The Theory Of Solitons And Their Application In Bose-Einstein Condensates

Soliton theory is one of the important branches of nonlinear science.It has crescent application in many fields.The soliton perturbation problem is an important part of the soliton theory.It exists in a large number of real nonlinear systems and can be roughly divided into two kinds.One is based on the inverse scattering transformation(IST)which has important academic value.But this technique is inconvenient to those who are not familiar with IST.Another is the direct method where the squared Jost solutions are employed as the basis for perturbation expansion after soliton eqation been linearized.Both of them are just applicable tc integral systems.Professor Jiaren Yan,had developed a direct approach of the perturbation thoery based on separating variable technique,which is applicable to both integrable and unintegrable systems.It is more simple and convenient in method and calculation.Here,We use an improved perturbation method,which is an improvement te a direct approach of perturbation theory for the nonlinear Schr(o|¨)dinger equation based on the separation of variables,to dispose the perturbation on kink solitons. At the same time,we developed the time-dependent perturbation thoery of soliton. Our method is also applied to dispose the soliton's perturbation problem in the system of the Bose-Einstein condensates.Bose-Einstien condensates have been an attractive subjects in recent decades. They not only offer the perfect macrosciopic quantum systems to investigate many foundmental problems in quantum mechanics but also have extensively application foregrounds such as in atom laser and quantum computation.The observations of dark and bright matter wave solitons,the study of matter wave solitons have been one of the hot issues in the fields of both low-temperature physics and condensed matter physics due to their potential application.In the framework of mean-field theory,the Bose-Einstein condensates is govonered by the Gross-Pitaeviskii equation. Based on the coupled Gross-Pitaeviskii equation,we discuss the interaction between the vector solitons.Our work is composed of two parts.The main contents as follows:The fitst part,we give the study to the theory of the solitons.Three main works are introduced.Firstly,we take the perturbtions of the kink soliton as example, illustrate an improvement to a direct approach of perturbation theory for the nonlinear Schr(o|¨)dinger equation based on the separation of variables.This method is put forward for solving the perturbations of dark solitons.We had used it to deal with the perturbations of bright,kink,and dark solitons.We find the results is consistent with that of the primary method,and prove it's validity.Secondly,we developed the time-dependent perturbation theory of KdV soliton,which is also based on a direct approach of perturbation theory of Professor Yan Jiaren.At last,we extend this theory from the first-order perturbation to the second-order perturbation and give their applications on one-component Bose-Einstien condensates. Our results are accordant with that of the perturbation based on the inverse scattering transformation.But it is more simple and convenient in method and calculation.The second part,we discuss the interaction between the solitons in two-componet Bose-Einstien condensates.We introduce three main works:The first work is the interaction between two bright solitons in coupled defocusing nonlinear Schr(o|¨)dinger equation.We discuss the relative motion of the two solitons from analytical analysis and numerical simulation.In the second work,we give categorys of vector solitons in dual-species Bose-Einstein condensates with an interspecies Feshbach resonance.We also discuss their stability and interaction.In the third work,we discuss the influence of the double species Bose-Einstein condensate with tunable interspecies interactions on the interaction of the bright vector solitons.At last,we give a simple summary and discussion to the above-mentioned works.Here,our main works are involved in chapters three,four and six.