【作者】 俞慧友；

【导师】 颜家壬；

【作者基本信息】 湖南师范大学 ， 凝聚态物理， 2009， 博士

【摘要】 孤子理论是非线性科学的一个重要分支,它在物理学的许多领域中有着日益广泛的应用。孤子的微扰是孤子理论中最有实用价值的重要内容。它大体可以分为两大类。一是建立在逆散射变换基础上的孤子微扰理论。它在理论上有着重要的学术价值,但其思路较迂回曲折,数学计算较繁。另一种直接微扰沦较为系统的方法是将孤子方程线性化后再按Jost函数的平方作微扰展开。这两种方法均只适用于可积系统。近年来,颜家壬教授发展了一种基于分离变量法的孤子微扰理论法,它适用于可积和非可积系统,而且思路和计算较为简便。在此,本文主要基于颜家壬教授的直接孤子微扰方法,通过改进,处理了扭结孤子的微扰问题。同时发展了孤子含时微扰理论。我们也应用这种系统的微扰方法处理了玻色-爱因斯坦凝聚中的孤子微扰问题。玻色-爱因斯坦凝聚也是近几十年来被广泛关注的课题。它不仅提供了一个研究量子力学基本问题的宏观系统,也在原子激光,量子计算等领域有着重要的应用前景。玻色-爱因斯坦凝聚中的暗,亮物质波孤子的成功观测及它们的潜在的应用前景,也使玻色-爱因斯坦凝聚中的物质波孤子成为了当前低温物理和凝聚态物理研究领域的研究热点之一。本人主要是在平均场理论的框架下,以耦合Gross-Pitaevskii方程为主要模型,讨论了其中的多种孤子相互作用问题。全文工作共分为两部分,主要内容如下:第一部分为孤子理论方面,主要介绍三个工作。一、以扭结孤子为例,阐述对基于分离变量法的孤子微扰理论的方法改进。该方法主要是针对暗孤子微扰问题的解决而改进的。我们已用它处理了亮孤子,扭结孤子,暗孤子问题。在亮孤子和扭结孤子微扰问题处理中,发现它的结果与原方法所得结果一致,证明了其有效和正确性。二、基于颜家壬教授的直接微扰理论,我们发展了KdV孤子含时微扰理论。三、将孤子微扰理论由一阶扩展到二阶微扰,并用于一分量玻色-爱因斯坦凝聚中的孤子实际问题。我们所得结果,与前人用逆散射所得结果一致。但方法更清晰,计算更简单。第二部分探讨两分量玻色-爱因斯坦凝聚中的孤子相互作用问题。主要介绍三个工作:一、耦合散焦非线性薛定谔方程中的孤子相互作用问题。从解析和数值模拟方面讨论了孤子之间的相对运动情况。二、对可调节的双种类玻色-爱因斯坦凝聚中的矢量孤子类型的分类,以及稳定性和相互作用情况的讨论。三、对可调节的双种类玻色-爱意斯坦凝聚中,可调种间相互作用,对亮亮孤子相互作用的影响的讨论。最后对本文做了简单的总结和对我们所研究的领域前景的展望。我们的研究工作集中在三、四、六章。更多还原

【Abstract】 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.更多还原