Aika Tsunematsu, Kenta Nakata, Yuta Tanaka, Ken-ichi Maruno
Integrable delay analogues of the two-dimensional Toda lattice equation are presented and their muti-soliton solutions are constructed by applying the delay reduction to the Gram determinant solution.
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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。
We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotka-Volterra, Toda lattice, and sine-Gordon equations and their multi-soliton solutions. It is carried out by applying a reduction and delay-differential limit to the discrete KP or discrete two-dimensional Toda lattice equations. Each of the delay-difference and delay-differential equations has the N-soliton solution, which depends on the delay parameter and converges to an N-soliton solution of a known soliton equation as the delay parameter ap…
This paper focuses on different reductions of 2-dimensional (2d-)Toda hierarchy. Symmetric and skew symmetric moment matrices are firstly considered, resulting in the differential relations between symmetric/skew symmetric tau functions and 2d-Toda's tau functions, respectively. Furthermore, motivated by the Cauchy two-matrix model and Bures ensemble from random matrix theory, we study the rank one shift condition in symmetric case and rank two shift condition in skew symmetric case, from which the C-Toda hierarchy and B-Toda hierarchy are found respectively, together with their special Lax ma…
We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the intermediate long wave hierarchy, and the remaining flows coincide with a certain limit of the flows of the fractional Volterra hierarchy which controls the special cubic Hodge integrals.
We apply the direct method of obtaining reductions to the Toda hierarchy of equations. The resulting equations form a hierarchy of ordinary differential difference equations, also known as delay-differential equations. Such a hierarchy appears to be the first of its kind in the literature. All possible reductions, under certain assumptions, are obtained. The Lax pair associated to this reduced hierarchy is obtained.
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painlevé equations. The Lax pair associated to this equation is obtained, also by reduction.
Fredholm determinant solutions to the modified KP hierarchy and 2D Toda lattice: isospectral and non-isospectral reductions
本文研究修正KP层次和二维Toda晶格的Fredholm行列式解，通过等谱约化构造了mKdV、修正Camassa-Holm和聚焦非线性薛定谔方程的解，并展示了非等谱约化得到变形Bessel行列式解。
Bihamiltonian structure of the $(n,1)$-type rational reductions of the 2D-Toda hierarchy
通过直接计算推导了$(n,1)$型二维Toda层次有理约化的局部双哈密顿结构，并构造了一个具有非平坦单位的$(n+1)$维半单广义Frobenius流形，其主层次包含无色散流。
Asymmetric rational reductions of 2D-Toda hierarchy and a generalized Frobenius manifold
该研究分析了两类非对称2D-Toda层级有理约化的局部双哈密顿结构，构造了对应(2,1)型的三维广义Frobenius流形，并建立了其与双分次Toda、约束KP层级的变换关联。
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.
One reduction of the modified Toda hierarchy
本文利用mToda族与第一mKP族共同的费米子形式，构造广义双分次mToda约化，讨论其与Toda约化的关系，并给出二者基于tau函数的等价表述。
The Volterra and Toda chains equations are considered. A class of special reductions for these equations are derived.
We show that one-variable reductions of the Pfaff-Toda integrable hierarchy in the dispersionless limit are described by a system of coupled elliptic Löwner (Komatu-Goluzin) equations.
This paper concerns the solutions of the full Kostant-Toda (f-KT) hierarchy in the Hessenberg form and their reductions to the $\ell$-banded Kostant-Toda ($\ell$-KT) hierarchy. We also study the f-KT hierarchy and the corresponding $\ell$-KT hierarchy on simple Lie algebras of type $A, B$ and $G$ based on root space reductions with proper Chevalley systems. Explicit formulas of the polynomial solutions for the $τ$-functions are also given in terms of the Schur functions and Schur's $Q$-functions.
We study finite-dimensional reductions of the dispersionless 2D Toda hierarchy showing that the consistency conditions for such reductions are given by a system of radial Loewner equations. We then construct their Hamiltonian structures, following an approach proposed by Ferapontov.
