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Generalization of the double reduction theory. (English) Zbl 1201.35014
Summary: Generalization of the double reduction theory to partial differential equations of higher dimensions is still an open problem. In this note we have attempted to provide this generalization to find invariant solutions for a nonlinear system of $q$th order partial differential equations with $n$ independent and $m$ dependent variables provided that the nonlinear system of partial differential equations admits a nontrivial conserved form which has at least one associated symmetry in every reduction. In order to give an application of the procedure we apply it to the nonlinear $\left(2+1\right)$ wave equation for arbitrary function $f\left(u\right)$ and $g\left(u\right)$.
##### MSC:
 35A25 Other special methods (PDE) 35B06 Symmetries, invariants, etc. (PDE) 35L70 Nonlinear second-order hyperbolic equations