Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics

Abstract

This is an updated, revised and extended translation from the Russian original work Symmetry Analysis and Exact Solutians of Equations of Nonlinear Mathematical Physics, Nauk:a, Kiev, © 1989 AU Rights Reserved

Nonclassical Symmetries for a Class of Reaction-Diffusion Equations: the Method of Heir-Equations

“…b Namely papers published within the first half of 2012. c In [25] this name was introduced for the first time. d The minus sign in front of F(t, x, u) was put there for the sake of simplicity: it could be replaced with a plus sign without affecting the following results.…”

The Bogolyubov renormalization group

“…In those cases when the embedding of the boundary-value problem for a DE leads to an integral formulation, it is required that the algorithms of group analysis should be extended to integro-differential systems of equations. Taking into account that recently some progress has also been made [54,55,56] in extending the range of applicability of the established methods of group analysis, one can say that the above combination turns out to be constructive enough also for integro-differential equations. We recall that the first embedding with a physical end in view was realized for the integral equation of radiative transfer [53].…”

Similarity Reductions of the Zabolotskaya-Khokhlov Equation with a Dissipative Term

“…The equation may be written as The infinitesimal symmetries and exact solutions of the Z-K equation have been investigated by many authors [2,3]. Recently, the general derivation of the Z-K equation was given by Taniuti [4].…”