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Nonclassical symmetry reductions and exact solutions of the Zabolotskaya–Khokhlov equation

Published online by Cambridge University Press:  16 July 2009

Peter A. Clarkson  and
Simon Hood
Peter A. Clarkson
Affiliation:
Department of Mathematics, University of Exeter, Exeter EX4 4QE, UK
Simon Hood
Affiliation:
Department of Mathematics, University of Exeter, Exeter EX4 4QE, UK
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Abstract

In this paper, new non-classical symmetry reductions and exact solutions for the 2+1- dimensional, time-independent and time-dependent, dissipative Zabolotskaya-Khokhlov equations in both cartesian and cylindrical coordinates, are presented. These are obtained using the Direct Method, which was originally developed by Clarkson & Kruskal (1989) to study symmetry reductions of the Boussinesq equation, and which involves no group theoretic techniques. In particular, we derive exact solutions of these Zabolotskaya-Khokhlov equations expressible in terms of elementary functions, Weierstrass elliptic and zeta functions, Weber-Hermite functions and Airy functions. Additionally, it is shown that some previously known solutions of these equations actually arise from non-classical symmetries.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 4 , December 1992 , pp. 381 - 414
Copyright
Copyright © Cambridge University Press 1992

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