本文關鍵詞： 非局域對稱 雙線性方法 可積離散 非線性系統(tǒng) 精確相互作用解 Pfaffian KP約化法 符號計算　出處：《華東師范大學》2016年博士論文　論文類型：學位論文

【摘要】：本文基于符號計算,研究了非線性科學中的對稱性、可積性、KP約化、可積離散及其相關應用問題。主要展開了四個方面的工作：研究了耦合非線性可積系統(tǒng)的非局域對稱和相關應用；利用Hirota雙線性方法發(fā)現了一類多分量孤子方程的Pfaffian形式的孤子解,并開發(fā)了檢驗Pfaffian瓜子解的程序包；基于KP理論研究了多分量耦合Yajima-Oikawa(YO)系統(tǒng)的多暗孤子解、混合孤子解和有理解；構造了耦合YO系統(tǒng)的半可積離散形式及其亮、暗孤子解,并提供了連續(xù)和半離散可積(復)Sp(m)-invariant massive Thirring models(SMTM)的Pfaffian形式的多孤子解。第一章為緒論部分,重點介紹了對稱理論、雙線性方法和符號計算的背景與發(fā)展現狀,并且闡明了本論文的主要工作。第二章研究了耦合的Hirota-Satsuma coupled Korteweg-de Vries(HS-cKdV)系統(tǒng)和modified Generalized Long Dispersive Wave(MGLDW)系統(tǒng)的非局域對稱和相關應用�；贚ax對,推導了由譜函數表示的非局域對稱。一方面,成功地將非局域對稱局域化,并考慮了局域對稱的有限變換和相似約化,得到了精確的孤立波和周期波,Painleve波,有理波等復合波的相互作用解。另一方面,構造了初始系統(tǒng)的負梯隊與有限維和無限維可積系統(tǒng)。第三章首先利用Hirota雙線性方法研究了HS-cKdV方程和Ito方程的多分量擴展系統(tǒng)。利用Pfaffian技巧,證明了孤子解滿足的雙線性方程即為Pfaffian恒等式。其次,基于雙線性方法和Pfaffian技術,開發(fā)了一個Maple程序包Pfafftest1:可以直接地計算一般形式的Pfaffian;利用三孤子解條件尋求cmKdV型和cdmKdV型的可積雙線性方程。第四章在KP理論基礎上,利用雙線性方法研究了多分量耦合YO系統(tǒng)的多暗孤子解,混合孤子解和有理解。首先,推導并證明了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彈性現象并在孤子之間沒有能量交換。然后,推導了一維多分量耦合YO系統(tǒng)的N-亮-暗孤子解。在這種混合型孤子中,只有在至少兩個短波分量為亮孤子時,這兩個短波分量中的兩孤子才可能產生非彈性碰撞現象。最后,構造了兩維和一維多分量YO系統(tǒng)的顯式行列式形式的有理解�；居欣斫饷枋隽司钟虻膌ump和怪波,其具有三種不同的類型：亮態(tài),亮-暗態(tài)和暗態(tài)。非基本型的怪波分成兩種類型：多怪波和高階怪波。特別地,考慮不同的參數要求,我們首次報道了兩維暗態(tài)和亮-暗態(tài)的怪波。第五章利用Hirota可積離散方法,構造了耦合YO系統(tǒng)的半可積離散形式。同時,基于半離散BKP族的Backlund變換,推導了半可積離散耦合YO系統(tǒng)的亮和暗孤子的Pfaffian形式解。提供了連續(xù)和半離散可積(復)SMTM系統(tǒng)的Pfaffian形式的多孤子解。雖然半可積離散的SMTM系統(tǒng)可以通過離散Lax對方法得到,但利用Hirota可積離散方法,推導了相同的離散格式。第六章對全文工作進行討論和總結,并對下一步要進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.
【學位授予單位】：華東師范大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O175
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本文編號：1449562