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

  • Peter A. Clarkson (a1) and Simon Hood (a1)
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.

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COPYRIGHT: © Cambridge University Press 1992
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