A family of real Hamiltonian forms (RHF) for the special class of affine 1+1 - dimensional Toda field theories is constructed. Thus the method, proposed in [Mikhailov;1981] for systems with finite number of degrees of freedom is generalized to infinite-dimensional Hamiltonian systems. We show that each of these RHF is related to a special Z_2-symmetry of the system of roots for the relevant Kac-Moody algebra. A number of explicit nontrivial examples of RHF of ATFT are presented.
It is shown that some special reduction of infinite 1D Toda lattice gives differential constraints compatible with the Kaup -- Broer system. A family of the travelling wave solutions of the Kaup -- Broer system and its higher version is constructed.
The 2D Toda hierarchy occupies a central position in the family of integrable hierarchies of the Toda type. The 1D Toda hierarchy and the Ablowitz-Ladik (aka relativistic Toda) hierarchy can be derived from the 2D Toda hierarchy as reductions. These integrable hierarchies have been applied to various problems of mathematics and mathematical physics since 1990s. A recent example is a series of studies on models of statistical mechanics called the melting crystal model. This research has revealed that the aforementioned two reductions of the 2D Toda hierarchy underlie two different melting cryst…
The problem of construction of integrable boundary conditions for the discrete Toda chain is considered. The restricted chains for properly chosen closure conditions are reduced to the well known discrete Painlevé equations $dP_{III}$, $dP_{V}$, $dP_{VI}$. Lax representations for these discrete Painlevé equations are found.
This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of "Landau-Ginzburg potentials" that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric polynomial. The other is a transcendental function, the logarithm of which resembles the waterbag models of the dispersionless KP hierarchy. They both satisfy a radial version of the Löwner equations. Consistency of these Löwner equations yields a radial version of the Gibbons-T…
Non-Abelian Toda lattice and analogs of Painlevé III equation
本文利用高对称性的稳态方程约化非阿贝尔二维 Toda 格，得到泵浦 Maxwell–Bloch 方程的非阿贝尔类似物，并通过自相似约化导出非阿贝尔 Painlevé III 方程。
We study a reduction of the Toda lattice in the limit of infintesimal lattice spacing. Using this reduction, we formulate a conjecture for the equivariant Gromov-Witten invariants of the sphere, which we prove in genus 0. This conjecture follows from recent work of Okounkov and Pandharipande (math.AG/0207233), combined with results from the sequel to this paper.
Ruijsenaars duality for B, C, D Toda chains
本文利用Hamiltonian约化方法构造了B、C、D型Toda链的Ruijsenaars对偶系统，证明其为rational Goldfish model的类似物，并给出了对偶Hamiltonian公式及高阶Hamiltonian的显式计算。
The Virasoro symmetries of the bigraded modified Toda hierarchy
针对修正Toda层级，本文构造其附加对称性并导出Adler-Shiota-van Moerbeke公式，同时证明$(N,M)$-BMTH的附加对称性构成Virasoro代数的子代数。
Orbifold generalizations of the ordinary and modified melting crystal models are introduced. They are labelled by a pair $a,b$ of positive integers, and geometrically related to $\mathbf{Z}_a\times\mathbf{Z}_b$ orbifolds of local $\mathbf{CP}^1$ geometry of the $\mathcal{O}(0)\oplus\mathcal{O}(-2)$ and $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ types. The partition functions have a fermionic expression in terms of charged free fermions. With the aid of shift symmetries in a fermionic realization of the quantum torus algebra, one can convert these partition functions to tau functions of the 2D Toda…
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.
