Study On Exact Solutions And Dynamic Properties Of Nonlinear Partial Differential Equations

Nonlinear partial differential equations are used to describe the problems in mechanics,control process,ecological and economic systems,chemical cycle systems and epidemiology.They are an important branch of modern mathematics.In this paper,we mainly use Hirota bilinear method,(G’/G)-expansion method,variable coefficient homogeneous balance method,three wave method and symbolic calculation method to study the exact solutions and dynamic properties of nonlinear partial differential equations,including the lump solution,the rogue wave solution and the periodic solution.The main content and arrangement of this paper are as follows:In Chapter 1,some important classifications of exact solutions of nonlinear partial differential equations are introduced,including solitary wave,rogue wave,lump wave and breather.Some basic methods are introduced,including Hirota bilinear method,Bell polynomial and Ba cklund transform.In Chapter 2,the methods and steps of solving the lump solution are introduced.Then,the lump solutions of the(3+1)-dimensional soliton equation are obtained by using this method.The interaction between the lump solution and the soliton and the interaction between the lump solution and the periodic solution are discussed respectively.The lump solutions of(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov equation are obtained,and the interaction between the bump solutions and soliton solutions is discussed.Then,the method of solving the lump solution is modified to make it suitable for solving nonlinear partial differential equations with variable coefficients.This work has not been discussed in other literatures.By using the modified method,the lump solution of the(3+1)-dimensional generalized Kadomtsev Petviashvili(KP)equation with variable coefficients is obtained,and its dynamic properties are analyzed.The lump solutions of the(2+1)-dimensional KP equation with variable coefficients are listed,and the interaction between the lump solution and single soliton and double soliton is discussed.In Chapter 3,a(2+1)-dimensional breaking soliton equation is studied,which describes the(2+1)-dimensional interaction between Riemann waves and long waves propagating along the y-axis.We obtain some new biperiodic soliton solutions of the(2+1)-dimensional breaking soliton equation by using a special ansatz function and Hirota bilinear form,and show the dynamic properties of the solutions through a large number of three-dimensional graphics.In Chapter 4,a(3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation is studied,which has important applications in fluid and plasma dynamics.The long wave propagating along the x-axis can be regarded as a model of incompressible fluid.Based on the(G’/G)-expansion method and symbolic computation,we obtain the exact solutions of the(3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation in the form of hyperbolic and trigonometric functions.The interaction between two solitary waves and specific local excitations is shown by some graphs.In Chapter 5,a(3+1)-dimensional generalized shallow water wave equation is studied,which is widely used in weather simulation,tidal wave,river and irrigation flow,tsunami prediction and so on.Based on the extended variable-coefficient homogeneous balance method and two new ansatz functions,we construct the self-Backlund transformation,non-traveling wave soliton type solutions and multi periodic soliton solutions of the(3+1)-dimensional generalized shallow water wave equation,including periodic cross-kink wave,periodic double solitary wave and breather type solutions of two solitary waves.In addition,the cross-kink triple soliton and cross-kink four soliton solutions are also discussed.In Chapter 6,the exact solutions of the new(3+1)-dimensional generalized KP equation,(2+1)-dimensional ITO equation and the new(2+1)-dimensional Korteweg-de Vries equation are studied by using the three wave method.Based on the three wave method,it is extended to nonlinear partial differential equations with variable coefficients.Taking the(3+1)-Dimensional generalized shallow water wave equation with variable coefficients as an example,a large number of new exact solutions are obtained.In Chapter 7,an improved symbolic calculation method is proposed.By using the improved symbolic computation method,the multi rogue wave solutions of the generalized(2+1)-dimensional Boussinesq equation and variable-coefficient KP equation are obtained.The dynamic characteristics of the obtained multi rogue wave solutions are shown in three-dimensional figures and contour maps.Compared with the original symbolic method,our method does not need to find the Hirota bilinear form of the nonlinear system.In Chapter 8,the main contents and innovative work of this paper are summarized,and the future research direction is prospected.