THE LIE SYMMETRY ANALYSIS, OPTIMAL SYSTEM, EXACT SOLUTIONS AND CONSERVATION LAWS OF THE (2+1)-DIMENSIONAL VARIABLE COEFFICIENTS DISPERSIVE LONG WAVE EQUATIONS

In this article, the (2+1)-dimensional variable coefficients dispersive long wave equations (vcDLWs) are studied by the Lie symmetry analysis method. The infinitesimal generators and geometric vector fields are given. Optimal system of the (2+1)-dimensional vcDLWs are analyzed by Olver's method. Based on the optimal system, the (2+1)-dimensional vcDLW equations are reduced to (1+1)-dimensional equations. A number of new exact solutions of vcDLW equations are derived. Some kink solutions and 2-soliton solutions are obtained by using $\left( {1/G'} \right)$-expansion method and $\left( {G'/G} \right)$-expansion method. Many different types of exact solutions can be obtained by changing the coefficient functions. By exploring the evolution of the solutions with function of the coefficients and time $t$, the dynamic behaviors of the solutions are analysed. At last, the conservation laws of the (2+1)-dimensional vcDLWs are derived based on the nonlinear self-adjointness.