Elliptic solutions of the Toda chain and a generalization of the Stieltjes–Carlitz polynomials

We construct new elliptic solutions of the restricted Toda chain. These solutions give rise to a new explicit class of orthogonal polynomials, which can be considered as a generalization of the Stieltjes–Carlitz elliptic polynomials. Relations between characteristic (i.e., positive definite) functions, Toda chain, and orthogonal polynomials are developed in order to derive the main properties of these polynomials. Explicit expressions are found for the recurrence coefficients and the weight function for these polynomials. In the degenerate cases of the elliptic functions, the modified Meixner polynomials and the Krall–Laguerre polynomials appear.     

Peakons,strings, and the finite Toda lattice

As is well‐known, the Toda lattice flow may be realized as an isospectral flow of a Jacobi matrix. A bijective map from a discrete string problem with positive weights to Jacobi matrices allows the pure peakon flow of the Camassa‐Holm equation to be realized as an isospectral Jacobi flow as well. This gives a unified picture of the Toda, Jacobi, and multipeakon flows, and leads to explicit solutions of the Jacobi flows via Stieltjes' determination of the continued fraction expansion of a Stieltjes transform. A simple modification produces a bijection from generalized strings, with positive and negative weights, to singular Jacobi matrices, and thus brings peakon/antipeakon flows into the same picture. © 2001 John Wiley & Sons, Inc.     

Wave functions of the Toda chain with boundary interaction

We give an integral representation of the wave functions of the quantum N-particle Toda chain with boundary interaction. In the case of the Toda chain with a one-boundary interaction, we obtain the wave function by an integral transformation from the wave functions of the open Toda chain. The kernel of this transformation is given explicitly in terms of -functions. The wave function of the Toda chain with a two-boundary interaction is obtained from the previous wave functions by an integral transformation. In this case, the difference equation for the kernel of the integral transformation admits a separation of variables. The separated difference equations coincide with the Baxter equation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 346–364, February, 2005.     

On powers of Stieltjes moment sequences,II

We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, and prove an integral representation of the logarithm of the moment sequence in analogy to the Lévy–Khintchine representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an application we construct a positive generating function for the orthonormal Hermite polynomials.     

Renormalization of the τ-functions for integrable systems: A model problem

We introduce a renormalization procedure for the τ-function of integrable systems. We illustrate the procedure using the supercritical Toda shock problem as a model problem. We start with a finite chain and take the limit of the solution as the number of particles N → ∞. This results in a new formula for the τ-function for the problem with an infinite chain. We apply the renormalized formula to rederive leading-order effects of the supercritical Toda shock problem. © 1998 John Wiley & Sons, Inc.     

Limit relation between toda chains and the elliptic SL(N, ?) top

We study a limit relation between the elliptic SL(N,?) top and Toda chains. We show that in the case of the nonautonomous SL(2, ?) top, whose equations of motion are related to the Painlevé VI equation, it turns out to be possible to modify the previously proposed procedure and in the limit obtain the nonautonomous Toda chain, whose equations of motion are equivalent to a particular case of the Painlevé III equation. We obtain the limit of the Lax pair for the elliptic SL(2, ?) top, which allows representing the equations of motion of the nonautonomous Toda chain as the equation for isomonodromic deformations.     

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.     

Integrable boundary conditions for (2+1)-dimensional models of mathematical physics

We consider the question of integrable boundary-value problems in the examples of the two-dimensional Toda chain and Kadomtsev-Petviashvili equation. We discuss the problems that are integrable from the standpoints of two basic definitions of integrability. As a result, we propose a method for constructing a hierarchy of integrable boundary-value problems where the boundaries are cylindric surfaces in the space of three variables. We write explicit formulas describing wide classes of solutions of these boundary-value problems for the two-dimensional Toda chain and Kadomtsev-Petviashvili equation.     

Finite-Dimensional Discrete Systems Integrated in Quadratures

We consider finite-dimensional reductions (truncations) of discrete systems of the type of the Toda chain with discrete time that retain the integrability. We show that for finite-dimensional chains, in addition to integrals of motion, we can construct a rich family of higher symmetries described by the master symmetry. We reduce the problem of integrating a finite-dimensional system to the implicit function theorem.     

Biorthogonal Laurent Polynomials,Toplitz Determinants,Minimal Toda Orbits and Isomonodromic Tau Functions

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish corresponding Christoffel-Darboux formulae. For all these classes of polynomials a 2 × 2 system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudo-measures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted Toplitz determinants of the moments of the pseudo-measure. In particular, the results imply that any (shifted) Toplitz (Hankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic tau function.     

Cluster characters and the combinatorics of Toda systems

We survey some connections between Toda systems and cluster algebras. One of these connections is based on representation theory: it is known that Laurent expansions of cluster variables are generating functions of Euler characteristics of quiver Grassmannians, and the same turns out to be true of the Hamiltonians of the open relativistic Toda chain. Another connection is geometric: the closed nonrelativistic Toda chain can be regarded as a meromorphic Hitchin system and studied from the standpoint of spectral networks. From this standpoint, the combinatorial formulas for the Hamiltonians of the open relativistic system are sums of trajectories of differential equations defined by the closed nonrelativistic spectral curves.     

