To the Theory of Generalized Burgers' Equations

Connections are given between partial differential equations describing nonlinear acoustic waves propagating in one space dimension in a dissipative medium. Equations governing low-frequency waves in tubes with varying cross section, high-frequency wave propagation through inhomogeneous media, the behaviour of spherical and cylindrical waves, the amplification of travelling waves, etc., can be reduced to the form of a generalized Burgers' equation (GBE). Approaches which can be used for finding exact and approximate analytical results are described. Cases where exact self-similar solutions exist are considered. An alternative form of GBE, which offers additional possibilities of analytical studies of 1D nonlinear waves is derived. New self-similar solutions are found. Generalized Burgers' equations of special forms, corresponding, for example, to an exponential concentrator or to a wave amplified during the propagation along a nonuniformly heated plate, are solved by the method of matched asymptotic expansions.