周期Toda格点的全局action -角变量

Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions

Symmetries of the periodic Toda lattice are expresssed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Jacobi matrices. Using these symmetries, the phase space of the lattice with Dirichlet boundary conditions is embedded into the phase space of a higher-dimensional periodic lattice. As an application, we obtain a Birkhoff normal form and a KAM theorem for the lattice with Dirichlet boundary conditions.

再论拟周期背景下Toda族的孤子解

Soliton Solutions of the Toda Hierarchy on Quasi-Periodic Backgrounds Revisited

利用双对易法和逆散射变换研究了准周期有限能隙背景下Toda族的孤子解。特别地，我们计算了准周期有限能隙背景上的孤子引起的相移。此外，我们考虑通过散射理论的短程扰动。详细分析了双对易法对散射数据的影响，建立了双对易法下的逆散射变换。

Combinatorial aspects of the conserved quantities of the tropical periodic Toda lattice

The tropical periodic Toda lattice (trop p-Toda) is a dynamical system attracting attentions in the area of the interplay of integrable systems and tropical geometry. We show that the Young diagrams associated with trop p-Toda given by two very different definitions are identical. The first definition is given by a Lax representation of the discrete periodic Toda lattice, and the second one is associated with a generalization of the Kerov-Kirillov-Reshetikhin bijection in the combinatorics of the Bethe ansatz. By means of this identification, it is shown for the first time that the Young diagr…

Graphs emerging from the solutions to the periodic discrete Toda equation over finite fields

The periodic discrete Toda equation defined over finite fields has been studied. We obtained the finite graph structures constructed by the network of states where edges denote possible time evolutions. We simplify the graphs by introducing a equivalence class of cyclic permutations to the initial values. We proved that the graphs are bi-directional and that they are composed of several arrays of complete graphs connected at one of their vertices.

Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

M. Toda in 1967 (\textit{J. Phys. Soc. Japan}, \textbf{22} and \textbf{23}) considered a lattice model with exponential interaction and proved, as suggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exact periodic and soliton solutions. The Toda lattice, as it came to be known, was then extensively studied as one of the completely integrable (differential-difference) non-linear equations which admit exact solutions in terms of theta functions of hyperelliptic curves. In this paper, we extend Toda's original approach to give hyperelliptic solutions of the Toda lattice in ter…

Action-angle map and duality for the open Toda lattice in the perspective of Hamiltonian reduction

An alternative derivation of the known action-angle map of the standard open Toda lattice is presented based on its identification as the natural map between two gauge slices in the relevant symplectic reduction of the cotangent bundle of $GL(n,{\mathbb R})$. This then permits to interpret Ruijsenaars' action-angle duality for the Toda system in the same group-theoretic framework which was established previously for Calogero type systems.

Stability of the Periodic Toda Lattice: Higher Order Asymptotics

In a recent paper we have considered the long time asymptotics of the periodic Toda lattice under a short range perturbation and we have proved that the perturbed lattice asymptotically approaches a modulated lattice. In the present paper we capture the higher order asymptotics, at least away from some resonance regions. In particular we prove that the decay rate is $O(t^{-1/2})$. Our proof relies on the asymptotic analysis of the associated Riemann-Hilbert factorization problem, which is here set on a hyperelliptic curve. As in previous studies of the free Toda lattice, the higher order asymp…

有限非周期Toda格的几何和拓扑观点

The Finite Non-periodic Toda Lattice: A Geometric and Topological Viewpoint

1967年，日本物理学家户田守和（Morikazu Toda）发表了开创性的论文，展示了最近邻粒子之间具有非线性相互作用的粒子链的孤子解。在随后的几十年里，户田的质点系被推广到不同的方向，每一个都有自己的分析、几何和拓扑特征，使它有别于其他粒子。这些被统称为户田晶格。本综述从几何和拓扑的角度描述和比较了有限非周期Toda格的几个版本。

周期Toda格的投影解法和代数几何解法

Solutions of the periodic Toda lattice via the projection procedure and by the algebra-geometric method

