Prolongation Structures Of The Camassa-Holm And Degasperis-Procesi Equations

It is a di?cult problem to solve nonlinear partial di?erential equations for a longtime,inverse scatting transformation is one of the most e?cient methods which solve kindsof nonlinear partial differential equations, the fundamental idea of that method is thattransform nonlinear problems to linear ones by using lax pair of partial di?erential equa-tions and spectral of ordinary di?erential equations. So far, prolongation structure theoryhas been a successful method to obtain the lax pair of equations,it fundamental idea isthat beginning with primitive nonlinear equations to obtain the lax pair ;then to checkthe integrability of the equations; and to solve the equations by using inverse scattingtransformation.In this thesis, we study on the prolongation structure of a partial differential equationwith a parameter u_t - u_xxt + (b + 1)uu_x = bu_xu_xx + uu_xxx ,and give the Lax pair ofCamassa-Holm equation and the Degasperis-Procesi equation.The whole thesis consists of three chapters. In the first chapter, we introduce theorign of soliton and the solution to it brieffy. In chapter 2, we introduce inverse scattingtransformation and the concept of Lax pair. In chapter 3, we mainly introduce continuouswedge di?erential knowledge which are fundamental tools of W-E continuous prolongationstructure theory, and the ideal of W-E prolongation structure theory . In the last chap-ter, we apply continuous prolongation structure to equation u_t - u_xxt + (b + 1)uu_x = bu_xu_xx + uu_xxx, and we can see the importance of continuous prolongation structure theoryin soliton or non-linear evolution equations; and calculate the Lax pair of Camassa-Holmequation and Degasperis-Procesi equation respectively.