Lie Symmetry Reductions, Exact Solutions and Conservation Laws of the Third Order Variant Boussinesq System

The research area of nonlinear partial differential equations (NLPDEs) has been very active for the past few decades. The study of the exact solutions of a nonlinear evolution equation (NLEE) plays an important role to understand the nonlinear physical phenomena which are described by these equations. In recent years several powerful and efficient methods have been developed for finding analytic solutions of NLEEs. Some of the most important methods found in the literature include the inverse scattering, the Hirota bilinear method, the Darboux transformation method, G′/G expansion method, homogeneous balance method, Adomian decomposition method, the functional variable method, the extended tanh function method, etc. [1–7]. One such NLEE is the third order variant Boussinesq system and is given by vt + vux + uvx + uxxx = 0, ut + vx + uux = 0. (1) This system was introduced as a model for water waves [8, 9] where u is the velocity and v the total depth, and the subscripts denote partial derivatives. In Ref. [10], the solitary wave solutions of the variant Boussinesq equations are obtained by using a homogeneous balance method. In Ref. [8], the authors constructed the soliton solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic wave solutions using the extended tanh method. In Refs. [9] and [11], the conservation laws for the variant constant and variable coefficients Boussinesq system are derived by the Noether approach. Since the end of the 19th century, the symmetry study which laid the foundations of Lie plays an important role in almost all the scientific fields. As mentioned in [12], theory of Lie groups for obtaining the group invariant solutions to NLPDEs is widely recognized as one of the most powerful methods. We observe a plenty of books and survey articles about Lie groups method [13, 14].