基于符号计算的可积系统的若干问题研究
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摘要
基于符号计算,本文分别从Bell多项式的角度、Riccati-型伪势的角度和非局域对称的角度对非线性数学物理中一些重要的非线性演化方程的可积性和对称性问题进行研究,并在符号计算平台Maple上开发了KdV-型方程可积性自动推演的程序包.另外,基于屠格式理论和源生成方法,研究了一类超可积方程族的自相容源和无穷守恒律.主要内容如下：
第一章,介绍与本文研究内容有关的可积系统与符号计算的研究背景和发展现状,并阐述本文的主要研究工作.
第二章,将Bell多项式方法推广到一些(1+1)-维、(2+1)-维和变系数的非线性演化方程可积性的研究.对于修正的广义Vakhnenko方程,系统地构造了其双线性形式、双线性Backlund变换、Lax对、N-孤子解和拟周期解,并通过图形对解进行了生动的刻画.进一步,通过引入辅助变量以及与之相关的辅助约束条件,研究了广义KdV-fKdV-型方程和变系数KdV-CBS-型方程的可积性,特别的,得到了它们相应的Darboux协变Lax对和无穷守恒律.
第三章,基于Bell多项式方法,设计了一个构造KdV-型方程双线性形式、双线性Backlund变换、Lax对和无穷守恒律的系统算法.基于该算法,在符号计算平台Maple上编写了实现该算法的程序包PDEBellII特别的,程序包PDEBellII也适用于mKdV-型方程双线性形式的自动推演.
第四章,基于Riccati-型伪势理论和非局域对称理论研究非线性演化方程的可积性和对称性.利用Riccati-型伪势方法,构造了变系数fKdV方程的Lax对、AKNS形式的Lax对、自-Backlund变换和奇异流形方程；利用五阶Lax方程的Riccati-型伪势和广义KP方程的Lax对,分别得到了它们相应的的非局域对称,通过对非局域对称进行局域化,进一步研究了它们相应的有限对称变换和对称约化.
第五章,研究了一类超可积方程族的自相容源和无穷守恒律.从一个超可积方程族出发,利用源生成方法构造了其自相容源；同时,通过引入两个新的变量,将谱问题转化为Riccati-型方程,得到其无穷守恒律.
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.
Based on symbolic computation, the integrablility and symmetry of some significant nonlinear evolution equations in nonlinear mathematical physics are investigated by Bell polynomial, by Riccati-typed pseudopotential and by nonlocal symmetry, respectively. Meanwhile, a software package which be used for investigating the integrablility of KdV-typed equation is developed. Furthermore, based on the Tu scheme and source generation procedure, the self-consistent sources and infinite conservation laws of a kind of super integrable equation hierarchy are obtained. The main work is carried out as follows:
In chapter1, an introduction is devoted to review the research background and the current situation related to the dissertation, which including integrable systems and sym-bolic computation. The main works of this dissertation are also illustrated.
In chapter2, the Bell polynomial approach is extended to investigate the integrablil-ity of some (1+1)-dimensional,(2+1)-dimensional and variable coefficient nonlinear evo-lution equations. For the modified generalised Vakhnenko equation, its corresponding bilinear representation, bilinear Backlund transformation, Lax pair, N-soliton solutions and quasiperiodic solution are systematically obtained. Meanwhile, relevant properties of the solutions are illustrated graphically. Moreover, the integrablility of the generalized KdV-fKdV-typed equation and variable coefficient KdV-CBS-typed equation are inves-tigated by introduce an auxiliary variable and impose a subsidiary constraint condition. In particular, the Darboux covariant Lax pairs and infinite conservation laws of the two equations are obtained.
In chapter3, based on the Bell polynomial approach, a systematic algorithm is pro-posed to obtain the bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws of the KdV-typed equation. Based on this algorithm, a corresponding software package PDEBellll in Maple is developed. In particular, the software package PDEBellll is also suitable for the seeking of the bilinear form of the mKdV-typed equation.
In chapter4, based on the Riccati-typed pseudopotential theory and nonlocal sym-metry theory, the integrablility and symmetry of some nonlinear evolution equations are investigated. The Lax pair, AKNS-typed Lax pair, auto-Backlund transformation and sin-gularity manifold equation of the variable coefficient fKdV equation are obtained by using the Riccati-typed pseudopotential theory. The nonlocal symmetries of the fifth order Lax equation and generalized KP equation are obtained by their Riccati-typed pseudopoten-tial and Lax pair. Meanwhile, their corresponding finite symmetry transformation and symmetry reduction are also investigated.
In chapter5, The self-consistent sources and infinite conservation laws of a kind of super integrable equation hierarchy are obtained. For a kind of super integrable equa-tion hierarchy, its self-consistent sources are obtained by source generation procedure; Meanwhile, by introduce two new variable, its corresponding infinite conservation laws are obtained by change the spectral problem to a Riccati-typed equation.
