## Maslov’s complex germ in the Cauchy problem for a wave equation with a jumping velocity.(English)Zbl 07500878

Summary: Using Maslov’s complex germ in the Cauchy problem for a wave equation, we consider the asymptotics of the solution of the Cauchy problem in which the velocity depends irregularly on a small parameter.

### MSC:

 35Q99 Partial differential equations of mathematical physics and other areas of application 35P25 Scattering theory for PDEs 35C20 Asymptotic expansions of solutions to PDEs 35L15 Initial value problems for second-order hyperbolic equations 35K15 Initial value problems for second-order parabolic equations
Full Text:

### References:

 [1] Maslov, V. P., Operator Methods (1973), Moscow: Izdat. MGU, Moscow · Zbl 0288.47042 [2] Maslov, V. P., The Complex WKB Method for Nonlinear Equations. I (1994), Basel: Birkhäuser, Basel [3] Belov, V. V.; Dobrokhotov, S. Yu., Semiclassical Maslov Asymptotics with Complex Phases. I. General Approach, Theoret. and Math. Phys., 92, 2, 843-868 (1992) [4] Dobrokhotov, S. Yu.; Shafarevich, A. I., Semiclassical Quantization of Invariant Isotropic Manifolds of Hamiltonian Systems, in: Topological Methods in the Theory of Hamiltonian Systems (eds. A.T. Fomenko, A.B. Bolsinov, A.I. Shafarevich), 41-114 (1998), Moscow: Faktorial, Moscow [5] Allilueva, A. I.; Shafarevich, A. I., Short-Wave Asymptotic Solutions of the Wave Equation with Localized Perturbations of the Velocity, Russ. J. Math. Phys., 27, 2, 145-154 (2020) · Zbl 1441.35150 [6] Allilueva, A. I.; Shafarevich, A. I., Reflection and Refraction of Lagrangian Manifolds Corresponding to Short-Wave Solutions of the Wave Equation with an Abruptly Varying Velocity, Russ. J. Math. Phys., 28, 2, 137-146 (2021) · Zbl 1468.35033 [7] Dobrokhotov, S. Yu., Application of Maslov’s Theory to Two Problems with Operator-Valued Symbol, Russ. Math. Surv., 39, 4, 125 (1984) · Zbl 0566.35038 [8] Buslaev, V. S., Adiabatic Perturbation of a Periodic Potential, Theoret. and Math. Phys., 58, 2, 153-159 (1984) · Zbl 0557.34053 [9] Dobrokhotov, S. Yu.; Shafarevich, A. I., Semiclassical Asymptotic Behavior of the Scattering of Wave Packets by the Rapidly Changing Potential Given by the Equation $$-2\vert \nabla\Phi\vert^2/{\rm ch}^2(\Phi (x)/h)+V_0(x)$$, Sov. Phys., Dokl., 32, 8, 633-635 (1987) · Zbl 0696.35125 [10] Shafarevich, A. I., Quasiclassical Scattering of Wave Packets on a Narrow Band in which the Potential Rapidly Changes, Math. Notes, 45, 1, 72-77 (1989) · Zbl 0689.35080 [11] Maslov, V. P.; Omel’yanov, G. A., Asymptotic Soliton-Form Solutions of Equations with Small Dispersion, Russian Math. Surveys, 36, 3, 73-149 (1981) · Zbl 0494.35080 [12] Calogero, F.; Degasperis, A., Spectral Transform and Solitons (1982), North-Holland: North-Holland Publishing Company, North-Holland · Zbl 0501.35072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.