Similarity Reductions and New Exact Solutions for B-Family Equations

In this paper, Lie-Group method is applied to the b-family equations which includes two important nonlinear partial differential equations Camassa--Holm (CH) equation and the Degasperis--Procesi (DP) equation. The complete Lie algebra of infinitesimal symmetries is established. Three nonequivalent sub-algebras of the complete Lie algebra are used to investigate similarity solutions and similarity reductions in the form of nonlinear ordinary equations (ODEs) for the b-family equations. The generalized He's Exp-Function method is used to drive exact solutions for the reduced nonlinear ODEs, some figures are given to show the properties of the solutions.