An Extension of the direct Method via an Application

Abstract

The Direct Method due to Clarkson and Kruskal [9] for finding
similarity reductions of partial differential equations is extended to obtain
similarity reductions of nonlinear partial differential equations with arbitrary
functions. The extension is illustrated by the nonlinear hyperbolic equation
with arbitrary wave velocity. The known results of symmetry reductions [2]
for this equation are recovered by an extension of the direct method.
Mathematics Subject Classification (1991): 35L05, 35L70, 58J90, 58J70
Keywords: similarity reductions, partial differential equations, symmetry,
direct method, wave equation, nonlinear second-order PDE of hyperbolic type,
differential equations, nonlinear

Quaestiones Mathematicae 24(1) 2001, 111–122