On the similarity solutions of the semi-linear hyperbolic equationu xt =f(u) via symmetry method

Abstract

Herein the physically and mathematically significant semi-linear hyperbolic equationuxt= f(u), where f(u) is an arbitrary smooth function ofu and which encompasses the Liouville equation, Phi-four equation, Sine-Gordon equation, Klein-Gordon equation, Mikhailov equation and double Sine-Gordon equation, has been analysed via the symmetry method as developed by Steinberg (Symmetry methods in differential equations, Technical Report No. 367, University of New Mexico, 1979). The infinitesimals, similarity variables, dependent variables and reduction to quadrature or exact solutions have been tabulated for the mentioned physical forms of f(u). Some interesting outcomes of this study are the deductions of the new exact solutions that does not seem to have been reported in the literature.