Prolongation Structure Of The Semi-Discrete MKdV Equation

The theory of prolongation structure that based on the di?erential geometry is animportant method in studying the soliton equation, and has a broad application. By usingthis theory, one can obtain the Lax pair from the original nonlinear partial di?erentialequation,then test the integrability of the original equation and solve the equation byinverse scattering transform.This dissertation is committed to establish and improve the theory of semi-discreteprolongation structure,and use this theory to discuss the semi-discrete MKdv equationand obtain its Lax pair.In the introduction of the first chapter, we will discuss the origin of the soliton equa-tion, Lax equation and the relevant theories of inverse scattering transform. In the secondchapter, we brie?y introduce the continuous di?erential calculus and theory of continuousprolongation structure. In the third chapter, we discuss the noncommutative semi-discretedi?erential calculus. In the fourth chapter, with the semi-discrete noncommutative dif-ferential calculus method, we establish and improve the theory of 1+1 dimensional semi-discrete prolongation structure, and specifically discuss the prolongation structure of the1+1 dimensional semi-discrete MKdV equation, then we obtain its Lax pair. At last, inthe fifth chapter, we give some summaries and prospects.