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Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data

Authors: S. Yu. Dobrokhotov, V. E. Nazaĭkinskiĭ and B. Tirozzi
Translated by: the authors
Original publication: Algebra i Analiz, tom 22 (2010), nomer 6.
Journal: St. Petersburg Math. J. 22 (2011), 895-911
MSC (2010): Primary 35L05
Published electronically: August 18, 2011
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Abstract | References | Similar Articles | Additional Information

Abstract: The Cauchy problem is considered for the two-dimensional wave equation with velocity $ c=\sqrt x_1$ on the half-plane $ \{x_1\geq 0,x_2\}$, with initial data localized in a neighborhood of the point $ (1,0)$. This problem serves as a model problem in the theory of beach run-up of long small-amplitude surface waves excited by a spatially localized instantaneous source. The asymptotic expansion of the solution is constructed with respect to a small parameter equal to the ratio of the source linear size to the distance from the $ x_2$-axis (the shoreline). The construction involves Maslov's canonical operator modified to cover the case of localized initial conditions. The relationship of the solution with the geometrical optics ray diagram corresponding to the problem is analyzed. The behavior of the solution near the $ x_2$-axis is studied. Simple solution formulas are written out for special initial data.

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Additional Information

S. Yu. Dobrokhotov
Affiliation: Institute for Problems in Mechanics, Russian Academy of Sciences, and Moscow Institute of Physics and Technology, Russia

V. E. Nazaĭkinskiĭ
Affiliation: Institute for Problems in Mechanics, Russian Academy of Sciences, and Moscow Institute of Physics and Technology, Russia

B. Tirozzi
Affiliation: Department of Physics, University of Rome “La Sapienza”, Italy

Keywords: Wave equation with degenerating velocity, asymptotic expansion, wave front, singular Lagrangian manifold, run-up
Received by editor(s): September 13, 2010
Published electronically: August 18, 2011
Additional Notes: Supported by RFBR (grants nos. 08-01-00726 and 09-01-12063-ofi-m) and by a joint project of the Department of Physics of the University of Rome “La Sapienza” and the Institute for Problems in Mechanics, Russian Academy of Sciences
Dedicated: To Vasiliĭ Mikhaĭlovich Babich
Article copyright: © Copyright 2011 American Mathematical Society