A Trail Equation Method For Solving Some Nonlinear Evolution Equations

As the bridge between the mathematical theory and the practical applications,the study of nonlinear evolution equations in the context of physical and mechanical problems is not only the main content of traditional applied mathematics,but also an important part of modern mathematics.Compared with the linear equation,the nonlinearity bring substantial difficulties in the mathematics study.Therefore,discussing nonlinear evolution equations is a challenging task,especially for nonlinear evolution equations,solving exact solutions is always the focus of the research.Although there are a lot of methods which have been put forward and developed,such as inverse scattering method,Backlund transformation method,Lie group method and some direct algebraic methods(Hirota bilinear method,mixing exponential method,homogeneous balance method,hyperbolic function expansion method,Jacobi elliptic function expansion method),there still exists a large amount of nonlinear evolution equations with the practical background which needs new methods to solve their exact solutions.The main task of this thesis is to develop and use trial equation method,which can be applied to five common classes of nonlinear evolution equations in physics and mechanics,obtain a series of the exact solutions,and describe the abundant wave propagation modes of these physical problems.The thesis includes the following three parts:Firstly,the traditional trial equation method is generalized to the complex trial equation method,and two classes of nonlinear Schr (?)dinger equations are studied,namely Schr (?)dinger equation with quadratic-cubic nonlinear terms and Schr (?)dinger equation with nonlocal parabolic law.For the first model,seven kinds of forms of exact solutions are obtained,such as optical soliton solution,discontinuous periodic solution,singular rational function solution,exponential function solution and Jacobi elliptic function solution,there are three new solutions which have not been obtained by other methods.For the second model,constructing the rich accurate envelope traveling wave solution,we can get the propagation mode of light waves in nonlocal parabolic media,which behaves as soliton behavior and periodic mode.According to the different selections of the parameters,the corresponding optical wave propagation mode is determined.Secondly,the traditional trial equation method is extended to coupled trial equation method,and two classes of shallow water wave motion equations are studied,the equations are Kaup-Boussinesq equation system and coupled Kaup-Boussinesq II equation system,respectively.For the first equation,by the fifth-order polynomial discriminant system,13 sets of accurate single traveling wave solutions can be obtained.For the second equation set,there are six sets of accurate single traveling wave solutions can be obtained by using the forth-order polynomial discriminant system.In particular,when the wave propagation velocity is taken as a special constant,two types of equations exist periodic solutions,and reveal the periodic dynamic behaviors.Finally,the traditional trial equation method is generalized to the variable coefficient trial equation method.Considering variable coefficient generalized Kd V-m Kd V combination equation,there are three conditions which the free parameter is adopted as 1,-1/2and 2 respectively.When the free parameter is adopted as 1,a fourth-order polynomial complete discriminant system can used to classify its solutions.When the free parameter is adopted as-1/2,the original equation can be transformed into a factor equation of rational form,then its solution can also be classified through a fourth-order polynomial complete discriminant system.When the free parameter is adopted as 2,the classification of the solution of factor equation can be transformed into a discriminant system of sixth-order polynomial,and then we obtain the exact solution of the original equation.