Asymptotics of solutions to systems of (pseudo)differential equations with localized right-hand sides

Abstract

Abstract The article describes a method for constructing semiclassical asymptotic solutions of multidimensional (pseudo)differential partial differential equations with localized right-hand sides. The method is based on the asymptotic reduction of vector problems (under certain additional conditions) to scalar (pseudo)differential problems, for which efficient asymptotic formulas have recently been constructed. The method is based on the Feynman–Maslov operator calculus. Even if the original problems are stated for differential equations, the operators in reduced problems, as a rule, turn out to be pseudodifferential. Further, nontrivial problems of calculating operator symbols in the reduced scalar problems arise in the reduction process. We discuss approaches to the transformation of these symbols so as to enable their effective use when solving real problems. The method is illustrated by several physical examples.