本文選題：孤立子　切入點(diǎn)：可積系統(tǒng)　出處：《華東師范大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：本文基于符號(hào)計(jì)算,分別利用推廣的Darboux變換、經(jīng)典的Darboux變換、雙Darboux變換和對(duì)稱性理論,研究了非線性數(shù)學(xué)物理中若干非線性模型的怪波解、孤子解及可積性.在符號(hào)計(jì)算軟件Maple平臺(tái)上開發(fā)了Darboux變換與怪波求解的自動(dòng)推演程序包.主要內(nèi)容及創(chuàng)新點(diǎn)包括：第一章,緒論部分.主要介紹了怪波、孤子與可積系統(tǒng)、Darboux變換及符號(hào)計(jì)算的背景與研究現(xiàn)狀,并闡明了本文的主要研究結(jié)果.第二章,研究了2+1維非線性模型的經(jīng)典的Darboux變換與孤子解.構(gòu)造了2+1維CDGKS方程和2+1維nKdV方程的經(jīng)典的Darboux變換,給出了兩個(gè)方程的N-孤子解的一般表達(dá)式,分析了亮、暗及扭結(jié)孤子的動(dòng)力學(xué)行為.第三章,研究了與2×2譜問題聯(lián)系的非線性模型的推廣的Darboux變換與怪波解.構(gòu)造了AB系統(tǒng)和Kundu-Eckhaus (KE)方程的推廣的Darboux變換,給出了兩個(gè)方程N(yùn)-階怪波解的統(tǒng)一表達(dá)式.對(duì)于AB系統(tǒng),發(fā)現(xiàn)了“四尖峰”型怪波.對(duì)于KE方程,通過數(shù)值計(jì)算,發(fā)現(xiàn)高次非線性項(xiàng)和Raman散射項(xiàng)只影響怪波的空間分布,而對(duì)怪波出現(xiàn)的時(shí)間和振幅沒有影響.第四章,討論了與3×3譜問題聯(lián)系的非線性模型的推廣的Darboux變換與怪波解.構(gòu)造了耦合Hirota方程、Manakov系統(tǒng)和三波共振(TWR)方程的推廣的Darboux變換,給出了三個(gè)不同類型方程的N-階怪波解的統(tǒng)一表達(dá)式.對(duì)于耦合Hirota方程,發(fā)現(xiàn)了向量形式的高階怪波、高階怪波與多個(gè)暗-亮孤子以及高階怪波與多個(gè)呼吸子相互作用的三種非線性波結(jié)構(gòu).對(duì)于Manakov系統(tǒng),討論了高階怪波與其它非線性波的相互作用性質(zhì).對(duì)于TWR方程,研究了組合型高階怪波的分類問題,并基于數(shù)值模擬的方法,對(duì)高階怪波解的穩(wěn)定性進(jìn)行了分析.第五章,討論了變系數(shù)與離散非線性模型的可積性與孤子解.獲得了2+1維變系數(shù)Gardner方程的Lax對(duì)和共軛Lax對(duì),構(gòu)造了方程的雙Darboux變換,得到了N-孤子解的一般表達(dá)式,并分析了亮、暗孤子和共振亮、暗孤子的動(dòng)力學(xué)行為.研究了四位勢(shì)Blaszak-Marciniak晶格方程的Lie點(diǎn)對(duì)稱、廣義對(duì)稱、守恒律和孤子解,從對(duì)稱的角度證明了模型的可積性.第六章,自動(dòng)推演程序包的開發(fā).在Maple平臺(tái)上開發(fā)了Darboux變換與怪波求解程序包DTRWSI,并以多個(gè)實(shí)例驗(yàn)證了程序包的實(shí)用性和有效性.第七章,總結(jié)與展望部分.對(duì)全文進(jìn)行了總結(jié),并就下一步工作做了展望.
[Abstract]:This paper based on symbolic computation, Darboux transform were used to promote the classical Darboux transform, Darboux transform and dual symmetry theory, study several nonlinear models in nonlinear mathematical physics strange wave solutions, soliton solutions and integrability. Maple software platform developed Darboux transform and wave solution of the automatic deduction procedure of blame wrapped in symbols. The main content and innovations include: the first chapter is the introduction part. This paper mainly introduced the strange wave, soliton and integrable system, the background and research status of Darboux transform and symbolic computation, and expounds the main research results of this paper. The second chapter, Darboux transformation and soliton of 2+1 dimensional nonlinear model of the classic the solution is constructed. Darboux transform 2+1 dimensional CDGKS equation and the 2+1 dimensional nKdV equation classic, the general expression of N- two soliton equations are given, analysis of the bright, dark and kink soliton dynamical behaviors . the third chapter studied with 2 * 2 spectral problem of contact nonlinear model of the generalized Darboux transform and strange wave solutions. To construct the AB system and Kundu-Eckhaus (KE) equation of the generalized Darboux transform, gives two equations of order N- wave solutions. Strange expressions for the AB system, found "four spike type strange wave. For the KE equation, through the numerical calculation, found that higher order nonlinear terms and Raman scattering only affects distribution of strange wave space, but have no influence on the strange wave amplitude and time. The fourth chapter discusses the nonlinear model with 3 * 3 spectral problem linked to the promotion of the Darboux transform and the blame wave solutions constructed coupled Hirota equation, Manakov system and three wave resonance (TWR) equation of the generalized Darboux transform, gives the N- order of three different types of equations of strange wave solutions. A unified expression for the coupled Hirota equation, found high order vector form strange wave, High order strange wave and multiple dark bright soliton and high-order strange waves with multiple breathers interaction of three kinds of nonlinear wave structure. For Manakov system, discusses the high order strange wave and other nonlinear wave interaction properties. For the TWR equation, the classification problem of combination type high order strange wave then, based on the method of numerical simulation, the stability of high order wave solutions of strange analysis. The fifth chapter discusses the variable coefficient nonlinear model and discrete Integrability and soliton solutions. The 2+1 dimensional Gardner equation with variable coefficients of Lax and Lax on the conjugate double Darboux transform, structural equation, general the expression of N- soliton solution is obtained, and the analysis of the bright, bright and dark soliton resonance, dynamics of dark solitons. On four potential Blaszak-Marciniak lattice equation Lie point symmetry, generalized symmetry, conservation and soliton solutions from the symmetric angle proves that the model can be Product. The sixth chapter, the development of the automatic deduction. The package is developed on the platform of Maple Darboux transform and strange wave solver package DTRWSI, and a plurality of examples to verify the practicality and validity of the package. In the seventh chapter, conclusion and outlook. The summary of the thesis, and prospected the next step of work.

【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O175.29
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本文編號(hào)：1621044