Two Exact Solutions Of The (2 +1)-dimensional Soliton Equations

The soliton theory is an important branch of applied mathematics and mathematical physics. It has important applications in fluid mechanics, classical and quantum fields theories etc. It is one of the most active fields in science. There are many methods to obtain solutions of nonlinear partial differential equations, specially the multi-dimensional soliton equations, in the soliton theory. It is very difficult to solve the multi-dimensional soliton equations due to their high nonlinearity. Usually one considers multi-dimensional problems to be solved in such a way as splitting into several lower-dimensional ones, which are easier to treat with the available tools, such as nonlinearization approach, Darboux transformation, the Inverse Scattering transformation, Backlund transformation, Hirota method, the algebra-geometric method and so on.The aim of the present paper is to solving the two (2+1)-dimensional soliton equations. With the help of Darboux transformation, the explicit solutions of the (1+1)-dimensional soliton equations can be obtained. Through the relation between the two (2+1)-dimensional soliton equations and the (1+1)-dimensional soliton equations, we get the explicit solutions of the two (2+1)-dimensional soliton equations and several interesting figures of the solitonian solutions are plotted. The two (2+1)-dimensional soliton equations are : the modified Gardner equationand the (2+1)-dimensioal mixed breaking soliton equationwhere There are three sections in the paper. The first section is an introduction to the history of the soliton theory and the Darboux transformation.In the second section, we study the Darboux transformation of the (1+1)-dimensional soliton equations, associated with the modified Gardner equation and the (2+1)-dimensioal mixed breaking soliton equation, and obtain their Darboux Matrix and the explicit solutions of the two...