The Complexion Solutions,Periodic Solutions And Rational Solutions Of Nonlinear Wave Equations

It is significance to study the numerous of the nonlinear phenomena in nature by using the exact solutions of nonlinear wave equations, which is also one of important topics in soliton theory. In this paper, with the aid of Hirota bilinear method, we will investigate two kinds of nonlinear wave equations, namely the generalized modified Vakhnenko equation and a(3+1)- dimensional nonlinear evolution equation.The complexiton solution and soliton solution of the generalized modified Vakhnenko equation are obtained by two approaches. The first one is the extended real parameters to the complex parameters which will construct its multiple complexiton solution. The second one is the three-wave approach, by which we find complexiton solutions and other new solutions. We use the graphics to display detailed structure of the obtained solutions, including nonsingular solutions of the complexiton solution, loop soliton solution, cusp soliton solution, line soliton solution, their interaction solutions and so on. The quasi-periodic solutions of the generalized modified Vakhnenko equation are established by the Riemann theta function method. And the connection between soliton solution and quasi periodic solutions are found by the progressive analysis of the periodic solutions.A(3+1)-dimensional nonlinear evolution equation is considered by using Hirota bilinear method. Some rational solutions of a(3+1)-dimensional nonlinear evolution equation are constructed under the long wave limit. We get high order lump solutions from the rational solutions, through appropriate choice of parameters. For examples, we will discuss the evolutions of 1-lump solution, 2-lump solution and the 3-lump solution.