In chapter6, the summary and discussion of this dissertation are given, as well as the outlook of future work is discussed.
第一章,介绍与本文研究内容有关的可积系统与符号计算的研究背景和发展现状,并阐述本文的主要研究工作.
第二章,将Bell多项式方法推广到一些(1+1)-维、(2+1)-维和变系数的非线性演化方程可积性的研究.对于修正的广义Vakhnenko方程,系统地构造了其双线性形式、双线性Backlund变换、Lax对、N-孤子解和拟周期解,并通过图形对解进行了生动的刻画.进一步,通过引入辅助变量以及与之相关的辅助约束条件,研究了广义KdV-fKdV-型方程和变系数KdV-CBS-型方程的可积性,特别的,得到了它们相应的Darboux协变Lax对和无穷守恒律.
第三章,基于Bell多项式方法,设计了一个构造KdV-型方程双线性形式、双线性Backlund变换、Lax对和无穷守恒律的系统算法.基于该算法,在符号计算平台Maple上编写了实现该算法的程序包PDEBellII特别的,程序包PDEBellII也适用于mKdV-型方程双线性形式的自动推演.
第四章,基于Riccati-型伪势理论和非局域对称理论研究非线性演化方程的可积性和对称性.利用Riccati-型伪势方法,构造了变系数fKdV方程的Lax对、AKNS形式的Lax对、自-Backlund变换和奇异流形方程；利用五阶Lax方程的Riccati-型伪势和广义KP方程的Lax对,分别得到了它们相应的的非局域对称,通过对非局域对称进行局域化,进一步研究了它们相应的有限对称变换和对称约化.
第五章,研究了一类超可积方程族的自相容源和无穷守恒律.从一个超可积方程族出发,利用源生成方法构造了其自相容源；同时,通过引入两个新的变量,将谱问题转化为Riccati-型方程,得到其无穷守恒律.
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.
Based on symbolic computation, the integrablility and symmetry of some significant nonlinear evolution equations in nonlinear mathematical physics are investigated by Bell polynomial, by Riccati-typed pseudopotential and by nonlocal symmetry, respectively. Meanwhile, a software package which be used for investigating the integrablility of KdV-typed equation is developed. Furthermore, based on the Tu scheme and source generation procedure, the self-consistent sources and infinite conservation laws of a kind of super integrable equation hierarchy are obtained. The main work is carried out as follows:
In chapter1, an introduction is devoted to review the research background and the current situation related to the dissertation, which including integrable systems and sym-bolic computation. The main works of this dissertation are also illustrated.
In chapter2, the Bell polynomial approach is extended to investigate the integrablil-ity of some (1+1)-dimensional,(2+1)-dimensional and variable coefficient nonlinear evo-lution equations. For the modified generalised Vakhnenko equation, its corresponding bilinear representation, bilinear Backlund transformation, Lax pair, N-soliton solutions and quasiperiodic solution are systematically obtained. Meanwhile, relevant properties of the solutions are illustrated graphically. Moreover, the integrablility of the generalized KdV-fKdV-typed equation and variable coefficient KdV-CBS-typed equation are inves-tigated by introduce an auxiliary variable and impose a subsidiary constraint condition. In particular, the Darboux covariant Lax pairs and infinite conservation laws of the two equations are obtained.
In chapter3, based on the Bell polynomial approach, a systematic algorithm is pro-posed to obtain the bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws of the KdV-typed equation. Based on this algorithm, a corresponding software package PDEBellll in Maple is developed. In particular, the software package PDEBellll is also suitable for the seeking of the bilinear form of the mKdV-typed equation.
In chapter4, based on the Riccati-typed pseudopotential theory and nonlocal sym-metry theory, the integrablility and symmetry of some nonlinear evolution equations are investigated. The Lax pair, AKNS-typed Lax pair, auto-Backlund transformation and sin-gularity manifold equation of the variable coefficient fKdV equation are obtained by using the Riccati-typed pseudopotential theory. The nonlocal symmetries of the fifth order Lax equation and generalized KP equation are obtained by their Riccati-typed pseudopoten-tial and Lax pair. Meanwhile, their corresponding finite symmetry transformation and symmetry reduction are also investigated.
In chapter5, The self-consistent sources and infinite conservation laws of a kind of super integrable equation hierarchy are obtained. For a kind of super integrable equa-tion hierarchy, its self-consistent sources are obtained by source generation procedure; Meanwhile, by introduce two new variable, its corresponding infinite conservation laws are obtained by change the spectral problem to a Riccati-typed equation.
In chapter6, the summary and discussion of this dissertation are given, as well as the outlook of future work is discussed.
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