Some Investigation On The Methods For Finding Solutions Of Nonlinear Evolution Equations And Realization Of Mathematics Mechanization

In this dissertation, under the guidance of mathematics mechanization and by means of computer algebraic and numerical systems, some problems in the theory of solitons are discussed and some methods for constructing the solutions of nonlinear evolution equations are presented. The methods presented are realized on the algebraic computation software Maple or MATLAB. The method of mathematics mechanization is applied to the related subjects and the operating systems of mathematics mechanization using MATLAB are set up. The description is as follows:Chapter 1 of this dissertation is devoted to reviewing the history and development of soliton theory, some methods for seeking exact solutions of nonlinear evolution equations and the mathematics mechanization, with an emphasis on some achievements on the subjects involved in this dissertation.Chapter 2 concerns the construction of exact solutions of nonlinear evolution equations under via the "AC=BD" theory introduced by Prof. H. Q. Zhang. The construction of the operators of C and D and some concrete algorithms for constructing the exact solution based on the division with remainder "AC=BD+R" in terms of the symbolic computation software Maple are introduced.In chapter 3, based on the ideas of unification methods, algorithm realization and mechanization for solving nonlinear evolution equations, we found a new theory of generalized hyperbolic functions and present some new methods to construct the solutions of nonlinear evolution equations. Its main contents are as follows:(1) A new definition of generalized hyperbolic functions, generalized hyperbolic function transformation and their properties are presented. Then some concrete formulae of generalized hyperbolic function transformation are also given for constructing the solutions of nonlinear evolution equations.(2) A new generalized hyperbolic function - B(?)cklund transformation method is presented and applied to construct the exact solutions of some nonlinear evolution equations. As a result, a lot of new exact solutions in more general forms are obtained. Then we investigate the long-playing traveling state of the solutions via computer simulation and find that the solutions are of long time stability.(3) A new method of partitioned long-playing traveling state of the generalized hyperbolic function solution of nonlinear evolution equations is presented. The validity of the method is tested by its application to investigating the long time stability of the solutions. In addition, two methods which revise long-playing traveling state of the generalized hyperbolic function solutions and the variable coefficient solutions of nonlinear evolution equations are given.(4) Based on the ideas of the WTC method and the homogeneous balance method for constructing B(?)cklund transformation, a new method and its mechanization algorithm are suggested for constructing the B(?)cklund transformation. The validity and reliability of the method are tested by its application to two nonlinear evolution equatious of higher order and higher dimension. In addition, a new theorem about the method is presented and proved.(5) We investigate in a wide context whether the different values of three parameters in the gener- alized hyperbolic function solution affect the local properties of the long-playing travelling state of the solutions, whether the different types of a seed solution of a nonlinear evolution equation under the same B(?)cklund transformation affect the shape and number of solitons, the seed solution affect on the main part of solution, long-playing travelling state of the variable (constant) coefficient non -travelling wave solution differs from travelling wave solution, and the long-playing travelling states of several types of generalized hyperbolic function solutions and so on by means of computer simulation. As a result, we find some new phenomena and suggest four guesses.In chapter 4, by use of the symbolic computation software Maple, the general projective Riccati equation method and the improved F- expansion method are improved, and the resulting new methods and theorems are devised to construct the exact solutions for nonlinear evolution equations.(1) A new generalized hyperbolic function-Riccati equation is constructed and two theorems, which shows that the equation possesses new and more general generalized hyperbolic function solution including the projective Riccati equation and general projective Riccati equation, are devised and proved by a mechanization method using Maple.(2) By means of the generalized hyperbolic function-Riccati equation, a new generalized hyperbolic function -Riccati method is presented and applied to some nonlinear evolution equations. As a result, new exact solutions in more general forms are obtained.(3) By devising two kinds of new and more general transformation, a new generalized F-expansion method and a new extended generalized F- expansion method are proposed for constructing the exact solutions of nonlinear evolution equations. These new methods are applied to some nonlinear evolution equations and more general exact solutions are obtained.In chapter 5, a new definition of N soliton-like solution, guess 5, and a new and more general transformation are given, the Exp- function method is developed, and a new Exp -B(?)cklund method and a new Exp -N soliton-tike method are presented. Some more general exact solutions including nontravelling wave solutions and traveling wave solutions of some nonlinear evolution equations are derived using these methods. The long-playing traveling state of the solutions is sought by means of computer simulation.In chapter 6, the algebraic methods for constructing the traveling wave solutions of nonlinear evolution equations are developed and the following new methods and theorems are presented.(1) A new and general transformation, and a new theorem which is proved in Maple are presented.(2) A new mechanization method to find the exact solutions of a first-order nonlinear ordinary differential equation with any degree is presented. The validity and reliability of the method are tested by its application to the first-order nonlinear ordinary differential equation with six degree, eight degree, ten degree, and twelve degree.(3) A new generalized algebraic method, an extended generalized algebraic method, and their algorithms are suggested based on a nonlinear ordinary differential equation with any degree. Some nonlinear evolution equations are chosen to illustrate our algorithm so that more families of new exact solutions are obtained, which contain both non-traveling and traveling wave solutions.In chapter 7, some numerical methods are improved, a new compound operation method consisting of numerical and analytical methods to solve nonlinear evolution equations and the improved Adams method are proposed, and the accuracy of numerical computation is raised. A new tactics of devising a new system for handling numerical computation orders, which can actualize synchronously that numerical solutions, sign solution, and their error estimates are outputted in a figure format or a table format, is presented. A great deal of the software on mathematics mechanization is developed and two systems for handling the computation and the figures orders on numerical analysis and advanced mathematics using MATLAB are set up so that the same type of problem is solved automatically.