Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data |
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Authors: | S Yu Dobrokhotov V E Nazaikinskii B Tirozzi |
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Institution: | 1. A. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences Moscow Institute of Physics and Technology, Moscow, Russia 2. Sapienza Università di Roma, Roma, Italy
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Abstract: | In a two-dimensional domain Ω ? R 2, we consider the wave equation with variable velocity c(x 1, x 2) degenerating on the boundary Γ = ?Ω as the square root of the distance to the boundary, and construct an asymptotic solution of the Cauchy problem with localized initial data. This problem is related to the so-called “run-up problem” in tsunami wave theory. One main idea (also used by the authors in earlier papers in the one-dimensional case and the two-dimensional case with c 2(x 1, x 2) = x 1) is that the (singular) curve Γ is a caustic of special type. We use this idea to introduce a generalization of the Maslov canonical operator covering the problem with degeneration and obtain efficient formulas for the asymptotic solutions. |
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