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The Generalized Burgers and Zabolotskaya-Khokhlov Equations: Transformations, Exact Solutions and Qualitative Properties

P. N. Sionoid and A. T. Cates
Proceedings: Mathematical and Physical Sciences
Vol. 447, No. 1930 (Nov. 8, 1994), pp. 253-270
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/52615
Page Count: 18
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The Generalized Burgers and Zabolotskaya-Khokhlov Equations: Transformations, Exact Solutions and Qualitative Properties
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Abstract

A point transformation between forms of the generalized Burgers equation (GBE) first given by Cates (1989) is investigated. Applications include generalizations of Scott's (1981) classification of long-time behaviour for compressive wave solutions of the GBE and the equivalence of the exponential and cylindrical forms of the GBE, yielding an exact solution for the exponential GBE. Applications to nonlinear diffractive acoustics are considered by using a similarity reduction of the dissipative Zabolotskaya-Khokhlov (DZK) equation (describing the evolution of nearly plane waves in a weakly nonlinear medium with allowance for transverse variation effects) onto the GBE. The result is that waves from parabolic sources may be described by the cylindrical GBE in the case of two dimensions, and by the spherical GBE in the three-dimensional, cylindrically symmetric case. Furthermore, results on the formation of shocks and caustics in the context of the ZK equation are presented, along with an exact solution to the DZK equation. Exact solutions with caustic singularities are studied, along with a possible mechanism for their control. Finally, results on the evolution of a shock approaching a caustic are given through the identification of a series of parameter regimes dependent on the diffusivity.

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