【作者】 王晓敏；

【导师】 张玲玲；

【作者基本信息】 太原理工大学 ， 生物医学工程， 2018， 博士

【摘要】 非线性薛定谔(Nonlinear Schrodinger简称NLS)方程是描述非线性现象的基本模型之一,孤子是NLS方程中色散作用和非线性作用的平衡结果,它能够在不改变形状、振幅以及速度等性质的情况下长距离传播。孤子理论在非线性光学、凝聚态物理及生物医学等领域被深入的研究,并得到了广泛的应用。近年来,随着科学技术的发展,多分量非线性系统备受关注。在许多实际问题中,需要将标准NLS方程演化为耦合NLS方程来更准确的刻画现实世界中特定的非线性现象。耦合NLS方程的孤子解通常被称为矢量孤子,它们能够表现出更丰富的现象和复杂的动力学性质。在生物医学方面,耦合NLS方程可以描述α-螺旋蛋白质中生物能量传递问题。蛋白质分子中的ATP水解作用释放的能量引起酰胺-I的振动,从而引起晶格的畸变,进而形成孤子。在蛋白质分子的非线性作用下,孤子沿着蛋白质分子链运动,实现能量的传递。由单个方程演化到多个方程耦合的方程组,由于方程数量的增加,导致求解过程更加复杂。因此对耦合NLS方程进行深入的研究,并得到方程精确的孤子解具有深刻的理论意义和广泛的应用价值。本文基于描述α-螺旋蛋白质中传递生物能量的动力学模型的诸多研究结果,对NLS方程在增加耦合个数、增加复杂项和变系数方面加以推广。将理论求解方法应用于较为复杂的耦合NLS方程中,从而得到这些方程的矢量孤子解,并采用渐近分析和图形化分析方法来讨论特定背景下孤子的传播动力学规律及相互作用性质。主要研究内容为:1.研究了具有四波混频项以及具有线性自耦合和交叉耦合项的2-耦合NLS方程模型,利用推广的广田双线性方法,得到了周期孤子解。分析了周期孤子解与一般矢量孤子解如:亮-亮孤子解、亮-暗(暗-亮)孤子解、暗-暗孤子解的关系。讨论了四波混频项以及耦合项对孤子传播的影响。利用渐近分析方法,从理论上严密论证了孤子的弹性碰撞机制。2.研究了描述准一维双组分玻色-爱因斯坦凝聚体的具有任意含时势的2-耦合NLS方程模型。利用广田双线性方法得到了非自治重叠N孤子解。分析了含时势对孤子传播性质的影响,并且讨论了多孤子之间的相互作用性质。3.研究了描述α-螺旋蛋白质中孤子传输动力学的常系数3-耦合NLS方程模型。通过合理的假设得到了模型的矢量孤子解,包括:2-重叠-1-暗孤子,1-亮-2-重叠孤子以及周期孤子。分析了孤子解的传播特性,并利用渐近分析和图形演化讨论了孤子之间的相互作用。4.研究了描述具有非均匀相互作用的α-螺旋蛋白质中孤子传输动力学的变系数3-耦合NLS模型。针对模型变系数的特点,利用合理的变换,得到模型的矢量孤子解。研究了变系数为双曲正割函数时孤子的传播特点。利用所得双孤子解,讨论孤子间相互作用性质,包括:无相互作用传播、周期相互作用和孤子碰撞。更多还原

【Abstract】 It is well known that the nonlinear Schrodinger(NLS)equation is one of the basic models for describing nonlinear phenomena and the soliton is the outcome of a delicate balance between dispersion and nonlinearity for the NLS equation,and it is capable of propagating over long distances without change of shape,amplitude and velocity.The soliton theory has been studied intensively in diverse areas of nonlinear optics,condensed matter physics and biomedical sciences.In recent years,with the development of technology,the investigations of multicomponent nonlinear systems have received much attention.In many practical problems,it is necessary to evolve the standard NLS equation into coupled NLS equations to more accurately depict the specific nonlinear phenomena in the real world.Solitary waves in coupled NLS equations are often called vector solitons and they have more rich phenomena and complex dynamics.The coupled NLS equations can be used to study the energy transfer in α-helical protein in biomedical field.The soliton is transport carrier of the hydrolysis energy of ATP in protein which causes the vibration of amide-I and distortion of the lattice.Then,under the nonlinear action of protein molecules,the soliton moves along the protein molecular chain and achieves energy transfer.Increasing from a single equation to coupled equations can lead to more complex for getting the exact solution of equations.Therefore,studying the coupled NLS equations deeply and attempts to get exact soliton solutions have profound theoretical significance and wide application value.Based on the results of many researches of the model describing bioenergy transfer in the α-helical protein,this dissertation is to generalize of the NLS equation from the aspects including by adding the number of couplings,adding the complex terms and variable coefficients.By using the theoretical approach,vector soliton solutions can be obtained and the dynamic properties of soliton solutions are discussed by asymptotic analysis and graphic simulation.The research results in this dissertation are list as follows:1.Investigations on 2-coupled NLS equations model with four-wave mixing terms and linear self-coupling and cross-coupling terms.The periodic soliton solutions were obtained by the developed Hirota bilinear method.We analyzed the relation of the periodic soliton solutions and general vector soliton solutions such as:bright-bright soli-ton solutions,bright-dark(dark-light)soliton solutions and dark-dark soliton solutions.How the four-wave mixing terms and coupled terms influence the propagate property of the solitons was discussed.Via the asymptotic analysis method,we have theoretically proved the mechanism of elastic collisions.2.Investigations on the 2-coupled NLS equations model with arbitrarily time-dependent potential for describing quasi-one-dimensional two-component Bose-Einstein condensates.We obtained the exact nonautonomous superposition N-soliton solutions analytically by the developed Hirota bilinear method.The effect of time-dependent potential on the propagation properties of soliton was analyzed and the interactions between multiple solitons were discussed.3.Investigations on the constant coefficient 3-coupled NLS equations model describ?ing the soliton dynamics in α-spiral protein.The vector soliton solutions of the model were obtained by reasonable assumptions,including:2-superposition-1-dark solitons,1-bright-2-superposition solitons and periodic solitons.We have analyzed the propagate properties of the soliton solutions and discussed the soliton interactions by asymptotic analysis and graphic simulation.4.Investigations on the variable coefficient 3-coupled NLS model which describing soliton dynamics in the three-spine α-helical protein with inhomogeneous effect.Consid-ering the coefficients of the model were variable,we obtained the vector periodic soliton solution of the model by using reasonable transformation.When the variable coefficient is a hyperbolic secant function,the propagate properties of solitons were discussed.In addition,we have gotten the two solitons and discussed the interactions as follows:prop-agation without interaction,propagation with periodic interaction and soliton collision.更多还原