The Darboux Transformation And Explicit Solution Of Constraint Of The Kadomtsev-petviashvili Equation

Soliton thery is an import subject of nonliearsience. There are at list two useful applications. First, it reflects a set of stable phenomena. Second, it provides a way to get the explicit solutions for nonliner partialdifferential equations. Therefore, it has been thinking highly of by physicists and mathematicans. This paper studies mainly the Darboux transformation, which can obtain explicit solutions of nonlinear partial differential equations.In section 1, we introduce the fundamental theory of Darboux transformation and Darboux martrix. In section 2, we consider a 3×3 spectral problem, according to the compatibility conditions and the zero curature equation yields the constraint of the Kadomtsev-Petviashvili equation. In the following paper, based on the theory introduced in section 1, with strict proof, we construct a Darboux transformation with multi-parameters. In section 3, from a trivial seed u = v = w = 0, we use this Darboux transformation obtain new solutions of the soliton equations respectively, then discuss the first two cases.