Alexander I. Aptekarev, Maxim Derevyagin, Hiroshi Miki, Walter Van Assche
In this paper we present multidimensional analogues of both the continuous- and discrete-time Toda lattices. The integrable systems that we consider here have two or more space coordinates. To construct the systems, we generalize the orthogonal polynomial approach for the continuous and discrete Toda lattices to the case of multiple orthogonal polynomials.
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Hermite-Padé approximation, multiple orthogonal polynomials, and multidimensional Toda equations
本文综述了Hermite-Padé逼近、多重正交多项式与多维Toda方程之间的联系，并介绍了向插值问题和非交换代数的推广。
The negative symmetry flows are incorporated into the Riemann-Hilbert problem for the homogeneous $A_m$-hierarchy and its $\hat{gl} (m+1, C)$ extension. A loop group automorphism of order two is used to define a sub-hierarchy of $\hat{gl} (m+1, C)$ hierarchy containing only the odd symmetry flows. The positive and negative flows of the $\pm 1$ grade coincide with equations of the multidimensional Toda model and of topological-anti-topological fusion.
In 1974 Jürgen Moser has shown that the classical Moment Problem plays a fundamental role for the theory of completely integrable systems, by proving that the simplest case of the finite Toda lattice is described exhaustively in its terms. In particular, the Jacobi matrix defined by the Flaschka variables corresponds to the Schrödinger operator related to the Korteweg-de Vries operator. In the present paper we use a recent breakthrough in the area of the multidimensional Moment Problem in order to develop a multidimensional generalization of the finite Toda lattice. Our construction is based o…
We associate to an arbitrary $\mathbb Z$-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati equations. The multidimensional extension of these equations is given. The generalisation of the associated Redheffer--Reid differential systems appears in a natural way. The connection between the Toda systems and the Riccati-type equations in lower and higher dimensions is established. Within this context the integrability problem for those equations is studied…
We study the interpolation analogue of the Hermite-Padé type I approximation problem. We provide its determinant solution and we write down the corresponding integrable discrete system as an admissible reduction of Hirota's discrete Kadomtsev-Petviashvili equations. Apart from the $τ$-function form of the system we provide its variant, which in the simplest case of dimension two reduces to the non-autonomous discrete-time Toda equations.
A box-ball system with more than one kind of balls is obtained by the generalized periodic discrete Toda equation (pd Toda eq.). We study the pd Toda equation in view of algebraic geometry. The time evolution of pd Toda eq. is linearized on an algebraic variety, and theta function solutions are obtained.
Toda and Laguerre-Freud equations for multiple discrete orthogonal polynomials with an arbitrary number of weights
本文将半经典多重离散正交多项式的研究扩展到任意数量权重的情形，并为多重广义Charlier族和多重广义Meixner II族推导出多重版本的Toda方程与Laguerre-Freud方程。
The discrete autonomous/non-autonomous Toda equations and the discrete Lotka-Volterra system are important integrable discrete systems in fields such as mathematical physics, mathematical biology and statistical physics. They also have applications to numerical linear algebra. In this paper, we first simultaneously obtain their general solutions. Then, we show the asymptotic behavior of the solutions for any initial values as the discrete-time variables go to infinity. Our two main techniques for understanding the distinct integrable systems are to introduce two types of discrete-time variable…
Combinatorial expressions are presented to the solutions to initial value problems of the discrete and ultradiscrete Toda molecules. For the discrete Toda molecule, a subtraction-free expression of the solution is derived in terms of non-intersecting paths, for which two results in combinatorics, Flajolet's interpretation of continued fractions and Gessel--Viennot's lemma on determinants, are applied. By ultradiscretizing the subtraction-free expression, the solution to the ultradiscrete Toda molecule is obtained. It is finally shown that the initial value problem of the ultradiscrete Toda mol…
In this paper we investigate flows on discrete curves in $\C^2$, $\CP^1$, and $\C$. A novel interpretation of the one dimensional Toda lattice hierarchy and reductions thereof as flows on discrete curves will be given.
We reformulate the singularity confinement of the discrete Toda equation. We prove the co-primeness property, which has been introduced in our previous paper (arXiv:1311.0060) as one of the integrability criteria, for the discrete Toda equation. We study three types of boundary conditions (semi-infinite, molecule, periodic) for the discrete Toda equation, and prove that the same co-primeness property holds for all the types of boundaries. (v2: typos corrected, final version to appear in J. Math. Phys.)
The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semi-infinite matrix that models the shifts by $\pm 1$ in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well known fact that moments of the semiclassical weights are logarithmic derivatives of ge…
We introduce a tropical analogue of Fay's trisecant identity for a special family of hyperelliptic tropical curves. We apply it to obtain the general solution of the ultra-discrete Toda lattice with periodic boundary conditions in terms of the tropical Riemann's theta function.