We introduce and study a two-parameter family of symmetry reductions of the two-dimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a product of an upper diagonal and the inverse of a lower diagonal formal difference operator. They subsume and generalize several classical 1 + 1 integrable hierarchies, such as the bigraded Toda hierarchy, the Ablowitz-Ladik hierarchy and E. Frenkel's q-deformed Gelfand-Dickey hierarchy. We establish their characterization in terms of block Toeplitz matrices for the associated factorization problem, an…
On integrable reductions of two-dimensional Toda-type lattices
本文研究二维Toda型格点的可积约化，证明其经适当截断边界条件得到的有限域约化具有Darboux可积性，并给出利用Lax对和Miura型变换构造完备特征积分组的算法。
In this paper, we construct the additional $W$-symmetry and ghost symmetry of two-lattice field integrable hierarchies. Using the symmetry constraint, we construct constrained two-lattice integrable systems which contain several new integrable difference equations. Under a further reduction, the constrained two-lattice integrable systems can be combined into one single integrable system, namely the well-known one dimensional original Toda hierarchy. We prove that the one dimensional original Toda hierarchy has a nice Block Lie symmetry.
In this paper we propose a new algorithm for obtaining the rational integrals of the full Kostant-Toda lattice. This new approach is based on a reduction of a bi-Hamiltonian system on gl(n,R). This system was obtained by reducing the space of maps from Z_n to GL(n,R) endowed with a structure of a pair of Lie-algebroids.
The Lie symmetries of a large class of generalized Toda field theories are studied and used to perform symmetry reduction. Reductions lead to generalized Toda lattices on one hand, to periodic systems on the other. Boundary conditions are introduced to reduce theories on an infinite lattice to those on semi-infinite, or finite ones.
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.
A N=4 supersymmetric matrix KP hierarchy is proposed and a wide class of its reductions which are characterized by a finite number of fields are described. This class includes the one-dimensional reduction of the two-dimensional N=(2|2) superconformal Toda lattice hierarchy possessing the N=4 supersymmetry -- the N=4 Toda chain hierarchy -- which may be relevant in the construction of supersymmetric matrix models. The Lax pair representations of the bosonic and fermionic flows, corresponding local and nonlocal Hamiltonians, finite and infinite discrete symmetries, the first two Hamiltonian str…
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.
It is well known that the compatible linear and quadratic Poisson brackets of the full symmetric and of the standard open Toda lattices are restrictions of linear and quadratic $r$-matrix Poisson brackets on the associative algebra ${\mathrm{gl}}(n,{\mathbb{R}})$. We here show that the quadratic bracket on ${\mathrm{gl}}(n,{\mathbb{R}})$, corresponding to the $r$-matrix defined by the splitting of ${\mathrm{gl}}(n,{\mathbb{R}})$ into the direct sum of the upper triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a quadratic Poisson structure on the cotangent bundle $T…
The extended Toda hierarchy of Carlet, Dubrovin and Zhang is reconsidered in the light of a 2+1D extension of the 1D Toda hierarchy constructed by Ogawa. These two extensions of the 1D Toda hierarchy turn out to have a very similar structure, and the former may be thought of as a kind of dimensional reduction of the latter. In particular, this explains an origin of the mysterious structure of the bilinear formalism proposed by Milanov.
The conformal affine Toda model coupled to the matter field (CATM) is obtained through a classical reduction of the $sl(2)^{(1)}$ affine two-loop WZNW model. After spontaneously broken the conformal symmetry by means of BRST analysis, we end up with an effective theory, the so called affine Toda model coupled to the matter (ATM). Further, using a bosonization technique we recover from this theory the sine-Gordon model plus a free massless scalar field. The ATM model is considered as a QCD-motivated integrable field theory, since it describes various features in the baryonic sector of the low-e…
In the present paper, we are concerned with the tau function and its connection with the Kadomtsev-Petviashvili (KP) theory for the massive Thirring (MT) model. First, we bilinearize the massive Thirring model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two-component KP-Toda hierarchy, we derive the multi-bright solution to the MT model by the KP hierarchy reductions. Then, we show that the discrete KP equation can generate a set of bilinear equations of a deformed KP-Toda hierarchy through Miwa transformation. By imposing constra…
The Toda lattice defined by the Hamiltonian $H={1\over 2} \sum_{i=1}^n p_i^2 + \sum_{i=1}^{n-1} ν_i e^{q_i-q_{i+1}}$ with $ν_i\in \{ \pm 1\}$, which exhibits singular (blowing up) solutions if some of the $ν_i=-1$, can be viewed as the reduced system following from a symmetry reduction of a subsystem of the free particle moving on the group $G=SL(n,\Real )$. The subsystem is $T^*G_e$, where $G_e=N_+ A N_-$ consists of the determinant one matrices with positive principal minors, and the reduction is based on the maximal nilpotent group $N_+ \times N_-$. Using the Bruhat decomposition we show th…
In this paper, we construct the additional symmetries of the fermionic $(2N,2M)$-Toda hierarchy basing on the generalization of the $N{=}(1|1)$ supersymmetric two dimensional Toda lattice hierarchy. These additional flows constitute a $w_{\infty}\times w_{\infty}$ Lie algebra. As a Bosonic reduction of the $N{=}(1|1)$ supersymmetric two dimensional Toda lattice hierarchy and the fermionic $(2N,2M)$-Toda hierarchy, we define a new extended fermionic $(2N,2M)$-Toda hierarchy which admits a Bosonic Block type superconformal structure.
