The Spectral Study Kp Equation

Backlund transformation was named after the Swedish geometer Albert Victor Backlund, who made the first study of transformations on the negative constant curvature surfaces in three dimensional Euclidean space. The basic idea of this method is to convert the problem of solving nonlinear partial differential equations to that of solving integrable systems constituting of ordinary differential equations. With the development of the theory of integrable systems, Backlund transformation has become a very important tool for solving nonlinear partial differential equations. The bilinear method, also called the direct method, was invented by Japanese mathematician Ryogo Hirota. It can be used to solve many nonlinear partial differential equations. The basic idea of this method is to transform a nonlinear partial differential equation into a bilinearized equation. The advantage of bilinear method is that the nonlinear partial differential equation discussed does not necessarily possess a Lax pair. So it can be used to study some non-integrable partial differential equations.In this thesis, for the non-isospectral KP equation we study the above mentioned two problems. The bilinearized non-isospectral KP equation isFirstly, we give some properties for the bilinear operator, which are the basis in the proof of the following theorems.Secondly, we study the local equivalence between non-isospectral KP equation and its bilinearized equation, and obtain the following three theorems.Theorem1. If f(x,y,t) is the solution of the bilinear non-isospectral KP equation, then is the solution of the non-isospectral KP equation.Theorem2. If u(x, y,t) is the solution of the non-isospectral KP equation, then is the solution of the bilinearized non-isospectral KP equation, where functions c(y,t)and r(y, t) satisfy some conditions.Theorem3. The non-isospectral KP equation and its bilinearized equation are local equivalent.Finally, we study Backlund transformation for the bilinearized non-isospectral KP equation, and give the following theorem.Theorem4. If g is the solution of the bilinear non-isospectral KP equation, then the following system on f is integrable; furthermore, f aslo satisfies the bilinear non-isospectral KP equation.By Theorem4, the above integrable system defines a Backlund transformation from the bilinearized non-isospectral KP equation to itself. As an application, by applying the Backlund transformation to the trivial solution g≡1, we generate one-soliton solution of the non-isospectral KP equation.