Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity: I

J. J. Stoker, Water Waves: The Mathematical Theory with Applications (John Wiley and Sons, New York, 1958) (reprinted in 1992). E. N. Pelinovskii, Hydrodynamics of Tsunami Waves (IPF RAS, Nizhnii Novgorod, 1996) [in Russian]. E. Pelinovsky and R. Mazova, “Exact Analytical Solutions of Nonlinear Problems of Tsunami Wave Run-Up on Slopes with Different Profiles,” Natur. Hazards 6, 227–249 (1992). S. Yu. Dobrokhotov, S. O. Sinitsyn, and B. Tirozzi, “Asymptotics of Localized Solutions of the One-Dimensional Wave Equation with Variable Velocity. I: The Cauchy Problem,” Russ. J. Math. Phys. 14(1), 28–56 (2007). S. Yu. Dobrokhotov and B. Tirozzi, “Localized Solutions of the One-Dimensional Nonlinear Shallow Water Equations with Velocity c = √x,” Uspekhi Mat. Nauk 65(1) (391), 185–186 (2010). G. F. Carrier and H. P. Greenspan, “Water Waves of Finite Amplitude on a Sloping Beach,” J. Fluid Mech. 4, 97–109 (1958). S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi, “Asymptotic Solutions of the Two-Dimensional Model Wave Equation with Degenerating Velocity and with Localized Initial Data,” Algebra i Analiz 22(6), 67–90 (2010). T. Vukašinac and P. Zhevandrov, “Geometric Asymptotics for a Degenerate Hyperbolic Equation,” Russ. J. Math. Phys. 9(3), 371–381 (2002). M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Selfadjoint Operators in a Hilbert Space (Leningrad Univ., Leningrad, 1980; D. Reidel Publishing Co., Dordrecht, 1987). V. S. Vladimirov, Equations of Mathematical Physics(Nauka, Moscow, 1981; “Mir”, Moscow, 1984). V. P. Maslov and M. V. Fedoryuk, Quasiclassical Approximation for the Equations of Quantum Mechanics (Nauka, Moscow, 1976; Reidel, Dordrecht, 1981). A. S. Mishchenko, B. Yu. Sternin, and V. E. Shatalov, Lagrangian Manifolds and the Method of Canonical Operator, (Nauka, Moscow, 1978; Lagrangian manifolds and the Maslov operator, Springer-Verlag, Berlin, 1990).