The reciprocal link between the reduced Ostrovsky equation and the $A_2^{(2)}$ two-dimensional Toda system is used to construct the $N$-soliton solution of the reduced Ostrovsky equation. The $N$-soliton solution of the reduced Ostrovsky equation is presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations and the $τ$-function of the reduced Ostrovsky equation are obtained from the period 3-reduction of the $B_{\infty}$ or $C_{\infty}$ two-dimensional Toda system, i.e., the $A_2^{(2)}$ two-dimensional Toda system. One of $τ$-functions of the $A_2…
Exact solutions and reductions of nonlinear Schrödinger equations with delay
本文研究带常延迟或变延迟的非线性 Schrödinger 方程，通过非对称约化、广义变量分离与函数约束构造求积形式及行波叠加精确解，为相关模型和数值方法提供基准。
This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hi- erarchy with the help of orthogonal polynomials theory and Toda-type equations. Starting from the symmetric reduction of Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the τ-function of the CKP hiearc…
Delay-difference and delay-differential analogues of the KdV and Boussinesq (BSQ) equations are presented. Each of them has the N-soliton solution and reduces to an already known soliton equation as the delay parameter approaches 0. In addition, a delay-differential analogue of the KP equation is proposed. We discuss its N-soliton solution and the limit as the delay parameter approaches 0. Finally, the relationship between the delay-differential analogues of the KdV, BSQ, and KP equations is clarified. Namely, reductions of the delay KP equation yield the delay KdV and delay BSQ equations.
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 …
Toda lattice hierarchy and the associated matrix formulation of the $2M$-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which abelianize the second KP Hamiltonian structure, we are able to obtain an unified formalism for the reduced $SL(M+1,M-k)$-KdV hierarchies interpolating between the ordinary KP and KdV hierarchies. The corresponding Lax operators are given as superdeterminants of graded $SL (M+1,M-k)$ matrices in the diagonal gauge and we describe th…
On the symmetry classification of integrable chains in 3D. Darboux-integrable reductions and their higher symmetries
本文提出了一种方法，用于识别和分类三维中具有离散和连续变量的可积非线性方程，并揭示了Toda型链的降维结构及其对称性特性。
We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda Lattice hierarchy (2DTL). This reduction implies that the RTC is gauge equivalent to the discrete AKNS hierarchy and, which is the same, to the two-component Volterra hierarchy while its forced (semi-infinite) variant is described by the unitary matrix integral. The integrable properties of the RTC hierarchy are revealed in different frameworks of: Lax representation, orthogonal polynomial systems, and $τ$-function approach. Relativistic Toda molecule hierarchy is also con…
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.
