Symmetry reductions and new exact invariant solutions of the generalized Burgers equation arising in nonlinear acoustics

Abstract

We perform a complete Lie symmetry classification of the generalized Burgers equation arising in nonlinear acoustics. We obtain seven functional forms of the ray tube area that allow symmetry reductions. We use symmetries to obtain reduced equations and exact solutions when possible. We also investigate the existence of potential symmetries for the generalized Burgers equation. It is found that only the classical Burgers equation admits true potential symmetries. We further obtain all conditional symmetries of the second kind and indicate a possible route for obtaining conditional symmetries of the first kind. The conditional symmetries of the second kind leads to symmetry reductions and exact solutions not obtainable from Lie point symmetries.