New symmetries and conservation laws for lossless KZK equation

Abstract

The Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation is widely used to describe intense acoustic beams. No general analytical solution has been obtained for this nonlinear equation, so numerical methods are usually employed. Known analytical approaches are associated with various approximations. An additional powerful mathematical tool is group analysis that enables the finding of general symmetrical properties of differential equations. These symmetries help to generalize known analytical and numerical solutions, to derive new solutions, and to obtain conservation laws. In this work, results of group classification of the KZK equation with arbitrary nonlinear term are presented. It is shown that the KZK equation can be written in Euler–Lagrange form. All the classical Lie symmetries (geometric symmetries) are found and some classes of Lie–Backlund symmetries are considered. The largest number of the symmetries corresponds to the cases of quadratic and cubic nonlinearities. Besides scaling and translation groups of symmetries, there exist additional transformation groups which are not evident from the physical point of view. Examples of obtaining new solutions and deriving reduced equations are presented. New conservation laws for intense acoustic beams are obtained using the Noether theorem. [Work supported by RFBR and CRDF.]