Group analysis approach for finding reciprocal transformations for the two-dimensional stationary gasdynamics

TL;DR: In this article, the authors proposed a method for finding classes of nonlocal equivalence transformations in continuum mechanics, and applied it to the two-dimensional stationary gas dynamics equations and showed that the method can be used to find all the reciprocal transformations of studied equations.

Abstract: Equivalence transformations play one of the important roles in continuum mechanics. These transformations reduce the original equations to simpler forms. One of the classes of nonlocal equivalence transformations is the class of reciprocal transformations. Despite the long history of applications of such transformations in continuum mechanics, there is no method of obtaining them. Recently such a method was proposed by the second author of the present paper. The method uses group analysis approach and it consists of similar steps as for finding an equivalence group of transformations. The new method provides a systematic tool for finding classes of reciprocal transformations (group of reciprocal transformations). As an illustration, the method was applied to the one-dimensional gas dynamics equations, and new reciprocal transformations were found. Similar to the classical group analysis this approach can be also applied for finding all reciprocal transformations (not only composing a group) of studied equations. The present paper provides this algorithm. As an illustration the method is applied to the two-dimensional stationary gas dynamics equations. Equivalence group, reciprocal equivalence group and completeness of all discrete reciprocal transformations are presented in the paper.