Partial Differential Equations V

In this paper we shall discuss the construction of formal short-wave asymp totic solutions of problems of mathematical physics. The topic is very broad. It can somewhat conveniently be divided into three parts: 1. Finding the short-wave asymptotics of a rather narrow class of problems, which admit a solution in an explicit form, via formulas that represent this solution. 2. Finding formal asymptotic solutions of equations that describe wave processes by basing them on some ansatz or other. We explain what 2 means. Giving an ansatz is knowing how to give a formula for the desired asymptotic solution in the form of a series or some expression containing a series, where the analytic nature of the terms of these series is indicated up to functions and coefficients that are undetermined at the first stage of consideration. The second stage is to determine these functions and coefficients using a direct substitution of the ansatz in the equation, the boundary conditions and the initial conditions. Sometimes it is necessary to use different ansiitze in different domains, and in the overlapping parts of these domains the formal asymptotic solutions must be asymptotically equivalent (the method of matched asymptotic expansions). The basis for success in the search for formal asymptotic solutions is a suitable choice of ansiitze. The study of the asymptotics of explicit solutions of special model problems allows us to "surmise" what the correct ansiitze are for the general solution.

The authors survey an important topic in PDE which is highly relevant for applications in physics

I. Equations with Rapidly Oscillating Solutions.- II. Asymptotic Expansion as t?3 ? of the Solutions of Exterior Boundary Value Problems for Hyperbolic Equations and Quasiclassical Approximations.- III. The Higher-Dimensional WKB Method or Ray Method. Its Analogues and Generalizations.- IV. Semiclassical Asymptotics of Eigenfunctions.- V. The Boundary Layer.- VI. The Averaging Method for Partial Differential Equations (Homogenization) and Its Applications.- Author Index.