The Darboux Transformation Of The (2+1)-dimensional KP Equation And Its Explicit Solutions

This paper mainly studies the Darboux transformation, which can obtain explicit solutions of (2+1)-dimensional KP equation. There are five sections in this article. In the first section, the fundamental theory of the Darboux transformation is mainly introduced. In the second section, based on a new 2×2 matrix spectral problem, we obtain the Lenard sequence and a hierarchy of nonlinear soliton equations. In the third section, first, the (2+1)-dimensional KP equation is decomposed into two (1+1)-dimensional soliton equations. Then, their one time Darboux transformation is constructed with the help of a gauge transformation. In the forth section, the one time Darboux transformation is generalized to N times and the strict proof is provided. In the fifth section, from the trivial seed u=0.u=1, the compatible solutions of the (1+1)-dimensional soliton equations can be obtained by applying the Darboux transformation, then, the explicit solutions of (2+1)-dimensional KP equation can be obtained.