The Research On The Dbar Problem And Its Analytical Solution For Several Classes Of Nonlinear Differential Equations

As far as we know,the soliton solutions of nonlinear partial differential equations play a vital role in revealing the physical significance of the equations.Therefore,it is of great significance to study the solutions of nonlinear differential equations to provide theoretical support for explaining the corresponding physical phenomena.This dissertation aims to use the Dbar-dressing method to study the soliton solutions and dynamic behaviors of several important types of partial differential equations.In the first chapter,this thesis mainly introduces the Dbar dressing method,including the research background,significance,current research status at home and abroad,and the general research content of this article.In the second chapter,mainly provides some of the theorems used in this thesis.In the third chapter,starting from a 5 ×5 local matrix Dbar problem,we successfully derive a hierarchy of the nonlinear evolution equations by Dbar-dressing method,which includes the nonlinear Schr（?）dinger equation（n=2）,vector modified Kortewegde Vries（KdV）equation（n=3）and Lakshmanan-Porsezian-Danielvia equation（n=4）via introducing a suitable recursion operator Λⁿ.In addition,we employ the Dbar-dressing method to find the N-soliton solutions of the vector modofied KdV equation.Finally,the influences of each parameters on interactions among solitons are discussed,and the influence of the characteristic lines on the relative position of the waves is also analyzed,and the method of controlling the propagation direction is given.In the fourth chapter,starting from two 2 x 2 matrix Dbar problems,we successfully derive two spatial and two time singular spectral problems via two linear constraint systems.We first derive the conservation laws of the couple nonlocal modified KdV equation by considering the temporal linear spectral problem.Then,the Dbar-dressing method is employed to study the relationship between the potential matrix Q and the solution of Dbar problem.At last,we successfully construct the N-soliton solutions of the equation.Concerning coupled nonlocal modified KdV equation,the results of the nonlocal mKdV can be degraded from the above conclusions.In the fifth chapter,the（2+1）dimensional Gardner equation is studied by developing the Dbar-dressing method.We first introduce the eigenfunction and Green’s function of the equation based on its Lax pair.By analyzing the Green’s function,we found that it does not jump on the real axis,but the characteristic function is not analytical on the z-plane,so we adopted the Dbar-dressing method.Analyzing the singularity of Green’s function through the Residue Theorem.Further,we understand that the Dbar problem can be used to establish scattering equations and solve inverse scattering problems.Therefore,we systematically construct solutions to the Gardner equation.Finally,the specific expression of the solution is given by determining the variation laws of scattering data with t.In the sixth chapter,starting from a 2×2 local matrix Dbar problem,we studies the positive flow short pulse equation,gives the Lax pairs of the positive flow short pulse equation,and successfully derives a hierarchy of nonlinear evolution equations by introducing a recursive operator Λⁿ.Finally,reconstruct the solution of the positive flow short pulse equation.Generally,when considering the Dbar problem,one should consider its dispersion part,which is divided into polynomial part Ω_p=α_nzⁿσ₃ and singularity part.Here,the α_n is a constant associated,but in this chapter,we are considering α_n is a function associated with x,t,which adds another layer of constraint to the spectral transformation matrix R,making it more difficult to handle,we fully adopt the concept of transformation to transform unfamiliar problems into familiar ones.Finally,we provided a brief summary of the research content of in the entire article and the prospects for further work.There are 32 figures and 134 references in this thesis.