Arithmetic aspects of discrete periodic Toda flows
本文通过Mumford的代数描述，提出离散周期Toda流的新线性化方法，并揭示其与周期盒球流的p-进描述及数论的联系。
We present a class of solutions of the two-dimensional Toda lattice equation, its fully discrete analogue and its ultra-discrete limit. These solutions demonstrate the existence of soliton resonance and web-like structure in discrete integrable systems such as differential-difference equations, difference equations and cellular automata (ultra-discrete equations).
We provide a representation of the $C^*$-algebra generated by multidimensional integral operators with piecewise constant kernels and discrete ergodic operators. This representation allows us to find the spectrum and to construct the explicit functional calculus on this algebra. The method can be useful in various applications since many discrete approximations of integral and differential operators belong to this algebra. Some examples are also presented: 1) we construct an explicit functional calculus for extended Fredholm integral operators with piecewise constant kernels, 2) we find a wave…
The entropic uncertainty measures of the multidimensional hydrogenic states quantify the multiple facets of the spatial delocalization of the electronic probability density of the system. The Shannon entropy is the most adequate uncertainty measure to quantify the electronic spreading and to mathematically formalize the Heisenberg uncertainty principle, partially because it does not depend on any specific point of their multidimensional domain of definition. In this work the radial and angular parts of the Shannon entropies for all the discrete stationary states of the multidimensional hydroge…
Toda and Laguerre-Freud equations and tau functions for hypergeometric discrete multiple orthogonal polynomials
本文研究离散多重正交多项式，通过tau函数和带状矩阵描述其性质，证明其提供Toda方程多分量可积扩展及Nijhoff-Capel方程的解，并探讨Laguerre-Freud方程在多种特例下的应用。
This paper presents a study of the discrete Toda equation $(τ_n^t)^2+τ_{n-1}^tτ_{n+1}^t=τ_n^{t-1}τ_n^{t+1}$, that was introduced in 1977. In this paper, it has been proved that the algebraic solution of the discrete Toda equation, obtained via the Lax formalism, is naturally related to the dual Grothendieck polynomial, which is a $K$-theoretic generalization of the Schur polynomial. A tropical permanent solution to the ultradiscrete Toda equation has also been derived. The proposed method gives a tropical algebraic representation of static solitons. Lastly, a new cellular automaton realization…
An explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete Toda lattice is presented. First, the time evolution is realized as a point addition on a hyperelliptic curve, which is the spectral curve of the periodic discrete Toda lattice, then the point addition is translated into curve intersections. Next, it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the periodic discrete Toda lattice. Finally, the formulation is lifted to the framework of tropical geometry, and a t…
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.
We characterize the law of the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice. The proof is via a multidimensional generalization of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion. It is based on a mapping which can be regarded as a geometric variant of the RSK correspondence.
We study the relation between the periodic Benjamin-Ono equation with discrete Laplacian and the two dimensional Toda hierarchy. We introduce the tau-functions tau_pm(z) for the periodic Benjamin-Ono equation, construct two families of integrals of motion {M_1,M_2,cdots}, {overline{M}_1,overline{M}_2,cdots}, and calculate some examples of the bilinear equations using the Hamiltonian structure. We confirmed that some of the low lying bilinear equations agree with the ones obtained from a certain reduction of the 2D Toda hierarchy.
In this paper, we discuss several concepts of the modern theory of discrete integrable systems, including: - Time discretization based on the notion of Bäcklund transformation; - Symplectic realizations of multi-Hamiltonian structures; - Interrelations between discrete 1D systems and lattice 2D systems; - Multi-dimensional consistency as integrability of discrete systems; - Interrelations between integrable systems of quad-equations and integrable systems of Laplace type; - Pluri-Lagrangian structure as integrability of discrete variational systems. All these concepts are illustrated by the di…
Dynamics of the ultra-discrete Toda lattice via Pitman's transformation
本文通过将超离散Toda晶格的配置编码为梯度交替为-1和1的分段线性路径，展示系统动力学可由Pitman变换的移位版本描述，并扩展至无限配置，对不变测度研究有重要意义。
A versatile method is described for the practical computation of the discrete Fourier transforms (DFT) of a continuous function $g(t)$ given by its values $g_{j}$ at the points of a uniform grid $F_{N}$ generated by conjugacy classes of elements of finite adjoint order $N$ in the fundamental region $F$ of compact semisimple Lie groups. The present implementation of the method is for the groups SU(2), when $F$ is reduced to a one-dimensional segment, and for $SU(2)\times ... \times SU(2)$ in multidimensional cases. This simplest case turns out to result in a transform known as discrete cosine t…
Non-Abelian discrete Toda chains and related lattices
该研究推导了具有拟行列式解与多形式Lax对的二维离散Toda格非阿贝尔类似物，探讨了其与多类低维格点非阿贝尔模型的关联，并研究了离散Toda方程到Somos-N序列再到离散Painlevé方程体系的非交换类比。
We rigorously derive the long-time asymptotics of the Toda shock wave in a middle region where the solution is asymptotically finite gap. In particular, we describe the influence of the discrete spectrum in the spectral gap on the shift of the phase in the theta-function representation for this solution. We also study the effect of possible resonances at the endpoints of the gap on this phase. This paper is a continuation of research started in [arXiv:2001.05184].
