Research On B(?)cklund Transformation?Nonlocal Symmetry And Analytical Solutions Of Some Nonlinear Differential Equations

In this thesis,several kinds of nonlinear differential equations are taken as research objects,and several types of nonlinear wave solutions with different characteristics are constructed by developing Hirota bilinear method.And analyzing its formation characteristics and propagation characteristics,and then explaining their important physical meaning.At the same time,this thesis also uses the symmetry theory to study the nonlocal symmetry,group invariant solutions and conservation laws of nonlinear differential equations.Studying these nonlinear differential equations can be used to describe nonlinear phenomena in many fields.The main research contents of this paper are as follows.Chapter 1 briefly introduces the background,significance and relevant theories of this field,and describes the main contents of this paper.Chapter 2 mainly based on the Hirota bilinear,simple Bell polynomial theory is introduced,to promote them to the(3+1)-dimensional variable-coefficient BKP(vcBKP)equation and the(3+1)-dimensional Kadomtsev-Petviashvil(KP)equation,the bilinear form respectively.B(?)cklund transformation of the equation is obtained by using its bilinear,and on this basis,the exponential wave solution and rational solution of the equation are obtained.Secondly,this chapter also extends Hirota's bilinear method to obtain a new Lump solution to the equation for the first time.Finally,the solution of the interaction between a Lump soliton and a block solution is discussed.The propagation characteristics of these solutions are also analyzed.In chapter 3,the Hirota bilinear method is generalized,the(2+1)-dimensional breaking soliton equation is studied,and the dynamic behavior is analyzed through the three-dimensional graph.We get a more general form of the lump solution,and use the obtained lump solution to give the motion path,and draw the wave propagation characteristics.Then,we continue to construct the lumpoff solution by inducing solitons to cut the lump solution.By selecting appropriate parameters,their propagation and evolution diagrams are drawn to analyze their dynamic behavior.Finally,the resonant soliton is used to cut the lump wave to generate a special rogue wave solution.According to the image simulation,we find that when the lump wave reached a large amplitude,it became a special rogue wave.The results show that the amplitude of the wave height is related to the lump wave and the induced soliton,and the special rogue wave can be predicted by tracking the locus of the lump wave.In chapter 4,based on the ansatz-solution form,two new types of precise brightdark soliton solutions with nonlinear Schr(?)dinger's governing equations and their existence conditions are studied for the first time.The appropriate parameters are used to simulate the time-varying propagation of bright and dark soliton waves with Maple software.The Gaussian soliton solution to this equation is also obtained for the first time.Secondly,this chapter also studies the bright soliton solution of(3+1)-dimensional modified Korteweg-de Vries-Kadomtsev-Petviashvil(mKdV-KP)equation and the conditions under which it exists.Finally,the bright and dark soliton solutions and their existence conditions of the(3+1)-dimensional variable-coefficient BKP(vc-BKP)equation are studied,and their propagation is analyzed.By selecting appropriate parameters and analyzing its propagation evolution diagram,it can be seen that the optical field energy of the fiber bright solitons is mainly concentrated near the center of the beam cross section.At this time,the center light of the beam is the strongest,and the light intensity gradually decreases along the propagation direction,and the light intensity approaches zero at infinity away from the center.On the contrary,the fiber dark soliton has a fading in the uniform background light.At this time,the energy at the center of the beam is the smallest.The intensity of the light at an infinite distance from the center of the beam tends to a constant value.Chapter 5 mainly studies the nonlocally symmetric,soliton-ellipse periodic wave interaction solutions and the conservation laws of the Jaulent-Miodek equation.First,by developing a truncated Painlev(?) expansion method,we obtain nonlocal symmetry and Schwarizan form symmetry groups.Then by introducing the extension transformation of the appropriate extension system,nonlocal symmetry can be localized to Lie point symmetry.The Riccati expansion method is used to verify that the equations are compatible Riccati solvable.The soliton-ellipse cosine function wave solution of the equation is constructed by introducing the Jacobian elliptic function.By selecting appropriate parameters,the dynamic behavior of soliton-elliptic periodic wave solutions is simulated.According to the analysis,when the phase changes,the interaction between the soliton and the peaks of the elliptic periodic wave is elastic.Finally,Ibragimov's new conservation law is used to obtain conservation law of the Jaulent-Miodek equation.Chapter 6 gives a brief summary of this paper and looks forward to the future worth of in-depth thinking and research.