Exact Wave Solutions For Three Kinds Of Nonlinear Coupled Equations

Many important phenomena and characteristic can be described by nonlinear coupled equations in differential field,such as physics and chemistry.It is of great significance for the study and analysis of the characteristic and phenomena by investigating the traveling wave solutions of these coupled equations.In this paper,the explicit expressions of nonlinear coupled equations are discussed by using the(G'/G)-expansion method,the homogeneous balance method and the auxiliary equation method.Taking the three following coupled equations for examples:1.The Gray-Scott System2.The Coupled KdV Equation3.The Coupled Variant-Boussinesq EquationBy investigating the traveling wave solutions of the above three nonlinear coupled equations,we obtain the following results:1.The coupled equations can be transformed into the equivalent coupled nonlinear ordinary differential equations by traveling wave transformation.We study the solutions of the equivalent coupled nonlinear ordinary differential equations by using different methods.2.By using the(G'/G)-expansion method and solutions of auxiliary equation method,Two exponential function solutions,four trigonometric function solutions and two hyperbolic function solutions of Gray-Scott systems have been obtained.The explicit expression of these solutions are of great significance for the study of chemical reaction procedure.3.By using the solutions of Riccati auxiliary equation and the homogeneous balance method,The exact wave solutions of the nonlinear coupled KdV equation in shallow water are discussed.The explicit expressions of bell solitary wave solutions,trigonometric function periodic solutions and some bounded traveling wave solutions have been obtained.4.By using the auxiliary equation method,Some explicit expressions of bounded traveling wave solutions for the nonlinear dispersion coupled Variant Boussinesq equation are given.These solutions are of great significance in analyzing the influence of the nonlinear terms and parameters of these coupled equations and enrich the theoretical knowledge of solutions for the Gray-Scott systems,coupled KdV equation and the coupled Variant-Boussinesq equation.