Algebro-geometric Solutions Of Two Continuous Soliton Equations

This paper mainly discussed about the algebro-geometric solutions of two soliton equations with clear physical meaning. This paper is divided into five chapters, as following:In the first chapter, we introduce the generation and development of soliton equation,and application in the various disciplines. Subsequently, we give an introduction to algebro-geometric solution of the development and the current research at home and abroad.In the second chapter,we elaborated the ideas of "AC=BD". Then we give the prove of Frobenius theorem and the solving of algebro-geometric solution from the point of "AC=BD".In the third chapter,we introduced the sources of Riemann surface and its basic theory. So, it laid the foundation for the calculation of algebro-geometric solution at the other chapters.In the fourth chapter, beginning with the spectral problems of D-AKNS equation, we got the D-AKNS equation of three variables using the trace identity. Subsequently, with the help of variables separation technology, the D-AKNS equation is decomposed into three ordinary differential equations which can be solved. Then, making use of the hyperelliptic curve, we introduce a concept of Riemann surface. So, we can constructed a set of Abel-Jacobi coordinates. On the Riemann surface, we straightened out the flow.Finally, using the inversion method and the theta function, we obtained algebro-geometric solution of the D-AKNS equation.In the fifth chapter, from a given spectral problem, we deduced Fokas-Lenells equation using of Lenard gradient sequences. And based on the finite order expansion of Lax matrix, this equation is decomposed into two ordinary differential equations which can be solved. Then, we introduce elliptic coordinates.Thus, the flow can be straighten out on the Abel-Jacobi coordi-nates. In the end, we get the algebro-geometric solution of Fokas-Lenells equation by Riemann surface and theta function.