在这篇简短的综述中，我们比较了不同的方法来构建周期Toda晶格的解决方案。我们给出了投影方法得出的两个配方，并将它们与克里切弗的代数几何构造进行了比较。

周期性Toda链的大偏差

Large deviations of the periodic Toda chain

本文研究了周期性Toda链在广义Gibbs测度下的谱测度的大偏差原理，建立了与系统自由能相关的速率函数，并证明了在动量受限和自由波动两种情况下的结果。

Dn型周期Toda晶格的分离变量

Separation of variables for the Dn type periodic Toda lattice

我们证明了分离变量的最一般的（Dn型）周期Toda格与2 × 2 Lax矩阵。它是通过找到相应的Baker-Akhiezer函数的适当归一化来实现的。分离变量的所有其他周期户田格与无限系列的根系如下采取适当的限制。

Stability of the periodic Toda lattice under short range perturbations

We consider the stability of the periodic Toda lattice (and slightly more generally of the algebro-geometric finite-gap lattice) under a short range perturbation. We prove that the perturbed lattice asymptotically approaches a modulated lattice. More precisely, let $g$ be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic background, the $n/t$-pane contains $g+2$ areas where the perturbed solution is close to a finite-gap solution in the same isospectral torus. In between there …

周期Toda格的Nekhoroshev定理

Nekhoroshev theorem for the periodic Toda lattice

具有$N$格点的周期Toda晶格是一个双参数$N-1$耦合谐振子族的全局辛同构。动作变量填充了$\R^{N-1}$的整个正象限。我们证明，在内部的正象限以及在附近的起源，户田哈密顿是严格凸的，因此Nekhoroshev定理适用于（几乎）所有部分的相空间。

超离散周期Toda格的热带雅可比矩阵和一般纤维同构

Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic

我们证明了超离散周期Toda格的一般等层集同构于与热带谱曲线相关的热带雅可比矩阵。这一结果意味着作者在前一篇文章中得到的θ函数解是完全解。我们还提出了一种方法来解决初始值问题。

Mutations of the cluster algebra of type $\boldsymbol{A^{(1)}_1}$ and the periodic discrete Toda lattice

A direct connection between two sequences of points, one of which is generated by seed mutations in the cluster algebra of type $A^{(1)}_1$ and the other by time evolutions of the periodic discrete Toda lattice, is explicitly given. In this construction, each of them is realized as an orbit of a QRT map and specialization of the parameters in the maps and appropriate choices of the initial points relate them. The connection with the periodic discrete Toda lattice enables us a geometric interpretation of the seed mutations in the cluster algebra of type $A^{(1)}_1$ as addition of points on an e…

量子周期Toda链本征函数的积分表示

Integral representation for the eigenfunctions of quantum periodic Toda chain

构造了N粒子情况下量子周期Toda链本征函数的积分表示。重积分是使用柯西留数公式计算的。这给出了再现Gutzwiller对N= 2，3和4粒子链所得到的特殊结果的表示。我们的方法解决这个问题的Gutzwiller和R-矩阵方法Sklyanin的思想与经典的结果在理论的Whittaker函数。特别是，我们计算Sklyanin的不变标积从Plancherel公式的惠特克函数导出Semenov-Tian-Shansky从而获得自然的解释Sklyanin措施的Harish-Chandra函数。

球面Whittaker反演定理与量子非周期Toda格

The spherical Whittaker Inversion Theorem and the quantum non-periodic Toda Lattice

本文为Whittaker反演定理的球面情形提供了自足证明，该情形有助于理解连续谱处理，并直接导出量子非周期Toda格的解。

Toda曲线和链的等周期变形，差分Korteweg-de Vries方程，以及SU(N)塞伯格-威滕理论

Isoperiodic deformations of Toda curves and chains, the difference Korteweg - de Vries equation, and $SU(N)$ Seiberg-Witten theories

本文研究了Toda曲线和链的等周期变形，差分Korteweg-de Vries方程，以及SU(N)塞伯格-威滕理论的奇异情形，并给出了受约束的Schlesinger系统的显式解。

Toda和Kac-van Moerbeke族的代数几何拟周期间隙解

Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies

结合代数几何方法和因式分解技术的有限差分表达式，我们提供了一个完整的和自包含的治疗所有实值的拟周期有限间隙解决方案的户田和卡茨-范Moerbeke的层次。为了得到我们的主要新结果，代数几何有限间隙的解决方案的卡茨-范Moerbeke的层次，我们采用特定的交换方法与三浦型变换，使我们能够转移整个类的解决方案（如有限间隙的解决方案）从户田的层次，其修改后的对应，卡茨-范Moerbeke的层次，反之亦然。

N=2 supersymmetric extensions of relativistic Toda lattice

N=2 supersymmetric extensions of both the periodic and non-periodic relativistic Toda lattice are built within the framework of the Hamiltonian formalism. A geodesic description in terms of a non-metric connection is discussed.

Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit

We consider the Fermi-Pasta-Ulam-Tsingou (FPUT) chain composed by $N \gg 1$ particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature $β^{-1}$. Given a fixed ${1\leq m \ll N}$, we prove that the first $m$ integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order $β$, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on…

周期Toda格超椭圆Jacobian实部的拓扑

Topology of the real part of hyperelliptic Jacobian associated with the periodic Toda lattice

本文研究由$2^{N-1}$不同系统组成的${\mathfrak sl}(N)$周期Toda格的等谱{ \it实}流形的拓扑.这些方程组的解包含爆破，并且这些奇点的集合定义了流形的一个导数。然后加上因子，流形紧化为与亏格为$g=N-1$的超椭圆黎曼曲面相关联的$(N-1)$维Jacobi簇的实部。研究了除子的实结构，给出了实Jacobian的仿射部分的拓扑结构，以及基于${\mathfrak sl}(N)$的Weyl群的符号表示的紧化流形的除子上的胶合规则.

广义费米离散Toda族

Generalized fermionic discrete Toda hierarchy

描述了广义费米Toda格族的双Hamilton结构和带谱参数的拉克斯对形式，以及在不同边界条件（包括周期边界条件）下的玻色子对称性和费米对称性.研究了它在不同基（包括正则基）上的两种约化--N=4和N=2超对称Toda格族.它的r-矩阵描述，monodromy矩阵，和光谱曲线进行了讨论。

Direct linearisation of the discrete-time two-dimensional Toda lattices

The discrete-time two-dimensional Toda lattice of $A_\infty$-type is studied within the direct linearisation framework, which allows us to deal with several nonlinear equations in this class simultaneously and to construct more general solutions of these equations. The periodic reductions of this model are also considered, giving rise to the discrete-time two-dimensional Toda lattices of $A_{r-1}^{(1)}$-type for $r\geq 2$ (which amount to the negative flows of members in the discrete Gel'fand--Dikii hierarchy) and their integrability properties.

非阿贝尔二维Toda晶格与带自相容源的矩阵sine-Gordon方程

The non-Abelian two-dimensional Toda lattice and matrix sine-Gordon equations with self-consistent sources

本文建立并求解了带自相容源的非阿贝尔二维Toda晶格，给出两族拟行列式解，并首次通过周期与拟周期约化构造带自相容源的矩阵sine-Gordon方程及其拟行列式精确解。

关于Toda格超可积性的一个注记

A note on the superintegrability of the Toda lattice

在作用角坐标系的框架下解释了非周期Toda晶格的超可积性。此外，还给出了第一积分的一种较简单的形式.

A geometric realization of the ultradiscrete periodic Toda lattice via tropical plane curves

This is a review article on a tropical geometric realization of the ultradiscrete periodic Toda lattice (UD-pTL). Time evolution of the UD-pTL is translated into an addition on the Picard group of its spectral curve, which is a tropical hyperelliptic curve of arbitrary genus depending on the system size. The addition on the Picard group can be realized by using intersection of several tropical plane curves, one of which is the spectral curve. In addition, the tropical eigenvector map, which maps a point in the phase space of the UD-pTL into a set of points on the spectral curve, can also be re…

通用层级的拟周期解

Quasi-periodic solutions of the universal hierarchy

该研究构建了包含多分量KP、Toda层级的通用层级的拟周期解，阐明其适配双线性形式的方式，并给出由黎曼θ函数与二次型指数函数乘积表示的tau函数形式。

Loop代数、规范不变量和一个新的完全可积系统

Loop algebras, gauge invariants and a new completely integrable system

对称为"Toda格“的可积系统族的大量研究的一个富有成效的激励原则是启发式假设，即仿射李代数中的周期Toda格直接类似于有限维李代数中的非周期Toda格。本文指出，类比是不完美的。由于有限维单李代数的结构理论的自然推广不是loop代数的结构理论，而是仿射卡茨-穆迪代数的结构理论，因此产生了矛盾。在本文中，我们使用这种自然的推广，构建自然模拟的非周期Toda格。令人惊讶的是，结果不是周期Toda晶格，而是周期Toda晶格相空间上的一个新的完全可积系统。这个可积系统纯粹是用李群理论的数据来描述的。交换函数正是规范不变函数，通过将圈代数的元素看作$S^1$上的丛上的联络而得到。

Toda格族与Zamolodchikov猜想

Toda Lattice Hierarchy and Zamolodchikov's Conjecture

本文证明了Zamolodchikov在研究二维聚合物时引入的Fredholm行列式$D(λ;t)$是具有2-周期约化条件的Toda格族孤子解的连续极限.