In this paper the relation between the cluster integrable systems and $q$-difference equations is extended beyond the Painlevé case. We consider the class of hyperelliptic curves when the Newton polygons contain only four boundary points. The corresponding cluster integrable Toda systems are presented, and their discrete automorphisms are identified with certain reductions of the Hirota difference equation. We also construct non-autonomous versions of these equations and find that their solutions are expressed in terms of 5d Nekrasov functions with the Chern-Simons contributions, while in the …
A q-discrete version of the two-dimensional Toda molecule equation is proposed through the direct method. Its solution, Bäcklund transformation and Lax pair are discussed. The reduction to the q-discrete cylindrical Toda molecule equation is also discussed.
Matrix Fourier-like integrals over the classical groups O_+(n), O_-(n), Sp(n) and U(n) are connected with the distribution of the length of the longest increasing sequence in random permutations and random involutions and the spectrum of random matrices. One of the purposes of this paper is to show that all those integrals satisfy the Painlevé V equation with specific initial conditions. In this work, we present both, new results and known ones, in a unified way. Our method consists of inserting one set of time variables t=(t_1,t_2,...) in the integrals for the real compact groups and two sets…
We consider an integrable conformally invariant two dimensional model associated to the affine Kac-Moody algebra SL(3). It possesses four scalar fields and six Dirac spinors. The theory does not possesses a local Lagrangian since the spinor equations of motion present interaction terms which are bilinear in the spinors. There exists a submodel presenting an equivalence between a U(1) vector current and a topological current, which leads to a confinement of the spinors inside the solitons. We calculate the one-soliton and two-soliton solutions using a procedure which is a hybrid of the dressing…
In this letter, we show that certain Fredholm determinant $D(λ;t)$, introduced by Zamolodchikov in his study of 2D polymers, is a continuum limit of soliton solution for the Toda lattice hierarchy with 2-periodic reduction condition.
First, we recall the algebro-geometric method of construction of finite field valued solutions of the discrete KP equation and next we perform a reduction of the dKP equation to the discrete 1D Toda equation. This gives a method of construction of solutions of the discrete 1D Toda equation taking values in a finite field.
Linear spectral transformations of orthogonal polynomials in the real line, and in particular Geronimus transformations, are extended to orthogonal polynomials depending on several real variables. Multivariate Christoffel-Geronimus-Uvarov formulae for the perturbed orthogonal polynomials and their quasi-tau matrices are found for each perturbation of the original linear functional. These expressions are given in terms of quasi-determinants of bordered truncated block matrices and the 1D Christoffel-Geronimus-Uvarov formulae in terms of quotient of determinants of combinations of the original o…
The hierarchy of the classical nonlinear integrable equations associated with relativistic Toda chain model is considered. It is formulated for the N-th powers of the quantum operators of the corresponding quantum integrable models. Following the ideas of the paper hep-th/9606144 it is shown how one can obtain such a system from 2D Toda lattice system. The reduction procedure is described explicitly. The soliton solutions for the relativistic Toda chain are constructed using results of solv-int/9304002 in terms of the rational tau-functions. The vanishing properties of these tau-functions are …
We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the Zakharov-Shabat type and the formulation through the generating equation for the dispersionless limit of logarithm of the tau-function. We show that the dispersionless constrained Toda hierarchy describes conformal maps of reflection-symmetric planar domains to the exterior of the unit disc. We also find finite-dimensional reductions of the hierarchy and show that they …
As a generalization of the integrable extended Toda hierarchy and a reduction of the extended multicomponent Toda hierarchy, from the point of a commutative subalgebra of $gl(2,\mathbb{C})$, we construct a strongly coupled extended Toda hierarchy(SCETH) which will be proved to possess a Virasoro type additional symmetry by acting on its tau-function. Further we give the multi-fold Darboux transformations of the strongly coupled extended Toda hierarchy.
We study a (2+1)-dimensional system that can be viewed as an infinite number of O(3) sigma-fields coupled by a nearest-neighbour Heisenberg-like interaction. We reduce the field equations of this model to an integrable system that is closely related to the two-dimensional relativistic Toda chain and the Ablowitz-Ladik equations. Using this reduction we obtain the dark-soliton solutions of our model.
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