非局域对称和双线性方法在非线性系统中的应用

发布时间：2018-01-20 22:07

  本文关键词： 非局域对称 双线性方法 可积离散 非线性系统 精确相互作用解 Pfaffian KP约化法 符号计算　出处：《华东师范大学》2016年博士论文　论文类型：学位论文

【摘要】：本文基于符号计算,研究了非线性科学中的对称性、可积性、KP约化、可积离散及其相关应用问题。主要展开了四个方面的工作：研究了耦合非线性可积系统的非局域对称和相关应用；利用Hirota双线性方法发现了一类多分量孤子方程的Pfaffian形式的孤子解,并开发了检验Pfaffian瓜子解的程序包；基于KP理论研究了多分量耦合Yajima-Oikawa(YO)系统的多暗孤子解、混合孤子解和有理解；构造了耦合YO系统的半可积离散形式及其亮、暗孤子解,并提供了连续和半离散可积(复)Sp(m)-invariant massive Thirring models(SMTM)的Pfaffian形式的多孤子解。第一章为绪论部分,重点介绍了对称理论、双线性方法和符号计算的背景与发展现状,并且阐明了本论文的主要工作。第二章研究了耦合的Hirota-Satsuma coupled Korteweg-de Vries(HS-cKdV)系统和modified Generalized Long Dispersive Wave(MGLDW)系统的非局域对称和相关应用。基于Lax对,推导了由谱函数表示的非局域对称。一方面,成功地将非局域对称局域化,并考虑了局域对称的有限变换和相似约化,得到了精确的孤立波和周期波,Painleve波,有理波等复合波的相互作用解。另一方面,构造了初始系统的负梯队与有限维和无限维可积系统。第三章首先利用Hirota双线性方法研究了HS-cKdV方程和Ito方程的多分量扩展系统。利用Pfaffian技巧,证明了孤子解满足的双线性方程即为Pfaffian恒等式。其次,基于双线性方法和Pfaffian技术,开发了一个Maple程序包Pfafftest1:可以直接地计算一般形式的Pfaffian;利用三孤子解条件寻求cmKdV型和cdmKdV型的可积双线性方程。第四章在KP理论基础上,利用双线性方法研究了多分量耦合YO系统的多暗孤子解,混合孤子解和有理解。首先,推导并证明了Gram型和Wronski型行列式形式的N-暗-暗孤子解。暗-暗孤子的碰撞只存在弹性现象并在孤子之间没有能量交换。然后,推导了一维多分量耦合YO系统的N-亮-暗孤子解。在这种混合型孤子中,只有在至少两个短波分量为亮孤子时,这两个短波分量中的两孤子才可能产生非弹性碰撞现象。最后,构造了两维和一维多分量YO系统的显式行列式形式的有理解。基本有理解描述了局域的lump和怪波,其具有三种不同的类型：亮态,亮-暗态和暗态。非基本型的怪波分成两种类型：多怪波和高阶怪波。特别地,考虑不同的参数要求,我们首次报道了两维暗态和亮-暗态的怪波。第五章利用Hirota可积离散方法,构造了耦合YO系统的半可积离散形式。同时,基于半离散BKP族的Backlund变换,推导了半可积离散耦合YO系统的亮和暗孤子的Pfaffian形式解。提供了连续和半离散可积(复)SMTM系统的Pfaffian形式的多孤子解。虽然半可积离散的SMTM系统可以通过离散Lax对方法得到,但利用Hirota可积离散方法,推导了相同的离散格式。第六章对全文工作进行讨论和总结,并对下一步要进行的研究工作做了展望。
[Abstract]:Based on symbolic computation, the symmetry and integrability of KP reduction in nonlinear science are studied in this paper. The main work of this paper is as follows: the nonlocal symmetry and related applications of coupled nonlinear integrable systems are studied; The soliton solutions in Pfaffian form for a class of multicomponent soliton equations are found by using the Hirota bilinear method, and the program package to test the Pfaffian soliton solutions is developed. Based on KP theory, the multi-dark soliton solution, mixed soliton solution and understanding of multi-component coupled Yajima-Oikawai Yo) system are studied. The semi-integrable discrete form of coupled YO system and its bright and dark soliton solutions are constructed. Continuous and semi-discrete integrable (SMTM) are also provided. The first chapter is the introduction. The background and development of symmetry theory, bilinear method and symbolic computation are introduced in detail. In the second chapter, we study the coupled Hirota-Satsuma coupled Korteweg-de Vries. HS-cKdV) system and modified Generalized Long Dispersive WaveMGLDW). Non-local symmetry and related applications of the system. Based on Lax pair. The nonlocal symmetry represented by spectral function is derived. On the one hand, the local symmetry is successfully localized, the finite transformation and similarity reduction of local symmetry are considered, and the exact solitary wave and periodic wave are obtained. Painleve waves, rational waves and other complex wave interaction solutions. On the other hand. The negative echelon and finite and infinite dimensional integrable systems of initial systems are constructed. In chapter 3, the multicomponent extended systems of HS-cKdV equation and Ito equation are studied by using Hirota bilinear method. With the Pfaffian technique. It is proved that the bilinear equation satisfied by soliton solution is Pfaffian identity. Secondly, based on bilinear method and Pfaffian technique. A Maple package, Pfafftest1, is developed: it can directly calculate the general form of Pfaffian; By using the condition of three-soliton solution, the integrable bilinear equations of cmKdV type and cdmKdV type are obtained. Chapter 4th is based on KP theory. The multi-dark soliton solution, mixed soliton solution and understanding of multi-component coupled YO system are studied by bilinear method. The N-dark dark soliton solutions in the form of Gram type and Wronski type determinant are derived and proved. The collision between dark and dark solitons has only elastic phenomena and there is no energy exchange between solitons. The N-bright dark soliton solution of a one-dimensional multicomponent coupled YO system is derived. In this hybrid soliton, only when at least two short-wave components are bright solitons. Only two solitons in these two short-wave components can produce inelastic collisions. The explicit determinant form of two-dimensional and one-dimensional multicomponent YO systems is constructed. The basic understanding describes the local lump and strange waves, which have three different types: bright state. Bright-dark and dark. Non-basic strange waves are divided into two types: multi-odd and high-order strange waves. In particular, different parameter requirements are considered. We report for the first time the strange waves of two dimensional dark states and bright dark states. In Chapter 5th the semi-integrable discrete form of coupled YO systems is constructed by using Hirota integrable discretization method. At the same time. Backlund transform based on semi-discrete BKP family. The Pfaffian forms of bright and dark solitons for semi-integrable discrete coupled YO systems are derived. The continuous and semi-discrete integrable (complex) solutions are provided. Multiple soliton solutions in Pfaffian form for SMTM systems, although semi-integrable discrete SMTM systems can be obtained by discrete Lax pair method. But by using Hirota integrable discretization method, the same discrete scheme is derived. Chapter 6th discusses and summarizes the full text work, and makes a prospect of the next research work.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175
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本文编号：1449562