【作者】 杨波；

【导师】 陈勇；

【作者基本信息】 华东师范大学 ， 软件工程， 2018， 博士

【摘要】 非局域可积非线性方程是当前可积系统领域的研究热点之一,基于Mathematica符号计算平台,我们研究了若干非局域可积模型,主要开展了三个方面的工作:首次构造了若干非局域可积方程的达布变换和精确解,其中包括孤子解,高阶孤子解,(1+1)-维的高阶怪波解,(1+2)-维的线型-多怪波解以及高阶怪波解;分析了精确解的动力学行为,包括了有限时间的爆破,长时间渐进行为以及解的相互作用等等;基于达布变换算法,构造了用于构造非局域可积方程精确解的NonlocSolve1.0程序包.论文的主要内容如下:第一章,绪论部分,从PT对称算子理论出发,简要介绍了非局域可积方程的发现和研究背景,以及达布变换方法和符号计算相关的研究背景和发展现状,并阐述了本文的选题和主要研究内容.第二章,首次构造了偏PT-对称以及全PT-对称非局域DS方程的达布变换,得到了多怪波解和高阶怪波解.在这两个非局域系统中,当时间趋于负无穷时,发现了基本型怪波解在某一个特定时间点产生奇性,其位置发生在空间平面的整个双曲线上.发现了若干基本型怪波的相互作用产生的多怪波解,该解的奇点通常是成对或者是以区间的形式出现的。特别地,首次被发现了三态合一的混合型怪波解,该解是由基本型怪波和暗型(Dark)-反暗型(Anti-Dark)的有理行波解的碰撞生成的.第三章,本章研究了时间反演的的非局域NLS和非局域DS方程.通过达布变换方法,构造了不同类型的怪波解。特别地,和以往经典的DS系统不同的是,发现了非局域DS系统的一个统一的双达布变换公式.分别研究了每个方程怪波解的动力学行为.对于非局域NLS方程,发现了(1+1)-维的怪波解.依据参数范围,这些解可分成两类,一类是全局有界的,另一类则是限时间爆破的.而对于非局域的DS系统,发现了(1+2)-维的线怪波解,这些解同样可以是有界的,也可以是在有限时间内沿着某空间平面上的某特定直线发生爆破.此外,在多怪波和高阶怪波解的动力学结构中,我们发现更加丰富的结构,其中大部分的结构在相应的局域可积方程中都没有出现过.第四章,研究了三种非局域NLS型可积方程的高阶孤子解,其中包括了PT-对称,时间反演以及时间-空间反演的非局域NLS方程.除了不同的扰动散射数据的对称关系,三个方程的广义高阶孤子均可从AKNS族的同一个Riemann-Hilbert解中约化得到.进一步地分析了这些高阶孤子的动力学行为.其中,高阶的基本型孤子可以是非奇性的,或者是重复爆破的.这些孤子用近乎相同的速度在不同的轨线上运动.此外,高阶多孤子解和高阶混合型孤子解的动力学行为揭示出不同于高阶基本型孤子的更加丰富的结构.第五章,我们基于Mathematica计算平台,首次开发了用于非局域可积方程精确解求解的NonlocSolve程序包,可求解NLS-型和DS-型的非局域可积方程的孤子解,怪波解及其有理解.通过多个实例的计算,检验了该程序包的实用性和高效性.第六章,总结和展望部分。更多还原

【Abstract】 The nonlocal integrable nonlinear evolution equations are one of the research hotspot in the filed of integrable system.Via using Mathematica software with symbolic computation,a study of several nonlocal nonlinear models is presented.Our thesis focuses on the following three main aspects: The Darboux matrix for several important nonlocal nonlinear integrable equations are constructed via a Darboux reduction technique,and the corresponding analytic solutions are also derived for those systems,which include the bright soliton solution,the high-order soliton solutions,the(1+1)-dimensional high-order rogue waves,the(1+2)-dimensional multi-and high-order line rogue wave solutions;Dynamics behavior for these exact solutions is further analysed,some novel behaviors are discussed,which includes the finite-time blowing-ups,the long-time asymptotic of solutions and the nonlinear interactions between solutions.Based on the Darboux’s algorithm scheme,the NonlocSolve 1.0 program package is developed,and it can be used to calculate the analytic solutions for several nonlocal integrable equations.The main contents of this thesis are as follows:In the first chapter,which is the introduction part of the whole thesis,we briefly introduced the background as well as the findings for the integrable nonlocal equations.The topic selection and the main content of the research are also elaborated.In the second chapter,the Darboux matrix for the partially and the fully PT-symmetric nonlocal DS equations are constructed for the first time.The multi-and the high-order rogue-wave solutions are derived.In both of these two nonlocal systems,the fundamental rogue waves can develop finite-time singularity on a whole hyperbolic curve in the spatial plane when time approaches negative infinity.Meanwhile,the multi-rogue waves are found,which are generated from nonlinear interactions between several fundamental rogue waves.In this case,their singularities are usually appear in pairs or in an timeinterval.In addition,the hybrid-pattern rogue waves are found for the first time,which are generated from the interactions between fundamental rogue waves and the dark-antidark rational traveling waves.In the third chapter,the time-reverse nonlocal NLS and DS equations are investigated.Using Darboux transformation method,different types of rogue-wave solutions are constructed.Particularly,a unified Darboux transformation for the time-reverse nonlocal DS equations are found,which is quite different from the classical DS system.Dynamics behaviors for those rogue-wave solution are further discussed.The(1+1)-dimensional rogue waves are found in the nonlocal NLS equation.According to the range of the parameters,these solutions can be classified into two categories,one is the globally bounded,the other has finite-time bowing-ups.For the nonlocal DS system,the(2+1)-dimensional line rogue waves are found to be either bounded or develops finitetime singularities along the certain curve in the spatial plane.In addition,richer solution patterns are found in these multi-and high-order rogue-wave solutions,most of them have not been reported before in the local counterparts.In the fourth chapter,the high-order solitons in three recently-proposed nonlocal NLS-type equations are presented,which includes the PT-symmetric,the time-reversal and the space-time-reverse nonlocal NLS equation.In addition to different symmetry relations on the scattering data,general solitons in the three equations ca be reduced from the same Riemann-Hilbet solutions of the AKNS hierarchy.Furthermore,dynamics behaviors for those high-order solitons are further analysed.It is shown that the high-order fundamental solitons can be nonsingular or repeatedly collapsing.These solitons moves along different trajectories with nearly velocities.Moreover,the high-order multi-solitons and the high-order hybird pattern solitons are also analyzed,which reveal some novel and rich dynamic structures that are quite differ from the fundamental pattern solitons.In the fifth chapter,the NonlocSolve program package is developed on the platform of Mathematica software.This program is devoted to construct exact solutions for the NLS-type and the DS-type nonlocal integrable equations.For instance,the soltion solutions,the rogue-wave solutions and some rational solutions.The feasibility and the high efficiency of this program package are verified through several particular cases.In the final chapter,a general summary to this thesis is presented as well as the prospect to our future research work.更多还原