Transition function for the Toda chain

We use the method of Λ-operators developed by Derkachov, Korchemsky, and Manashov to derive eigenfunctions for the open Toda chain. Using the diagram technique developed for these Λ-operators, we reproduce the Sklyanin measure and study the properties of the Λ-operators. This approach to the open Toda chain eigenfunctions reproduces the Gauss-Givental representation for these eigenfunctions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 371–390, March, 2007.     

Solutions for Toda system on Riemann surface with boundary

In this paper, we study the solutions for Toda system on Riemann surface with boundary. We prove a sufficient condition for the existence of solution of Toda system in the critical case.     

Integrability conditions for an analogue of the relativistic Toda chain

We consider a class of discrete-differential equations that contains the relativistic Toda chain and is characterized by one arbitrary function of six variables. We derive three conditions that allow testing the integrability of any given equation in this class. In deriving these conditions, we use higher symmetries distinguishing the equations that are integrable via the inverse scattering method. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 66–80, April, 2007.     

Toda Chains in the Jacobi Method

We use the Jacobi method to construct various integrable systems, such as the Stäckel systems and Toda chains, related to various root systems. We find canonical transformations that relate integrals of motion for the generalized open Toda chains of types B _n, C _n, and D _n.     

Toda chain,Stieltjes function,and orthogonal polynomials

We discuss relations between the theory of orthogonal polynomials, Hankel determinants, and the unrestricted one-dimensional Toda chain. In particular, we show that the equations of motion for the Toda chain are equivalent to a Riccati equation for the Stieltjes function. We consider some examples of the Stieltjes function with an explicit (hypergeometric and elliptic) time dependence in detail. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 81–108, April, 2007.     

Soliton solutions of an integrable boundary problem on the half-line for the discrete toda chain

We write formulas for soliton solutions of the discrete Toda chain and pose the integrable boundary value problem for this chain. We find conditions for the parameters (discrete spectrum points, transmission coefficients, and the corresponding factors) whereby solutions of the integrable boundary value problem are selected from all soliton solutions. As a result, we construct two hierarchies of soliton solutions of the specified problem with even and odd soliton numbers and find an explicit form of the conditions for the parameters. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 387–397, September, 2006.     

Eigenvectors of the Baxter-Bazhanov-Stroganov τ<Superscript>(2)</Superscript>(<Emphasis Type="Italic">t</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>) model with fixed-spin boundary conditions

We give explicit formulas for the eigenvectors of the transfer matrix of the Baxter-Bazhanov-Stroganov (BBS) model (N-state spin model) with fixed-spin boundary conditions. We obtain these formulas from the formulas for the eigenvectors of the periodic BBS model by a limit procedure. The latter formulas were derived in the framework of Sklyanin’s method of separation of variables. In the case of fixed-spin boundaries, we solve the corresponding T-Q Baxter equations for the functions of separated variables explicitly. As a particular case, we obtain the eigenvectors of the Hamiltonian of the Ising-like ℤ_N quantum chain model. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 94–108, April, 2008.     

Symmetries of nonlinear hyperbolic systems of the Toda chain type

We consider hyperbolic systems of equations that have full sets of integrals along both characteristics. The best known example of models of this type is given by two-dimensional open Toda chains. For systems that have integrals, we construct a differential operator that takes integrals into symmetries. For systems of the chosen type, this proves the existence of higher symmetries dependent on arbitrary functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 344–355, May, 2008.     

Fermionic construction of the partition function for multimatrix models and the multicomponent Toda lattice hierarchy

We use p-component fermions, p = 2, 3,..., to represent (2p−2)N-fold integrals as a fermionic vacuum expectation. This yields a fermionic representation for various (2p−2)-matrix models. We discuss links with the p-component Kadomtsev-Petviashvili hierarchy and also with the p-component Toda lattice hierarchy. We show that the set of all but two flows of the p-component Toda lattice hierarchy changes standard matrix models to new ones. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 265–277, August, 2007.     

Darboux-Nijenhuis variables for open generalized Toda chains

We consider the possibility of using the Sklyanin method to construct Darboux-Nijenhuis variables of special form in the example of generalized open Toda chains associated with classical root systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 440–456, September, 2007.     

Canonically conjugate variables for the Volterra lattice with periodic boundary conditions

The Volterra lattice is considered. This dynamical system is known to be Hamiltonian with respect to two compatible Poisson brackets (quadratic and cubic). For each of the two brackets, a set of canonically conjugate variables is found by using the spectral theory of the Jacobi operator.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 115–128, July, 1998.The author wishes to express his gratitude to A. P. Veselov for setting the problem and for useful discussions.     

<Emphasis Type="Italic">N</Emphasis>-soliton train and generalized complex Toda chain for the Manakov system

We analyze the dynamical behavior of the N-soliton train of the Manakov system and of the vector NLS equation in the adiabatic approximation. We prove that the dynamics of the N-soliton train in both cases are described by a generalized version of the complex Toda chain model. This fact can be used to predict the asymptotic regimes of the N-soliton train provided the initial soliton parameters are given. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 391–404, June, 2007.     

Inverse scattering transform for the Toda hierarchy with quasi-periodic background

We provide a rigorous treatment of the inverse scattering transform for the entire Toda hierarchy in the case of a quasi-periodic finite-gap background solution.

     

Numerical analysis of the Toda lattice equations

Numerical solutions to three systems of integrable evolutionary equations from the Toda lattice hierarchy are analyzed. These are the classical Toda lattice, the second local dispersive flow, and the second extended dispersive flow. Special attention is given to the properties of soliton solutions. For the equations of the second local flow, two types of solitons interacting in a special manner are found. Solutions corresponding to various initial data are qualitatively outlined.     

Toda equation and special polynomials associated with the Garnier system

We prove that a certain sequence of τ-functions of the Garnier system satisfies Toda equation. We construct algebraic solutions of the system by the use of Toda equation; then show that the associated τ-functions are expressed in terms of the universal character which is a generalization of Schur polynomial attached to a pair of partitions.     

The rapidly decreasing solution of the Cauchy problem for the Toda lattice

We prove the existence of a rapidly decreasing solution of the Cauchy problem for the Toda lattice. We find a class of initial data guaranteeing the existence of such a solution.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 5–12, January, 2005.     

Geometric Properties of W-Algebras and the Toda model

The W-algebra minimal models on hyperelliptic Riemann surfaces are constructed. Using a proposal by Polyakov, we reduce the partition function of the Toda field theory on the hyperelliptic surface to a product of partition functions: one of a free field theory on the sphere with inserted Toda vertex operators and one of a free scalar field theory with antiperiodic boundary conditions with inserted twist fields.     

Meromorphic functions with bounded boundary rotation

In this article, we generalize the class of meromorphic functions with bounded boundary rotation and related classes. Characterizations and some properties of these classes of functions are given.     

Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.     

On the Integrability of Deformation Quantized Toda Lattice

In the present paper, we are concerned with deformation quantization of irregular Poisson structures. Translating Toda lattice equation into Hamiltonian formalism equation, we also study the global integrability of deformation quantized Toda lattice.