The Study Of Solving Several Kinds Of Nonlinear Evolution Equations

Nonlinear evolution equations have extremely important application backgrounds in many fields such as physics,chemistry,etc.Obtaining their exact solutions is a key step in quantitative analysising and qualitative analysising the complex phenomena occurring in the fields of engineering technology,and validation and confirmation of models.Therefore,solutions of nonlinear evolution equations has been one of the key issues in this field of study.Each chapter of this paper starts by looking for the symmetry of nonlinear evolution equations.The classical Lie group method and the modified CK direct reduction method are selected as the main research methods.In addition,the Riccati auxiliary equation method,the power series expansion method,and(G'/G)-expansion method was used as an aid to study a new coupled ZK system,variable coefficient Benjamin-Bona-Mahony-Burg--ers(abbreviated BBMB)equation,(2+1)-dimensional extended Zakharov-Kuznetsov(abbreviated as In the ZK)equation,the corresponding symmetries and some new exact solutions for these types of nonlinear evolution equations are obtained.The first-order differential invariants of the variable-coefficient BBMB equations are also obtained.Moreover,by using the corresponding symmetry,the conservation laws of the new coupled ZK system and the extended(2+1)-dimensional ZK equation are obtained respectively.In chapter one,through the modified direct reduction method presented by Clarkson and Kruskal,we establish a new relation between the old solution of the new coupled ZK system and its new solution,and then get some new exact solutions based on the known solutions.Finally,the infinite conservation laws of the equations are obtained with the corresponding Lie symmetry.In chapter two,we discussed the variable coefficients BBMB equation.Firstly,the Lie symmetry method is performed for it and the continuous equivalence transformations are obtained.Starting with the equivalent algebra,we construct its first order differential invariant.After calculation,it is found that there are eight first-order invariants that are independent of each other.Finally,the general variable coefficient BBMB equations are mapped to the constant coefficient BBM equation or Burgers equation or BBMB equationby the given equivalent transformation.Reference to the solutions of the above kinds of constant coefficient equations,a series of new exact solutions of those variable coefficient equations are obtained.And the special variable coefficients BBM equation,Burgers equation of the exact solution of the images are made.In chapter three,by the application of the classical Lie group method,the symmetry group theorem and reduction equations of the extended Zakharov-Kuznetsov equation.By solving these reduction equations,combined with the homogeneous balance method,power series expansion method and the method of Riccati auxiliary equation,we find a great many of exact solutions,including periodic solutions,rational function solutions,exponential progression solutions and so on.At the same time,through the symmetry and its adjoint equations,the conservation laws of this equation are given.In summary,the main task of this paper is to study the solving problem of nonlinear evolution equations based on the classical Lie group method: to choose the appropriate equivalent transformation,and then to properly reduce and solve the original equation.The application of the classical Lie group method theory to the problem of solving nonlinear evolution equations with variable coefficients is a major feature of this paper.By constructing the differential invariant of the variable coefficient evolution equation,it finds a reasonable equivalent transformation between it and ordinary differential equations,in order to reduce the difficulty of solving the variable coefficient evolution equation,so that we can relatively easily find the solution of some space-time variable coefficient equation.In addition,in order to better explain the nature of the solution,we have made relevant three-dimensional image.