New Solutions Of MKDV Hierarchy And MNW Hierarchy And New Algorithm For Solutions Of KdV-type Equation

From the view of mathematics,the soliton is a special kind of solution of nonlinear partial differential equations.With the development of the soliton theory,searching for solitary wave solutions of nonlinear partial differential equations is a meaningful work in soliton theory,which has important theoretical and practical application value,is also a very interesting task.Hirota bilinear method,F-expansion method and exp-function method are three constructive methods for solving nonlinear partial differential equations,such methods don’t depend on the spectral problem or Lax pair of the considered equations,this makes the solving process more direct.Based on the introduction of the research history and development situation of soliton theory,on the one hand this dissertation extends Hirota bilinear method and F-expansion method to new mKdV hierarchy and MNW hierarchy,respectively.On the other hand,this dissertation uses a proposed new algorithm of exp-function method to solve a KdV-type equation.The main work of this dissertation includes:The second chapter solves a new variable-coefficient mKdV hierarchy,which includes the constant-coefficient mKdV hierarchy as the special case,by using Hirota bilinear method.First of all,by introducing the effective transformations,the variable-coefficient mKdV hierarchy is bilinearized and its bilinear forms are obtained.Secondly,based on the obtained bilinear forms new single-soliton solution,double-soliton solution and three-soliton solution of the variable-coefficient mKdV hierarchy are constructed by means of the truncated-expansion technique.At the same time,the general mathematical expression of N-soliton solution is summarized.Besides,evolution behavior characteristics of some obtained solutions are simulated by figures.Based on the detailed steps of F-expansion method for constructing exact solutions ofnonlinear partial differential equations,the third chapter extends the F-expansion method to a new MNW hierarchy.As a result,a lot of new Jacobi elliptic function solutions are obtained.In the limit cases when the module tends to 1 and 0 respectively,many new hyperbolic function solutions and trigonometric function solutions are derived from the obtained Jacobi elliptic function solutions.For convenient discussion of the local spatial structures and the dynamical evolutions of the obtained precise solutions of,this chapter also inserts some two-dimensional and three-dimensional figures.The fourth chapter first summarizes a direct algorithm proposed in this dissertation of the exp-function method,the advantage of this algorithm lies in compared with the original exp-function method the degree of "middle expression expansion" appearing in the process of calculating is smaller.Secondly,in order to find further applications of the algorithm,this dissertation considers a KdV-type equation with variable coefficients.As a result,exact solitary wave solutions containing multiple coefficient functions of this KdV-type equations are obtained.It is shown that the proposed new algorithm of the exp-function method has its advantage and wide range of applications.