Prolongation Structure Of Semi-discrete Sine-Gordon Equation

The solving of nonlinear partial di?erential equations has always been a di?cultproblem, and inverse scattering transform is one of the e?ective methods in solving a broadrange of nonlinear partial di?erential equations. Its basic idea is to convert the nonlinearproblem into the linear problem by the Lax pair of nonlinear partial di?erential equationsand the spectral theory of ordinary di?erential equations. The theory of prolongationstructure is a relatively successful and systematical method in obtaining the Lax pairof equations up to now. Its basic idea is to get the Lax pair firstly from the originalnonlinear evolution equation, check the integrability of the equation, and get the solutionof the equation by the method of inverse scattering transform.This dissertation is committed to establish and improve the theory of semi-discreteprolongation structure, and use this theory to obtain the Lax pair of the Sine-Gordonequation.In the first chapter, we introduce the origin of the soliton theory and the research ofthe soliton question. In the second chapter, we introduce the inverse scattering transformand the Lax equation. In the third chapter, we establish and improve the theory of 1+1dimensional semi-discrete prolongation structure. In the fourth chapter, using the theoryof previous chapter, we discuss the 1+1 dimensional semi-discrete Sine-Gordon equation,and get its Lax pair.