth sources with a thin (∼ ε) transition layer. The fundamental problem addressed here is to describe the dynamics of solutions with sources with such transition layers. The solutions have internal transition layers of size ∼ ε. The limiting solution has a conormal singularity along the characteristic hypersurface. Our analysis employs conormal and ε∂ estimates. The technical difficulty is that the obvious ansatz motivated by the cases of wave trains and short pulses yields overdetermined equations for correctors to the leading approximation. This is so even in the linear case. If one does not choose specially adapted coordinates, the transport equations differ from those describing the propagation of singularities and oscillations.