Researching On The Rial Function Method And Some New Solutions Of Kinds Of Nonlinear Evolution Equationsand Its Properties

Some methods for solving the equations' solutions are widely used in Applied Mathematics, for example, Euler's method of undetermined exponential function, method of variation of constants and method of undetermined coefficients are the solving methods with the "trial"[1]property, which are called trial function method. Homogeneous balance method[2], hyperbolic tangent function expansion method[3], Jacobi elliptic function expansion method[4],[5] and auxiliary equation method[6]?[9] are the trial function method with two characteristics of constructive and the nature of mechanization. Trial function method has been widely used in solving nonlinear evolution equations[1],[10]?[26].In this paper, the hyperbolic tangent function expansion method is improved, and the multisoliton solutions of dispersive long-wave equation, wave dispersion water wave equation and (2+1)-dimensional dispersive long wave equations are presented by using symbolic computation system Mathematica. Auxiliary equation method is improved, and the method of combining function transformation and auxiliary equation method is given, and the new complexion solutions of(2+1)-dimensional potential Burgers system, (2+1 )-dimensional asymmetric Nizhnik-Novikov-Veselov system, (3+1)-dimensional Jimbo-Miwa equations and (3+1)-dimensional breaking soliton equation are presented.The discovery of localized excitations of high-dimensional integrable system is one of the important and difficult research in soliton theory[27].Some known excitations mode include Peakon solutions,compacton solutions and invisible solitons and their collision properties,fission fusion of soliton, chaotic soliton excitation, fractal soliton modes of excitation, folded solitary wave and folds. In order to find the localized excitations and the special structure of the nonlinear evolution equation,the solution of the nonlinear evolution equation is numerically simulated by using symbolic computation system Mathematica.In the first chapter, the generation and the development history of soliton theory are briefly introduced, some methods of solving the solutions of the nonlinear evolution equations and the main work of this paper are encapsulated.In the second chapter, based on the hyperbolic tangent function expansion method given in literature [28]and[29],an improved hyperbolic tangent function expansion method is given, and the multisoliton solutions of dispersive long-wave equation, wave dispersion water wave equation and (2+1)-dimensional dispersive long wave equations are obtained by using symbolic computation system Mathematica. The new multisoliton solutions are series of exponential function multiply by the sum of trigonometric functions and hyperbolic functions polynomials,and the nature of the solutions of the nonlinear evolution equations is analyzed.In the third chapter, based on the results given in literature[30]and[31], the method of combining function transformation and auxiliary equation is given, the new complexion solutions of some kinds of nonlinear evolution equations are obtained, and the nature of the solutions of the nonlinear evolution equations is analyzed by using symbolic computation system Mathematica.1. The method of combining function transformation and Riccati equation is given, with the aid of known solutions of the Riccati equation and related results of solutions, the new complexion solutions of(2+1)-dimensional potential Burgers system are presented, which are infinite sequences composed by the exponential function, trigonometric functions and hyperbolic functions polynomials each other.2. The method of combining function transformation and the second elliptic equation is given, with the aid of known solutions of the second elliptic equation and related results of solutions, the new complexion solutions of (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov system are presented, which are infinite sequences composed by hyperbolic function and the arbitrary one of the Jacobi elliptic function,Riemann ? function and trigonometric functions, and two-soliton solutions and double-periodic solutions are included.3. Based on the Painleve analysis, the method of combining function transformation and the second elliptic equation is given, and the new complexion solutions of a (3+1 )-dimensional nonlinear evolution equation are presented, which are infinite sequences.In the fourth chapter, two methods of solving the nonlinear evolution equations' solutions is given, the new complexion solutions of(3+1)-dimensional Jimbo-Miwa equation and (3+l)-dimensional breaking soliton equation are presented, and the nature of the solutions of the nonlinear evolution equations is analyzed by using symbolic computation system Mathematica.1.The method of combining transformation containing arbitrary function with variables z and t and the second elliptic equation is given,and the new complexion solutions of (3+1)-dimensional Jimbo-Miwa equation are presented, which are composed by the Riemann ? function,Jacobi elliptic function, hyperbolic function and trigonometric functions.These solutions contain arbitrary function with variables z and t.2. The formal solution given in literature [32]and[33] is improved,the new complexion solutions of (3+1 )-dimens ional breaking soliton equation are presented by using symbolic computation system Mathematica, which are composed by the exponential function,trigonometric functions and hyperbolic function, and the nature of the solutions of the nonlinear evolution equations is analyzed by using symbolic computation system Mathematica.In the fifth chapter, the main work of this paper and the research work to be done in the future are summarized.