Toda格的一个三哈密顿扩张

A trihamiltonian extension of the Toda lattice

定义了Kupershirt代数的子空间上的一个新的Poisson结构。这种泊松结构，连同其他两个已知的，允许构建一个trihamilton递归的扩展周期户田格与$n$粒子。一些明确的例子的建设和第一个积分发现，以这种方式。

广义Toda场论的点对称性II对称性的应用

Point Symmetries of Generalized Toda Field Theories II Applications of the Symmetries

研究了一类广义Toda场论的Lie对称性，并利用其进行对称性约化。约简一方面导致广义Toda格，另一方面导致周期系统。边界条件的引入，以减少理论上的一个无限的晶格上的半无限，或有限的。

Toda格的双Hamilton几何与分离变量

Bihamiltonian geometry and separation of variables for Toda lattices

我们讨论的双哈密顿几何的户田格（周期和开放）。利用双汉密尔顿流形变量分离的一些最新结果，我们表明这些系统可以通过经典的Hamilton-Jacobi方法在所谓的Darboux-Nijenhuis坐标中显式积分。

关于Toda晶格的Whitham调制方程及其色散冲击的定量表征

On the Whitham modulation equations for the Toda lattice and the quantitative characterization of its dispersive shocks

本文结合已有工作，完整定量描述Toda晶格的周期行波解与Whitham调制方程，填补关键细节，并用于定量表征其色散冲击，最后通过数值模拟验证分析。

相对论Toda链

Relativistic Toda chain

调查是相对论性周期户田链，超本地Weyl代数相关联的每个网站。我们在这里考虑的外尔$q$被限制在单位圆内。该模型的量子Lax算符由六顶点$R$ -矩阵交织而成。这两个独立的巴克斯特的$Q$ -运营商被明确地构建为seria在当地Weyl发电机。还计算了$Q$ -s的算子值wronskian。

Toda格族的几类解

Some Classes of Solutions to the Toda Lattice Hierarchy

我们应用类似的Zakharov-Shabat敷料方法，以获得无限矩阵的解决方案，户田格族。使用算子变换，我们将其中的一些积分算子和Fredholm行列式的解决方案。其他人被转换成一类运营商解决方案的$l$周期Toda的层次。

非Abel Toda格的拟行列式解和矩阵sine-Gordon方程的扭结解

Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation

考虑了广义非Abel Toda格的两类解。这些解决方案表示的拟行列式，通过达布和二元达布变换。作为应用这些解决方案的一个例子，我们考虑的2-周期约化矩阵sine-Gordon方程。特别地，我们研究了极化扭结解的相互作用性质。

Toda格和Holt系统时间的正则变换

Canonical transformations of the time for the Toda lattice and the Holt system

对于Toda格和Holt系统，我们考虑了扩展相空间的正则变换的性质，这些性质保持了可积性。分离的变量在时间变化下是不变的。另一方面，时间的映射引起作用角变量的变换和贝克隆变换的母函数的移位。

二维非定域Toda格点的达布变换、无穷小对称性和守恒律

Darboux Transformations, Infinitesimal Symmetries and Conservation Laws for Nonlocal Two-Dimensional Toda Lattice

将达布变换技术应用于二维周期A_{n-1} Toda晶格的非定域伙伴。该系统被证明是承认的表示作为直接和双重超定线性系统的兼容性条件与量化的谱参数。给出了达布变换技术在这类线性方程组上的推广。给出了超定线性方程组的解与其在奇点邻域的级数展开式之间的关系。得到了非局部Toda格的解、无穷小对称的无穷族和守恒律的解。

Toda格与Riemann型极小曲面

Toda lattice and Riemann type minimal surfaces

本文通过Allen-Cahn方程将Toda格与极小曲面关联，运用可积系统技术寻找新的平衡配置，从而构造出亏格为$j(j+1)/2-1$的新单周期极小曲面，推广了亏格为零的Riemann极小曲面。

