ИСТИНА

Let X be a compact manifold with smooth boundary. Consider the wave equation on X in which the squared velocity is a smooth function on X positive in the interior of X and vanishing (with first order) on the entire boundary. We are interested in short-wave asymptotics of solutions of this equation and of other PDE with this type of degeneration. To this end, we represent X as the quotient of a closed manifold M by a semifree circle action. The equations in question can be lifted to M, where the asymptotic solutions can be written by standard methods. The solutions of the original equations are just the fiberwise constant solutions of these new equations. Now the nonstandard phase space corresponding to degenerate PDE of this kind can be defined as the Marsden–Weinstein symplectic reduction of the cotangent bundle of M by the circle action. The surprisingly simple implementation of this approach provides a complete analysis of asymptotic solutions of the original equations and simple efficient formulas for these solutions. The results have applications to the theory of run-up of long waves on a shallow beach (including tsunami waves generated by a localized source, waves trapped by the coast, or seiches).