Researches On Solving And Integrability Of Nonlinear Differential Equations And Ultra-discrete Equations

In this dissertation, some topics are studied based on the mathematical mechanization and AC= BD model, including exact solutions, hyperelliptic functions solutions and quasi-period solutions of nonlinear evolution equations and discrete equations, Lax integrability and tropical Riemann theta functions solutions of ultra-discrete equations.In Chapter 1, we introduce the history and development of mathematical mechaniza-tion, soliton theory, integrability of nonlinear evolution equations and algebraic geometry solutions, soliton celluar automaton, solutions of the mathematical and physical equations, as well as the main work of this dissertation.In Chapter 2, the main content and idea are AC= BD model and C-D pair theory. According to these, the mechanized construction of the transformations of two differential equations is presented. Based on the research on a class operator of "D", we get more solutions of this "Dv=0", so we can get more exact solutions of some nonlinear evolution equations.In Chapter 3, based on the theory of hyperelliptic functions and algebraic curve, we extend the direct method of solving the nonlinear evolution equations. By the identities of the hyperelliptic functions of two genus and three genus, we obtain the solutions of two genus and three genus hyperelliptic functions of some classes of the nonlinear evolution equations.In Chapter 4, first, we extend the method of combined the bilinear with the Rie-mann theta functions to the Riemann theta functions with rational character, and give the quasi-periodic solutions of some nonlinear equation(s) which has (have) above two de- pendent variables and apply this method to the differential difference equations. Second, we give quasi-periodic solutions of two discrete equations by the direct method based on the identities of the Riemann theta functions with rational character.In Chapter 5, we obtain the ultradiscrete equation of the discrete mKdV equation and prove its Lax integrability by the ultradiscretization. And By the tropicalization, we give the tropical spectral curve of the ultradiscrete rndKP equation and show that its solution can be got by the tropical Riemann theta function.