非交换可积系统、拟行列式和Marchenko引理

Nonabelian Integrable Systems, Quasideterminants, and Marchenko Lemma

我们发现显式（多孤子）解决方案的nonabelian可积系统，如周期Toda场方程，朗缪尔方程，薛定谔方程的功能与值在任何关联代数。作者在前一篇文章（q-alg/9701008）中用前两位作者引入的拟行列式表示了A、B、C型根系的非交换Toda场方程的解。为了找到多孤子解的周期户田方程和其他nonabelian系统，我们使用这些想法相结合的重要引理，这是由于马尔琴科。

峰、R-矩阵与Toda-格

Peakons, R-Matrix and Toda-Lattice

本文用$r$ -矩阵方法，从Lax表示出发，研究了一类哈密顿系统的可积性，在特殊情况下，描述了N个"峰”（Camassa-Holm方程的特解）的运动.在一般情况下，$r$ -矩阵是一个动态的，并有一个有趣的，但复杂的结构。然而，对于哈密顿量中的相关参数的特定选择（对应于纯"peakons”情况的一个），$r$ -矩阵变得基本恒定，并且简化为与有限（非周期）Toda晶格有关的一个。有趣的后果，这样的财产进行了讨论，并得出一个可积的时间离散。

延迟Lotka-Volterra和延迟Toda格方程的Lax对构造及其约化为延迟Painleve方程

Construction of the Lax pairs for the delay Lotka-Volterra and delay Toda lattice equations and their reductions to delay Painleve equations

本文构造延迟Lotka-Volterra和延迟Toda格方程的Lax对及守恒量，通过空间2-周期约化推导延迟Painleve方程，并给出其自洽版本的行列式解。

Construction of exact solutions to the Ruijsenaars-Toda lattice via generalized invariant manifolds

The article discusses a new method for constructing algebro-geometric solutions of nonlinear integrable lattices, based on the concept of a generalized invariant manifold (GIM). In contrast to the finite-gap integration method, instead of the eigenfunctions of the Lax operators, we use a joint solution of the linearized equation and GIM. This makes it possible to derive Dubrovin type equations not only in the time variable $t$, but also in the spatial discrete variable $n$. We illustrate the efficiency of the method using the Ruijsenaars-Toda lattice as an example, for which we have derived a …

Toda晶格的爆破及其与Bruhat胞的相交

Blow-ups of the Toda lattices and their intersections with the Bruhat cells

研究了实分裂半单李代数$\mathfrak g$上非周期Toda格解的奇点集（blow-ups）的拓扑.爆破的集合被称为Painlevé因子。Toda格的等谱流形通过同伴嵌入被紧化，同伴嵌入将流形映射到与底层李代数$\mathfrak g$相关的旗流形上. Painlevé因子则由紧致流形与旗流形中的Bruhat胞腔的交集给出。本文给出了二秩李代数、$A_2,B_2, C_2, G_2$型和$A_3$型的Painlevé因子的拓扑的显式描述.利用Mumford系统和最初引入的周期Toda格的极限矩阵得到了这些结果。我们还给出了余维为1的Painlevé因子的Lie理论描述，并对一般情况提出了几种解决方法.

Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm

The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight functions in the orthogonality condition. In this paper we extend this mechanism to a new class of two-variable orthogonal polynomials where the variables are related via an elliptic curve. This leads to a `Higher order Analogue of the Discrete-time Toda' (HADT) equation for the associated Hankel determinants, together with its Lax pair, which is derived from t…

Toda格的奇异结构与某些紧李群的上同调

Singular structure of Toda lattices and cohomology of certain compact Lie groups

研究了实分裂半单李代数$\mathfrak g$的Toda格的奇点（爆破）.证明了Toda格沿轨道的爆破点总数由Chevalley群$K({\mathbb F}_q)$的点数给出，该Chevalley群$K({\mathbb F}_q)$与有限域${\mathbb F}_q$上具有$\check{\mathfrak g}={\rm Lie}(\check G)$的群$\check G$的极大紧子群$K$相关。这里$\check{\mathfrak g}$是${\mathfrak g}$的朗兰兹对偶。Toda格的爆破由$τ$ -函数的零点集给出。例如，A型Toda晶格的爆破由与矩形Young图相关的Schur多项式的零点决定。这些Schur多项式是幂零Toda格的$τ$ -函数。然后，我们猜想，爆破的数量也给出了这些舒尔多项式的实根的数量为一个特定的变量。我们还讨论了与仿射Kac-Moody代数相关的旗流形的实上同调的周期Toda格的情况。