A Kind Of Sub-equation Method And Its Applications To Nonlinear Evolution Equations

In this paper eight systems of nonlinear evolution equations are studied by using the F-expansion method, which is a kind of sub-equation method (mean by subsidiary ordinary differential equation method) and can be thought of as a generalization or concentration of the Jacobi elliptic functions expansion since F here stands for every one of various Jacobi elliptic functions, and abundant doubly periodic wave solutions expressed by various Jacobi elliptic functions of each system are obtained. Meanwhile the F-expansion method is extended in two respects that on the one hand, besides F-expansion we use (F, G) expansion and (F, F~-1) expansion; on the other hand, besides Jacobi elliptic equations we consider other ODEs being the sub-equation. As the results of the researching we have obtained more 22 kinds of solution to the dispersive long wave equations in 2 + 1 dimensions than those obtained in [10]; more 21 kinds of solution to the coupled nonlinear Klein-Gordon equations than those obtained in [24]; 22 kinds of solution to the coupled KdV system of equations than those obtained in [10]; 38 kinds of solution to the variant shallow water wave equations than those obtained in [25]; 17 kinds of solution to the long-short wave interaction equations than those obtained in [18]; 29 kinds of solution to the Drinfel'd-Sokolov-Wilson equations; 36 kinds of solution to the coupled nonlinear Klein-Gordon-Zakharov equations than those obtained in [24]. To our knowledge, a part of these solutions are new results. In the limit cases when the modulus m→ 1 and m → 0, then the solitarywave solutions and the trigonometric function solutions are also obtained,respectively. Last, by using other sub-equation, we have obtained the solitary wave solution of generalized Drinfel'd -Sokolov equations.