### Abstract

We suggest a new asymptotic representation for the solutions to the multidimensional wave equations with variable velocity with localized initial data. This representation is the generalization of the Maslov canonical operator based also on a simple relationship between fast decaying and fast oscillating solutions, and on boundary layer ideas. It establishes the connection between initial localized perturbations and wave profiles near the wave fronts including the neighborhood of backtracking (focal or turning) and self intersection points. We show that wave profiles are related with a form of initial sources and also with the Lagrangian manifolds organized by the rays and wavefronts. In particular we discuss the influence of such topological characteristics like the Maslov and Morse indices to metamorphosis of the profiles after crossing the focal points. We apply these formulas to the problem of a propagation of tsunami waves in the frame of so-called ``piston model''. Finally we suggest a fast asymptotically-numerical algorithm for simulation of tsunami wave over nonuniform bottom. Different scenarios of the distribution of the waves are considered, the wave profiles of the front are obtained in connection with the different shapes of the source and with the diverse rays generating the fronts. It is possible to use suggested algorithm to predict in real time the zones of the beaches where the amplitude of the tsunami wave has dangerous high values. In this connection we also discuss the following questions: the problems of the regularization of the wave field near focal points; ill-possed problems appearing in the geometry of the wavefronts; the inverse problem connected with the possibility of reconstruction of the source via the measurement of the tsunami wave profile on the shelf etc. This work was done together with S.Sekerzh-Zenkovich, B.Tirozzi, B.Volkov and was partially supported by RFBR grant N 05-01-00968 and Agreement Between University "La Sapienza", Rome and Institute for Problems in Mechanics RAS, Moscow.

Bibliography

[1] S.Yu. Dobrokhotov, S.Ya Sekerzh-Zenkovich, B. Tirozzi, T.Ya. Tudorovskiy, The description of tsunami waves propagation based on the Maslov canonical operator, Doklady Mathematics, 2006, v.74, N 1, pp. 592-596

[2] S.Yu. Dobrokhotov, S.Ya Sekerzh-Zenkovich, B. Tirozzi, T.Ya. Tudorovskiy, Asymptotic theory of tsunami waves: geometrical aspects and the generalized Maslov representation, Publications of Kyoto Research Mathematical Institute, Vol.4, page 118-153, 2006, ISSN 1880-2818. [3] S.Dobrokhotov, S.Sekerzh-Zenkovich, B.Tirozzi, B.Volkov Explicit asymptotics for tsunami waves in framework of the piston model, Russ. Journ. Earth Sciences, 2006, v.8, ES403, pp.1-12

[4] S.Dobrokhotov, S.Sinitsyn, B.Tirozzi, Asymptotics of Localized Solutions of the One-Dimensional Wave Equation with Variable Velocity. I. The Cauchy Problem, Russ. Jour.Math.Phys., v.14, N1, 2007, pp.28-56