Pseudopotential, Lax Pair, Darboux Transformation And Exact Solutions For Partial Differential Equations

The rapid development of the soliton theory,many scholars interested in thestudy of it.Problem of solving nonlinear partial differential equations is important anddifficult in the research of the soliton theory. Furthermore, it is so extensive to solvethe method of partial differential equations. Darboux transformation is an effectivemethod to workout the nonlinear partial differential equation. It makes correspondingchanges among Lax pair invariant norms, while finds the affiliation of the equation’ssolution. Ultimately, we obtain the freshly exact solution to the equation. This essaymainly studies the method proposed by Nucci for the Sharma-Tasso-Olever equationpseudopotential and Lax pair, and the exact solutions of the generalized KdV equationDarboux transformation.The first chapter is the introduction, introducing the system of the historicalbackground and development of the theory of soliton.The second chapter is based on the method proposed by Nucci, getting thepseudopotential and Lax pair of Sharma-Tasso-Olever equation.The third chapter is the first from the spectral problem of the generalized KdVequation in AKNS system, after a series of discussion, we have obtained three kindsof Darboux transformation of the equation, and given their proving with the properchoice of the trivial solutions of the equations, then the exact equation.