The Traveling Wave Solutions For A Generalized KdV Equation With Two Power Nonlinearities

The shallow water wave equations,such as the Camassa-Holm equation,originning from Modern Mechanics and Physics,are becoming one of important research topics in nonlinear partial differential equations.Searching the exact solutions of a fixed equation with practical background is of particular interest for mathematicians,physicists and engineers etc.In order to search the exact solutions of nonlinear partial differential equations,a series of methods are formulated,such as the first integral method,the symmetry reduction method,the Backlund transformation,the Darboux transformation method,the function expansion method,the inverse dispersion transform method and so on.But,due to the form and characteristic of each nonlinear partial differential equation are different,there is no unified method,which can give all exact solutions of a non-linear partial differential equation.In the article,according to the characteristics of the generalized KdV equation with two power nonlinearities,we obtain some new travelling wave solutions by the trial function expansion methods.In Chapter 1,we recall the history of travelling waves of nonlinear par-tial differential equations,and summarize some function expansion methods for searching the exact solutions used in this article.We state our main results for two classes of generalized KdV equations.In Chapter 2,by applying the tanh function expansion method,we study the generalized KdV equation with two power nonlinearities,UCnder some values of m and k,we obtained four classes new travelling waves solutions.In Chapter 3,by using the complex transformation-elliptic function expan-sion method,we investigate the fractional generalized KdV equation with two power nonlinearities.Under some values of m and k,we obtained some new periodic solutions and solitary wave solutions.