Singularly Perturbed And Soliton Solutions Of Schr(?)dinger Equation And Navier-stokes Equation

Soliton is an important part of nonlinear science.Soliton can be described by nonlinear dispersive partial differential equation,the balance between dispersion effect and nonlinear effect is the main reason for soliton.The earliest discovery of solitons comes from the shallow water wave equation.In fact,the soliton can be found by many equations that describing water wave.Navier-Stokes equation is often used to describe the motion of viscous incompressible fluid.It is difficult to find the exact solution of general Navier-Stokes equation.However,combining continuity equation and equation of state,many equations containing soliton can be derived,such as KdV equation,KP equation and nonlinear Schrodinger equation.If the viscosity coefficient of N-S equation is small,the derived equation will also have small parameters,that is,singularly perturbed equation.Because the soliton is the result of both dispersion effect and nonlinear effect,the dispersion term with small parameter will affect the shape and position of soliton.In this paper,the singularly perturbed KdV-Burgers equation,singularly perturbed KP equation and singularly perturbed nonlinear Schrodinger equation are discussed.By using the method of singularly perturbed expansion,the formal asymptotic solution is constructed,and the first term expression of internal and external solutions is obtained.Then,the existence and uniqueness of higher-order solutions of internal and external solutions are proved.Finally,the uniform validity of the solution is proved by the remainder estimation.The main contents are as follows:1.Discussing a class of KdV burgers equations with large Reynolds number and weak dispersion,which is mathematically expressed as a class of singularly perturbed KdV burgers equations.The interaction between the nonlinear term and the dispersion term in KdV Burgers equation forms a stable forward propagating soliton,through mathematical analysis,the propagation path and propagation speed of soliton are described.By using the singularly perturbed expansion method,the asymptotic solution of the problem is constructed.First of all,the degenerate solution is obtained by using the Riemann-ershawn method,and the simple wave is obtained:There is a velocity difference between any point of the simple wave shape and the initial point,which makes the wave form continuously distorted in the process of propagation,and finally forms the shock wave surface,namely discontinuity.There is a jump in the velocity of particles on both sides of it,and it changes with time;Secondly,constructing a modified traveling wave transformation by substituting variables at the discontinuous surface of the degenerate solution.obtaining soliton solutions of the expansion of internal solutions and proving the existence and uniqueness of the internal and external solutions.Finally,the residual term is estimated by the existence of uniformly bounded inverse operator,and the uniform efficiency of the asymptotic solution is obtained.The results show that the perturbations of the KdV-Burgers equation with large Reynolds number and weak dispersion are concentrated near the discontinuities of the degenerate solutions.The soliton links the particles on both sides,and its propagation path is not a linear form of time and space,but propagates along the discontinuity of the degenerate solution,forming a stable waveform.2.Discussing a class of two-dimensional KdV Equations with weak dispersion,which is mathematically expressed as a class of singularly perturbed KP equations.By using the singularly perturbed expansion method,constructing the asymptotic solution of the problem.First of all,by using the Riemann-ershawn method,the degenerate solution is solved,and the simple wave is obtained.The simple wave will produce shock wave and form discontinuity in the process of propagation.Secondly,a modified traveling wave transformation is constructed by replacing variables at the discontinuity of the degenerate solution.The expansion of internal solution is obtained.For the equation with the first term of internal solution,the single soliton solution is obtained by direct integration method,and the 2-soliton solution is obtained by using Hirota bilinear method and singular perturbation expansion method.Finally,the L2 estimator of the asymptotic solution is obtained by estimating the remainder term,and the uniform validity of the asymptotic solution is proved.3.Discussed a class of nonlinear Schrodinger equation with out of band potential.The interaction between nonlinear effect and dispersion effect in NLS equation forms stable forward propagating optical soliton,which plays an important role in the field of communication.By using the singularly perturbed expansion method,constructing the asymptotic solution of the problem.Firstly,two kinds of expansion methods are used to solve the external solution,and the expression of the external solution is obtained.The first term of the external solution has a critical point,and the critical point is only related to the external potential.Secondly,the internal solution is constructed at the initial time,and the correction term between the external first term and the initial value is obtained.Thirdly,the internal solution is constructed at the critical point and initial time of the external solution.By using the traveling wave transformation method,the first term of the internal solution is solved by traveling wave transformation method,and the soliton is obtained,and the existence of the external solution and the internal solution is proved.Finally,the residual term is estimated by using the extremum principle,and the uniform validity of the asymptotic solution is proved.The results show that the difference between the initial value and the first term of the external solution of the nonlinear Schrodinger equation with external potential does not lead to soliton.The soliton will be generated at the extreme point of the external potential,which shows that the influence of the external potential on the original NLS equation is to change the position of the soliton.In the process of research,we synthetically applied the knowledge of ordinary differential equation,partial differential equation,mathematical and physical equation,nonlinear acoustics,mathematical analysis,singular perturbation theory and so on,and obtained soliton related conclusions...