The Darboux Transformation And Exact Solution Of (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 solution of nonlinear partial differential equations, specially the multi-dimensional soliton equations, in the soliton theories. It is very difficult to solve the multi-dimensional soliton equations due to their high nonlinearity. Usually one considers multi-dimensional problem 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 (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 soliton solutions are plotted. The two (2+1) dimensional soliton equations are:MKP-type equations.There are four 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 consider a spectral problem associated with the MKdV equation. The Lenard pairs K, J is introduced by means of a map. The hierarchy of MKP equation is generated.In the third section. We study the Darboux transformation of the (1+1) dimensional soliton equations, associated with the MKP equation and (2+1) dimensional MKP-type soliton equations, and obtain their Darboux matrix and the Explicit solution of the two (2+1) dimensional soliton equations.In the last section, takingωfor constant, we obtain the explicit solution of the two (2+1) dimensional soliton equations and several interesting figures of the soliton solutions are plotted.