The Method of Weak Asymptotics

This method for constructively constructing solutions to nonlinear partial differential equations was proposed by the author, V.M. Shelkovich, and G.A. Omelyanov, and is a continuation of some of V.P. Maslov's works (in particular, on three algebras associated with singular solutions of nonlinear equations). It differs from the standard approach to constructing an asymptotic solution in that the smallness of the residual arising during construction is understood not in the classical sense, but in the weak sense—that of generalized functions. This approach allows one to define a "multiplication table" of approximations of generalized functions by considering nonlinearities in the weak sense, and, based on this approach:
- introduce the concept of finitely generated asymptotic algebras of generalized functions in the weak sense;
- define a generalized solution in the form of an integral identity for problems with a small parameter that allows a weak limiting transition to a limiting solution describing the propagation and interaction of nonlinear waves;
- solve some problems of nonlinear wave interaction.
The construction of the definition of a generalized solution is based on the construction of classical asymptotics and models the solvability conditions of the linearized equation for the correction to the principal term of the asymptotic solution. Examples: free boundary problems, solitons, delta shock waves.