Study On The Soliton Solution And The Stability Of The Dispersion Equation With Typical Single And Double Singular Lines

Nonlinear phenomena are ubiquitous in nature,and the nonlinear science of nonlinear phenomena is closely related to all kinds of disciplines.Many complex problems can be modeled by nonlinear systems,so the research of nonlinear systems is particularly important.Soliton theory is an important branch of nonlinear research,and it is a hot topic in nonlinear science.The study of solitary wave solution of nonlinear system is helpful to understand the motion change in the system,thus revealing the essential law that concealed in the surface phenomena,which has great application values in the field of physics and engineering technology.In the past few decades,with the development of computer hardware and software technology,the research methods in the field of the applied mathematics and engineering have been innovated,our computing ability has been greatly improved,and our drawing ability has also been strengthened,which can be viewed from all directions and from multiple angles,and can also enter into the micro field in part of the image.It also greatly improves the ability of solving and drawing about nonlinear evolution equations,which makes us go further in the study of solitons.In this paper,the exact traveling wave solution of the nonlinear dispersive wave equation is studied.By using the bifurcation method of dynamic system theory and the geometric singular perturbation theory,the nonlinear evolution equation with singular lines is discussed and studied.The abundant solitary wave solutions with parameter variation in the interior of the equation are shown.The explicit expression of the solution is given,and the three-dimensional drawing is made.At the same time,the stability of the partial solitary wave solution under the disturbance of time delay is studied and the corresponding results are obtained.The specific work is as follows:In Chapter 1 and 2,the introduction and basic theory are given.The research background,research progress and current situation of nonlinear evolution equation are summarized.The soliton theory and its main research methods and the first integration method of dynamic system used in this paper are introduced.At the sametime,the elliptic integral function which is often used in the process of solving the exact solution is introduced.In Chapter 3,we study the two-component Degasperis-Procesi equation with single singular line.The singular traveling wave system is transformed into a regular dynamic system by time scale transformation.Because of the typicality of the two components Degasperis-Procesi equation with single singular line,the most detailed analysis and discussion of the equation are carried out,and the exact solitary wave solution and image are fully displayed.Through the discussion of the range of parameters variation,we get the abundant exact traveling wave solutions,including kink and anti kink solutions,compacton solutions,anti-compacton solutions,peakon solutions,valleyon solutions,periodic compacton solutions,periodic anti-compacton solutions,periodic peakon solutions,periodic valleyon solutions,loop solutions,antiloop solution,periodic loop solution,some unbounded solutions and new discontinuous solutions and periodic solutions appearing in the second variable txv),(.The dynamic properties of these solutions correspond to the conditions satisfied by the parameters,and in the process of parameter continuous change,we can see how the solution changes correspondingly.In Chapter 4,the traveling wave solutions of two-component Degasperis-Procesi equation with two singular lines are studied qualitatively.At this time,the first integral is no longer rational.We use the qualitative theory of differential equation to transform singular system into regular system.According to the qualitative properties of the phase diagram orbit of the regular system of two-component DP equation,we can judge the abundant solitary wave solutions of the equation,For example,kink and anti kink solutions,compacton solutions,anti-compacton solutions,peakon solutions,valleyon solutions,periodic compacton solutions,periodic anti-compacton solutions,periodic peakon solutions,periodic valleyon solutions,loop solutions,anti-loop solutions,periodic loop solutions,etc.,and under the condition that parameters take some special values,the exact expressions of solitary wave solutions are obtained.In Chapter 5,we study the traveling wave solution of the generalized osmosisdispersion K(2,2)equation.By using the bifurcation method of dynamic system theory,we analyze its dynamic properties,discuss the phase diagram orbit of the system,and obtain loop solutions,periodic loop solutions,kink and anti kink solutions,peakon solutions,periodic sharp wave solutions,smooth solitary wave solutions,periodic smooth solitary wave solutions and some unbounded solutions of the equation.At the same time,through the dynamic behavior of the system,the generation mechanism of the spiking solitary wave solution is discussed,and the changes of the periodic spiking wave solution and the smooth solitary wave solution are obtained when the parameters change.They are transformed into spiking solitary wave solution together.Finally,it is shown that the dispersive perturbation term does not change the distribution of the original solution.In Chapter 6,the traveling wave solution of the generalized dispersion Degasperis-Procesi equation is studied.The phase diagram orbit of the system is analyzed by the bifurcation method of the dynamic system theory,and the abundant and accurate solutions of the generalized dispersion Degasperis-Procesi equation are obtained,such as loop solutions,periodic loop solutions,kink and anti kink solutions,peakon solutions,periodic sharp wave solutions,smooth solitary wave solutions,periodic smooth solitary wave solutions and some unbounded solutions.At the same time,the generation mechanism of the spiking solitary wave solution is discussed.Finally,the comparison of the solutions shows that the dispersion disturbance term does not change the distribution of the original solution.In Chapter 7,we study the existence of kink and anti kink solutions of Schr?dinger equation with time-delay disturbance.When the distributed delay kernel is strong,the equation with time-delay disturbance is transformed into a four-dimensional ordinary differential system without delay.Because the delay coefficient is small enough,the four-dimensional ordinary differential system is a standard singularly perturbed system.By using singular perturbation theory and Melnikov function method,it is proved that there are kink wave and anti kink wave solutions for Schr?dinger equation with time delay under the condition of(27)(27)(27)?10,(10)-(28)Oc)(1 ?.In Chapter 8,we study the existence of periodic wave solutions of Schr?dinger equation with time-delay disturbance.By using singular perturbation theory and Melnikov function method,it is proved that there are kink wave and anti kink wave solutions for Schr?dinger equation with time delay under some conditions.In Chapter 9,we summarize the whole paper and puts forward the prospect.