The six articles in this EMS volume provide an overview of a number of contemporary techniques in the study of the asymptotic behavior of partial differential equations. These techniques include the Maslov canonical operator, semiclassical asymptotics of solutions and eigenfunctions, behavior of solutions near singular points of different kinds, matching of asymptotic expansions close to a boundary layer, and processes in inhomogeneous media. Asymptotic expansions are one of the most important areas in the theory of partial differential equations. Readers should find the wide variety of approaches of interest.

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 six articles in this EMS volume provide an overview of a number of contemporary techniques in the study of the asymptotic behavior of partial differential equations. These techniques include the Maslov canonical operator, semiclassical asymptotics of solutions and eigenfunctions, behavior of solutions near singular points of different kinds, matching of asymptotic expansions close to a boundary layer, and processes in inhomogeneous media. Asymptotic expansions are one of the most important areas in the theory of partial differential equations. Readers should find the wide variety of approaches of interest.

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.