We construct integrable discrete nonautonomous quad-equations as Bäcklund auto-transformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by using transformations of discrete equations which are invertible on their solutions. In this way we obtain integrable examples of different types: discrete analogs of the sine-Gordon equation, the Liouville equation and the dressing chain of Shabat. For Liouville type equations we construct general solutions, using a specific linearization. For sine-Gordon type …
Bi-Hamiltonian structure and Lax pair formulation with the spectral parameter of the generalized fermionic Toda lattice hierarchy as well as its bosonic and fermionic symmetries for different (including periodic) boundary conditions are described. Its two reductions --N=4 and N=2 supersymmetric Toda lattice hierarchies-- in different (including canonical) bases are investigated. Its r-matrix description, monodromy matrix, and spectral curves are discussed.
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.
In order to study the invariant measures of discrete KdV- and Toda-type systems, this article focusses on models, discretely indexed in space and time, whose dynamics are deterministic and defined locally via lattice equations. A detailed balance criterion is presented that, amongst the measures that describe spatially independent and identically/alternately distributed configurations, characterizes those that are temporally invariant in distribution. A condition for establishing ergodicity of the dynamics is also given. These results are applied to various examples of discrete integrable syst…
We study the Toda equations in the continuous level, discrete level and ultradiscrete level in terms of elliptic and hyperelliptic $σ$ and $ψ$ functions of genera one and two. The ultradiscrete Toda equation appears as a discrete-valuation of recursion relations of $ψ$ functions.
We introduce a so-called `coprimeness-preserving non-integrable' extension (another terminology is `quasi-integrable' extension) to the two-dimensional Toda lattice equation. We believe that this equation is the first example of such discrete equation defined over a three-dimensional lattice. We prove that all the iterates of the equation are irreducible Laurent polynomials of the initial data and that every pair of two iterates is co-prime, which indicate confined singularities of the equation. By reducing the equation to two- or one-dimensional lattices, we obtain coprimeness-preserving non-…
The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated to an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov--Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov--Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method …
We present a set of uniform polynomial equations that provides multidimensional on-lattice higher-order models of the lattice Boltzmann theory, while keeping compact the number of discrete velocities. As examples, we explicitly derive two- and three-dimensional on-lattice models applicable to describing thermal compressible flows of the accuracy levels of the Navier-Stokes equations with smaller numbers of discrete velocities in comparison to the existing models. We demonstrate the accuracy and stability of the three-dimensional model by using the Riemann problem.
We will give a short introduction to discrete or lattice soliton equations, with the particular example of the Korteweg-de Vries as illustration. We will discuss briefly how Bäcklund transformations lead to equations that can be interpreted as discrete equations on a $\mathbb Z^2$ lattice. Hierarchies of equations and commuting flows are shown to be related to multidimensionality in the lattice context, and multidimensional consistency is one of the necessary conditions for integrability. The multidimensional setting also allows one to construct a Lax pair and a Bäcklund transformation, which …
An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and non-local families of R-matrix solutions to the modified Yang-Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.
We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is a natural extension of the ideas known by the names "Lagrangian multiforms" and "Pluri-Lagrangian systems", which have previously been established in both the fully discrete and fully continuous cases. The main feature of these ideas is to capture a hierarchy of commuting equations in a single variational principle. Our main example to illustrate the new semi-discrete theory of Lagrangian multiforms is the Toda lattice. This ODE describes the evolution in continuous time …
Discrete non-commutative hungry Toda lattice and its application in matrix computation
本文提出一种基于非交换可积系统与矩阵值正交多项式的离散非交换饥饿Toda格，用于块Hessenberg矩阵的特征值计算，并给出收敛性分析与数值验证。
We propose a new integrable generalization of the Toda lattice wherein the original Flaschka-Manakov variables are coupled to newly introduced dependent variables; the general case wherein the additional dependent variables are vector-valued is considered. This generalization admits a Lax pair based on an extension of the Jacobi operator, an infinite number of conservation laws and, in a special case, a simple Hamiltonian structure. In fact, the second flow of this generalized Toda hierarchy reduces to the usual Toda lattice when the additional dependent variables vanish; the first flow of the…
Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings
本文研究了KdV和Toda型离散可积系统的双无穷解，通过路径编码方法证明了初始值问题的唯一解，并揭示了系统行为与经典Pitman变换的关联。
A connection between matrix orthogonal polynomials and non-abelian integrable lattices is investigated in this paper. The normalization factors of matrix orthogonal polynomials expressed by quasi-determinant are shown to be solutions of non-abelian Toda lattice in semi-discrete and full-discrete cases. Moreover, with a moment modification method, we demonstrate that the Bäcklund transformation of non-abelian Toda given by Popowicz is equivalent to the non-abelian Volterra lattice, whose solutions could be expressed by quasi-determinants as well.
