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On multiplicative exponential asymptotics of solutions to wave equations. (English. Russian original) Zbl 0815.34047
Differ. Equations 29, No. 11, 1722-1732 (1993); translation from Differ. Uravn. 29, No. 11, 1984-1995 (1993).
The paper is devoted to finding the solutions of a wave equation in the form of their so-called asymptotic expansions. The latters are functional series in negative powers of a large parameter entering in the equation. Conventionally, this approach is called to be the WKB-method generalized by V. Maslov. The new authors’ results in this field consist in constructing the WKB-asymptotics for solutions of wave equations in the case where there exist both oscillations and exponential decays of the solution in some domains of physical space. This analysis effectively uses the theory of resurgent functions developed in the last years. It allows the authors to construct multiplicative power asymptotics for the solution of the Cauchy problem for the wave equation. The main theorem proves the existence of the WKB-asymptotics in question. References, 16 in number, very well reflect the considered topic.
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
35E15 Initial value problems for PDEs and systems of PDEs with constant coefficients
35L05 Wave equation