从Rosochatius型可积系统到孤子方程
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摘要
本文研究连续与离散的Rosochatius型有限维可积系统的构造,拉直与求解,并揭示它们与孤子方程的关系.主要利用代数曲线的工具以及母函数的技术.
对于连续情形,具体研究了Neumann-Rosochatius系统与Garnier-Rosochatius系统.大致思路和结果如下：首先,从变形的Lax矩阵出发,借助于母函数的技术,构造了相应的Rosochatius类型的Hamilton系统族.接着对这族Rosochatius型Hamilton系统进行“批处理”,通过引入代数曲线与Abel-Jacobi坐标,一举将整族Rosochatius流拉直,用准确无误的线性无关性证明了守恒积分的函数独立性.从而证明了整族Rosochatius系统的Liouville完全可积性.与此同时,也获得了该族系统的直接求积—获得了Rosochatius型可积系统在Jacobi簇上的线性形式解.最后,揭示了Rosochatius型可积系统与KdV方程的关系,即它们构成了KdV方程的一个新的可积分解.在此基础上,根据两个Rosochatius流的相容解,获得了KdV方程的拟周期解.
对于离散情形,我们提出并研究了辛映射的Rosochatius变形.我们给出了一个构造辛映射的Rosochatius变形的方法.通过该方法,获得了Toda辛映射,Volterra辛映射,以及Ablowitz-Ladik辛映射的Rosochatius变形.并且找到了它们的Lax表示,进而用r-矩阵方法证明了它们的可积性.不仅如此,我们还以Rosochatius型Toda辛映射为例、研究了它们与离散的孤子方程—Toda方程的关系.通过构造特殊的亚纯函数并借助于Abel定理,我们成功拉直了Rosochatius型辛映射.最后,通过Jacobi反演与Riemannθ函数,我们获得了Toda方程的拟周期解.
In this thesis, the Rosochatius-type systems (both continuous and discrete) are inves-tigated by virtue of algebraic-geometric tools and generating function technique. These systems are straightened out in the Jacobi variety of the associated hyperelliptic curve. Also the relation between these systems and the soliton equations are revealed. As an ap-plication, the quasi-periodic solution of the KdV equation and Toda equation is obtained in the context of the Rosochatius hierarchy.
In the continuous case, we first introduce a hierarchy of Hamiltonian systems re-lated to the Rosochatius-type system (Rosochatius hierarchy for short) based on the deformed Lax matrix. Then we establish the integrability of the Rosochatius hierarchy and straighten out these systems in the Abel-Jacobi coordinates of the associated hyper-elliptic curve. Next, we reveal the relation between these systems and the KdV equation, namely, the first two Rosochatius flows constitute a new integrable decomposition of KdV equation. At last, we calculate the quasi-periodic solution of the KdV equation by the Riemann-theta function and Jacobi inversion.
In the discrete case, we propose a method of generating the integrable Rosochatius-type deformations for the integrable symplectic maps. Making use of the method, we obtain the integrable Rosochatius-type deformations of the Toda symplectic map, the Volterra symplectic map and the Ablowitz-Ladik symplectic map. We find their Lax rep-resentations and thus establish their integrabilities in the sense of Liouville with the aid of r-matrix method. Furthermore, we take the Rosochatius-type Toda symplectic map as an illustrative example to show the relation between the Rosochatius-type symplectic maps and the discrete soliton equations. We succeeded in straightening out the Rosochatius-type Toda symplectic map. Based on the new Rosochatius-type integrable decomposition of the Toda equation, we obtain the quasi-periodic solution of the Toda lattice equation.
对于连续情形,具体研究了Neumann-Rosochatius系统与Garnier-Rosochatius系统.大致思路和结果如下：首先,从变形的Lax矩阵出发,借助于母函数的技术,构造了相应的Rosochatius类型的Hamilton系统族.接着对这族Rosochatius型Hamilton系统进行“批处理”,通过引入代数曲线与Abel-Jacobi坐标,一举将整族Rosochatius流拉直,用准确无误的线性无关性证明了守恒积分的函数独立性.从而证明了整族Rosochatius系统的Liouville完全可积性.与此同时,也获得了该族系统的直接求积—获得了Rosochatius型可积系统在Jacobi簇上的线性形式解.最后,揭示了Rosochatius型可积系统与KdV方程的关系,即它们构成了KdV方程的一个新的可积分解.在此基础上,根据两个Rosochatius流的相容解,获得了KdV方程的拟周期解.
对于离散情形,我们提出并研究了辛映射的Rosochatius变形.我们给出了一个构造辛映射的Rosochatius变形的方法.通过该方法,获得了Toda辛映射,Volterra辛映射,以及Ablowitz-Ladik辛映射的Rosochatius变形.并且找到了它们的Lax表示,进而用r-矩阵方法证明了它们的可积性.不仅如此,我们还以Rosochatius型Toda辛映射为例、研究了它们与离散的孤子方程—Toda方程的关系.通过构造特殊的亚纯函数并借助于Abel定理,我们成功拉直了Rosochatius型辛映射.最后,通过Jacobi反演与Riemannθ函数,我们获得了Toda方程的拟周期解.
In this thesis, the Rosochatius-type systems (both continuous and discrete) are inves-tigated by virtue of algebraic-geometric tools and generating function technique. These systems are straightened out in the Jacobi variety of the associated hyperelliptic curve. Also the relation between these systems and the soliton equations are revealed. As an ap-plication, the quasi-periodic solution of the KdV equation and Toda equation is obtained in the context of the Rosochatius hierarchy.
In the continuous case, we first introduce a hierarchy of Hamiltonian systems re-lated to the Rosochatius-type system (Rosochatius hierarchy for short) based on the deformed Lax matrix. Then we establish the integrability of the Rosochatius hierarchy and straighten out these systems in the Abel-Jacobi coordinates of the associated hyper-elliptic curve. Next, we reveal the relation between these systems and the KdV equation, namely, the first two Rosochatius flows constitute a new integrable decomposition of KdV equation. At last, we calculate the quasi-periodic solution of the KdV equation by the Riemann-theta function and Jacobi inversion.
In the discrete case, we propose a method of generating the integrable Rosochatius-type deformations for the integrable symplectic maps. Making use of the method, we obtain the integrable Rosochatius-type deformations of the Toda symplectic map, the Volterra symplectic map and the Ablowitz-Ladik symplectic map. We find their Lax rep-resentations and thus establish their integrabilities in the sense of Liouville with the aid of r-matrix method. Furthermore, we take the Rosochatius-type Toda symplectic map as an illustrative example to show the relation between the Rosochatius-type symplectic maps and the discrete soliton equations. We succeeded in straightening out the Rosochatius-type Toda symplectic map. Based on the new Rosochatius-type integrable decomposition of the Toda equation, we obtain the quasi-periodic solution of the Toda lattice equation.
引文
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