Researchs Of The Darboux Transformation And N-soliton Solution For Extended Form Of MKP Equation With Variable-coefficient Based On Symbolic Computation

The Kadomtsev-Petviashvili(KP)equation has been considered as a ubiquitous and important physical model which describes nonlinear wave motion.The celebrated KP-like equations with wide applications in physics and engineering have been researched at home and abroad.After the extended form of modified Kadomtsev-Petviashvili equation with constant coefficient researched by A.M.Wazwaz,the extended form of modified Kadomtsev-Petviashvili equation with variable-coefficient has been considered as a wor-thy field of study.Using Mathematica symbolic computation software,this thesis focuses on construction the Lax pairs of the extended form of modified Kadomtsev-Petviashvili e-quation and the relative Darboux transformation.Moreover,we research on Mathematica algorithm aiming at this equation's two dimensions:the construction of Darboux trans-formation and the integrability proved by automatic deducing N-soliton solution.The main work is as follows:Chapter 1 introduces the theoretical background and the main features of Mathe-matica symbolic computation software.Analyzes the current state of the research about the celebrated KP-like equations.Briefly expounds the research content and innovation points of this thesis.Chapter 2 based on the singularity analysis and the Painlev(?) property,uses the singu-lar manifold method to obtain the two Painlev(?) branches of this equation.Then with the auto-B(?)cklund transformations between two pairs of solutions,we derive two Lax-pairs for this equation.Chapter 3 constructs the N-times Darboux transformation.Then the N-soliton so-lution formula is given,which contains 2n free parameters and two arbitrary functions.Choosing arbitrary functions and the different seed solution,we can obtain several type-s of one-soliton solutions,two-soliton solutions or three-soliton solutions.In order to avoid the singularity of the solutions,the regularity conditions are discussed seriously.Moreover,we construct the generalized Darboux transformation matrix by considering a special limiting process and find a rational-type solution for this equation.Chapter 4 according to the Darboux transformations of KP-like equations,such as KP-?,KP-?,mKP and extended form of mKP,the algorithm of automatic deducing N-soliton solution of KP-like equations is presented.Chapter 5 is the summarization of this thesis and the prospect of the future work.