A Simplified Algorithm for Finding Partially Invariant Solutions of Quasilinear Systems

Abstract

The method of partially invariant solutions of PDE systems was introduced by Ovsiannikov as a generalization of the classical similarity analysis. It offers a possibility to calculate exact solutions possessing a higher degree of freedom than similarity solutions. Ovsiannikov's algorithm, however, is somewhat hard to apply because one has to deal with three equation systems derived from the original PDE system. By means of the two-dimensional Euler equations, we show how the algorithm can be essentially simplified if classical similarity solutions are already known. Further, we prove a necessary criterion for the simplified algorithm to be senseful.Key words: Hydrodynamics, Nonlinear systems, Group theory.Similarity analysis is a well-known and useful method for finding exact group-invariant solutions of PDE systems [1,2]. As a generalization, Ovsiannikov has introduced the theory of partially invariant solu tions which have a higher degree of freedom than classical similarity solutions [1]. The algorithm de scribed by Ovsiannikov yields, in general, two reduced PDE systems. It can well be applied to quasilinear first-order systems, e.g. the ideal M H D equations [3]. Martina and Winternitz calculated partially invariant solutions of nonlinear Klein-Gordon and Laplace equations [4]. But compared to the great number of publications on classical similarity solutions there ex ist very few examples of partially invariant solutions which are not actually invariant ones (we will call such solutions proper partially invariant) in the literature. This may be a result of the method's complexity.In this paper, we develop a simplified algorithm for finding partially invariant solutions, which can be ap plied whenever classical similarity solutions are known. As a detailed similarity analysis has been done for many PDE systems of mathematical physics; this al gorithm might offer a chance for finding a lot of fur ther exact solutions. The algorithm, however, is not always senseful. Often it yields only solutions which do not differ essentially from the known similarity solutions. In order to avoid useless calculations we prove for the case of quasilinear first-order systems a necessary criterion for the algorithm to yield new, that Reprint requests to Dr.