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.     

Stieltjes representation of the 3D Bruggeman effective medium and Padé approximation

The paper deals with Bruggeman effective medium approximation (EMA) which is often used to model effective complex permittivity of a two-phase composite. We derive the Stieltjes integral representation of the 3D Bruggeman effective medium and use constrained Padé approximation method introduced in [39] to numerically reconstruct the spectral density function in this representation from the effective complex permittivity known in a range of frequencies. The problem of reconstruction of the Stieltjes integral representation arises in inverse homogenization problem where information about the spectral function recovered from the effective properties of the composite, is used to characterize its geometric structure. We present two different proofs of the Stieltjes analytical representation for the effective complex permittivity in the 3D Bruggeman effective medium model: one proof is based on direct calculation, the other one is the derivation of the representation using Stieltjes inversion formula. We show that the continuous spectral density in the integral representation for the Bruggeman EMA model can be efficiently approximated by a rational function. A rational approximation of the spectral density is obtained from the solution of a constrained minimization problem followed by the partial fractions decomposition. We show results of numerical rational approximation of Bruggeman continuous spectral density and use these results for estimation of fractions of components in a composite from simulated effective permittivity of the medium. The volume fractions of the constituents in the composite calculated from the recovered spectral function show good agreement between theoretical and predicted values.     

Structure of Stieltjes classes of moment-equivalent probability laws

A Stieltjes class is a one-parameter family of moment-equivalent distribution functions constructed by modulation of a given indeterminate distribution function F, called the center of the class. Members of a Stieltjes class are mutually absolutely continuous, and conversely, any pair of moment-equivalent and mutually absolutely continuous distribution functions can be joined by a Stieltjes class. The center of a Stieltjes class is an equally weighted mixture of its extreme members, and this places restrictions on which distributions can belong to a Stieltjes class with a given center. The lognormal law provides interesting illustrations of the general ideas. In particular, it is possible for two moment equivalent infinitely divisible distributions to be joined by a Stieltjes class, and random scaling can be used to construct new Stieltjes classes from a given Stieltjes class.     

Tau-function formulation for bright,dark soliton and breather solutions to the massive Thirring model

In the present paper, we are concerned with the link between the Kadomtsev–Petviashvili–Toda (KP–Toda) hierarchy and the massive Thirring (MT) model. First, we bilinearize the MT model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two-component KP–Toda hierarchy, we derive multibright solution to the MT model. Then, considering a set of bilinear equations of the single-component KP–Toda hierarchy, multidark soliton and multibreather solutions to the MT model are constructed by imposing constraints on the parameters in two types of tau function, respectively. The dynamics and properties of one- and two-soliton for bright, dark soliton and breather solutions are analyzed in details.     

A generalized Stieltjes criterion for the complete indeterminacy of interpolation problems

The main result of this paper is a generalized Stieltjes criterion for the complete indeterminacy of interpolation problems in the Stieltjes class. This criterion is a generalization to limit interpolation problems of the classical Stieltjes criterion for the indeterminacy of moment problems. It is stated in terms of the Stieltjes parameters M _j and N _j . We obtain explicit formulas for the Stieltjes parameters. General constructions are illustrated by examples of the Stieltjes moment problem and the Nevanlinna-Pick problem in the Stieltjes class.     

ℤ-graded loop Lie algebras, loop groups, and Toda equations

We consider Toda equations associated with twisted loop groups. Such equations are specified by ℤ-gradings of the corresponding twisted loop Lie algebras. We discuss the classification of Toda equations related to twisted loop Lie algebras with integrable ℤ-gradings. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 451–476, March, 2008.     

Continued fractions and the polygamma functions

In a previous study we have shown that the polygamma functions (derivatives of the logarithm of the gamma function) relate to Stieltjes transforms in the square of the argument. These transforms in turn may be converted to Stieltjes continued fractions; in the background is a determined Stieltjes moment problem.In the present study we use the Hamburger form of the Stieltjes integral to produce a set of real monotonic increasing and monotonic decreasing approximants to each of the real and imaginary parts of a polygamma function when the argument is complex. The approximants involve rational fractions which appear to be new.Special attention is given to ln Γ(z) and the psi function.     

Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures

We introduce a criterion that a given bi-Hamiltonian structure admits a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bi-Hamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bi-Hamiltonian structures. Near, a generic point, the bi-Hamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general results on geometry of webs. This demonstrates the possibility of applying a geometric language to problems on bi-Hamiltonian integrable systems; such a possibility may be no less important than the particular results proved in this paper. Based on these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bi-Hamiltonian Lie coalgebra. We also state general conjectures about the geometry of more general "homogeneous" finite-dimensional bi-Hamiltonian structures. The class of homogeneous structures is shown to coincide with the class of systems integrable by Lenard scheme. The bi-Hamiltonian structures which admit a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda lattice.     

On the relation between measures defining the Stieltjes and the inverted Stieltjes functions

A compact formula is found for the measure of the inverted Stieltjes function expressed by the measure of the original Stieltjes function.