Modified Generalized Riccati Equation Of Rational Expansion Method Solves Nonlinear Evolution Equations

Long wave propagation models on the surface of the nonlinear dispersion medium, has received extensive attention of mathematicians and physicists since the 1960s.The abstract model of it is available to build the nonlinear evolution equation, so the study of solving nonlinear evolution equations is of great significance. Tanh function, extended tanh function, and generalized Riccati equation of rational expansion methods are used to solve several model equations in this article. Through the comparative analysis methods of exponential function and generalized Riccati equation of rational expansion method, we propose a new function expansion method for solving nonlinear equations, and solve the two important nonlinear evolution equations by this method. The main research works of this article are as follows:1. we introduce the major steps of tanh function method, expanded tanh function, tanh function, Exp-function and generalized Riccati equation of rational expansion methods for solving nonlinear evolution equations.2. The IBq and BBM-Burgers equations are modified, the standard tanh method and the extended tanh method has been used to obtain solutions in the form of preset of modified IBq equation, and the generalized Riccati equation rational expansion method also has been used to obtain solutions in the form of preset of modified BBM-Burgers equation by combining with the new solutions of the generalized Riccati equation those are obtained by Exp-function method.3. By summarizing a class of equation with a specific form of the nonlinear and the highest derivative terms, we can directly use the simplified method of Exp-function when solving the equation by Exp-function method. Some new solutions of KdV equation by Exp-function method is deduced in detail, and two new periodic solutions are given by singularity analysis.4. This article modify the generalized Riccati equation rational expansion method by combining with Exp-function method and the conclusion of balancing the nonlinear term UsU(r) and the highest order derivative term U(n) in Exp-function. In combination with the new solutions of the generalized Riccati equation those are obtained by Exp-function method, we solve KdV-mKdV nonlinear evolution equation and the KdV equation with variable coefficients about time by using the modified generalized Riccati equation rational expansion method.