A nonautonomous version of the ultradiscrete hungry Toda lattice with a finite lattice boundary condition is derived by applying reduction and ultradiscretization to a nonautonomous two-dimensional discrete Toda lattice. It is shown that the derived ultradiscrete system has a direct connection to the box-ball system with many kinds of balls and finite carrier capacity. Particular solutions to the ultradiscrete system are constructed by using the theory of some sort of discrete biorthogonal polynomials.
A periodic box-ball system (pBBS) is obtained by ultradiscretizing the periodic discrete Toda equation (pd Toda eq.). We show the relation between a Young diagram of the pBBS and a spectral curve of the pd Toda eq.. The formula for the fundamental cycle of the pBBS is obtained as a colloraly.
We present the q-deformed multivariate hypergeometric functions related to Schur polynomials as tau-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of the hypergeometric functions are the higher times of those hierarchies. The discrete Toda lattice variable shifts parameters of hypergeometric functions. The role of additional symmetries in generating hypergeometric tau-functions is explained.
As it is shown in previous works, discrete periodic operators with defects are unitarily equivalent to the operators of the form $$ {\mathcal A}{\bf u}={\bf A}_0{\bf u}+{\bf A}_1\int_0^1dk_1{\bf B}_1{\bf u}+...+{\bf A}_N\int_0^1dk_1...\int_0^1dk_N{\bf B}_N{\bf u},\ \ {\bf u}\in L^2([0,1]^N,\mathbb{C}^M), $$ where $({\bf A},{\bf B})(k_1,...,k_N)$ are continuous matrix-valued functions of appropriate sizes. All such operators form a non-closed algebra ${\mathscr H}_{N,M}$. In this article we show that there exist a trace $\pmbτ$ and a determinant $\pmbπ$ defined for operators from ${\mathscr H}_…
Coprimeness property was introduced to study the singularity structure of discrete dynamical systems. In this paper we shall extend the coprimeness property and the Laurent property to further investigate discrete equations with complicated pattern of singularities. As examples we study extensions to the Somos-$4$ recurrence and the two-dimensional discrete Toda equation. By considering their non-autonomous polynomial forms, we prove that their tau function analogues possess the extended Laurent property with respect to their initial variables and some extra factors related to the non-autonomo…
The iterative methods to diagonalize matrices and many-body Hamiltonians can be reformulated as flows of Hamiltonians towards diagonalization driven by unitary transformations that preserve the spectrum. After a comparative overview of the various types of discrete flows (Jacobi, QR-algorithm) and differential flows (Toda, Wegner, White) that have been introduced in the past, we focus on the random XXZ chain with random fields in order to determine the best closed flow within a given subspace of running Hamiltonians. For the special case of the free-fermion random XX chain with random fields, …
We introduce a class of recursions defined over the $d$-dimensional integer lattice. The discrete equations we study are interpreted as higher dimensional extensions to the discrete Toda lattice equation. We shall prove that the equations satisfy the coprimeness property, which is one of integrability detectors analogous to the singularity confinement test. While the degree of their iterates grows exponentially, their singularities exhibit a nature similar to that of integrable systems in terms of the coprimeness property. We also prove that the equations can be expressed as mutations of a see…
By introducing some special bi-orthogonal polynomials, we derive the so-called discrete hungry quotient-difference (dhQD) algorithm and a system related to the QD-type discrete hungry Lotka-Volterra (QD-type dhLV) system, together with their Lax pairs. These two known equations can be regarded as extensions of the QD algorithm. When this idea is applied to a higher analogue of the discrete-time Toda (HADT) equation and the quotient-quotient-difference (QQD) scheme proposed by Spicer, Nijhoff and van der Kamp, two extended systems are constructed. We call these systems the hungry forms of the h…
High rank solutions to the 2D Toda Lattice System are constructed simultaneously with the effective calculation of coefficients of the high rank commuting ordinary difference operators. Our technic is based on the study of discrete dynamics of Tyurin Parameters characterizing the stable holomorphic vector bundles over the algebraic curves (Riemann Surfaces).
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