Study On Analytic Solutions For Three New Kinds Of Auto-coupled KdV Equations With Variable Coefficents

The nonlinear scientifc problems are widely applied in various felds of natural scienceand social science. The nonlinear evolution equation is an important mathematical modelfor describing the nonlinear scientifc problems. The research on fnding and analyzingthe exact solutions of nonlinear wave equation can help us understand the motion laws ofthe nonlinear systems under some nonlinear interactions, and explain the correspondingnatural phenomena reasonably. In this thesis, based on the shallow water wave KdVequation, three kinds auto-coupled KdV equations with variable coefcients are proposed.By applying an extended Jacobi elliptic function expansion method and extended variable-coefcient mapping method to discuss this three new kinds of auto-coupled KdV equationswith variable coefcients. We obtain the following main results:Firstly, two kinds of linear symmetrically auto-coupled KdV equations with variablecoefcients are discussed by an extended Jacobi elliptic function expansion method. Thesolitary wave solutions and periodic solutions are obtained by using the method for thetwo kinds of equations. During the process, in order to obtain successfully the solutionsfor the frst-order linear symmetrically auto-coupled KdV equations, the variable coef-cients of the equations must satisfy certain linear relationship, while the solutions for thehigher-order linear symmetrically auto-coupled KdV equations can be used without anyrestrictions about the variable coefcients.Secondly, a kind of nonlinear asymmetrically auto-coupled KdV equations variablecoefcients are discussed by the extended variable-coefcient mapping method. By solvingthe nonlinear diferential algebraic equations which are derived from solving the nonlinearevolution equations, many Jacobi elliptic function solutions, hyperbolic function solutionsshock wave solutions, solitary solutions and trigonometric periodic solutions for the auto-coupled KdV equation with variable coefcients are derived. By selecting the appropriateparameter values, some exact solutions of the other forms are also obtained.