Exact Solutions, Symmetry Reductions, Painlevé Test and Bäcklund Transformations of A Coupled KdV Equation**Supported by the National Natural Science Foundation of China under Grant Nos. 11675084 and 11435005, Ningbo Natural Science Foundation under Grant No. 2015A610159 and granted by the Opening Project of Zhejiang Provincial Top Key Discipline of Physics Sciences in Ningbo University under Grant No. xkzwl1502, and the

Abstract

A coupled KdV equation is studied in this manuscript. The exact solutions, such as the periodic wave solutions and solitary wave solutions by means of the deformation and mapping approach from the solutions of the nonlinear ϕ4 model are given. Using the symmetry theory, the Lie point symmetries and symmetry reductions of the coupled KdV equation are presented. The results show that the coupled KdV equation possesses infinitely many symmetries and may be considered as an integrable system. Also, the Painlevé test shows the coupled KdV equation possesses Painlevé property. The Bäcklund transformations of the coupled KdV equation related to Painlevé property and residual symmetry are shown.