Prolongation Structures Of Difference Equations

Diferential geometry play an important role in mathematical physics.The prolon-gation structure theory that based on this method is an important method in studyingthe soliton equation,and it has a broad application.The theory is a relatively successfuland systematical method in obtaining the Lax pair of equation up to now. Its basic ideais getting the Lax pair from the original nonlinear evolution equation.Then verify theintegrability of the equation,and the last is to get the solution of the equation by inversescattering transform.This paper is committed to establishing and improving the discrete prolongationstructure theory,In addition,using this theory to discuss diference equation and obtainingits Lax pair.The whole paper consists of four chapters.In the first chapter,we will introduce theorigin of the soliton theory,and the research of the soliton theory and its broad appli-cation.In the second chapter,we will introduce the integrability of the inverse scatteringtransform and the Lax pair.In the third chapter,Based on the Noncommutative difer-ential we established the prolongation structure theory of diference equations. and inthe last chapter,by using the theory of previous chapter we will discuss the diferenceequation(p q+u2n,m+1un+1,m)(p+q un+1,m+1+un,m)=p q2and obtain its Laxpair.