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.     

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 lattice with a special self-consistent source

We describe a method for integrating the Toda lattice with a self-consistent source using the inverse scattering method for a discrete Sturm-Liouville operator with moving eigenvalues. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 305–315, February, 2008.     

Generalized dressing method for the extended two-dimensional Toda lattice hierarchy and its reductions

In this paper, we solve the extended two-dimensional Toda lattice hierarchy (ex2DTLH) by the generalized dressing method developed in Liu-Lin-Jin-Zeng (2009). General Casoratian determinant solutions for this hierarchy are obtained. In particular, explicit solutions of soliton-type are formulated by using the τ-function in the form of exponential functions. The periodic reduction and one-dimensional reduction of ex2DTLH are studied by finding the constraints. Many reduced systems are shown, including the pe...     

Two-periodic waves and asymptotic property for generalized 2D Toda lattice equation

In this paper, two-periodic wave solutions are constructed for the (2 + 1)-dimensional generalized Toda lattice equation by using Hirota bilinear method and Riemann theta function. At the same time, we analyze in details asymptotic properties of the two-periodic wave solutions and give their asymptotic relations between the periodic wave solutions and the soliton solutions.     

The Camassa–Holm Equation on the Half-Line: a Riemann–Hilbert Approach

We consider the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation u _t −u _txx +2u _x +3uu _x =2u _x u _xx +uu _xxx on the half-line x≥0. In this article, we aim to provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann–Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated to initial and boundary values of the solution. The construction requires more boundary data than those needed for a well-posed IBV problem. Their dependence is expressed in terms of an algebraic relation to be satisfied by the spectral functions. This RH formulation gives us the long-time asymptotics of a solution of the CH-equation. Dedicated to Gennadi Henkin in great admiration.     

Moment asymptotics for the parabolic Anderson problem with a perturbed lattice potential

The parabolic Anderson problem with a random potential obtained by attaching a long tailed potential around a randomly perturbed lattice is studied. The moment asymptotics of the total mass of the solution is derived. The results show that the total mass of the solution concentrates on a small set in the space of configuration.     

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.     

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.     

The Algebraic Integrability of the Quantum Toda Lattice and the Radon Transform

We study the maximal commutative ring of partial differential operators which includes the quantum completely integrable system defined by the quantum Toda lattice. Kostant shows that the image of the generalized Harish-Chandra homomorphism of the center of the enveloping algebra is commutative (Kostant in Invent. Math. 48:101–184, 1978). We demonstrate the commutativity of the ring of partial differential operators whose principal symbols are -invariant. Our commutative ring includes the commutative system of Kostant (Invent. Math. 48:101–184, 1978). The main tools in this paper are Fourier integral operators and Radon transforms.      

2D Toda lattice equation with self-consistent sources: Casoratian type solutions,bilinear Bäcklund transformation and Lax pair

The two-dimensional Toda lattice equation with self-consistent sources is proposed based on its bilinear forms. Casoratian-type solutions and Bäcklund transformation (BT) for the bilinear forms are presented. Starting from the BT, a Lax pair is derived for the 2D Toda lattice with self-consistent sources.     

First Colonization of a Spectral Outpost in Random Matrix Theory

We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N ^−2γ where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature. The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity at the rate N ^−γ , whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.      

Topology of the Real Part of the Hyperelliptic Jacobian Associated with the Periodic Toda Lattice

We study the topology of the isospectral real manifold of the periodic Toda lattice consisting of 2 ^N–1 different systems. The solutions of these systems contain blow-ups, and the set of these singular points defines a divisor of the manifold. With the divisor added, the manifold is compactified as the real part of the (N–1)-dimensional Jacobi variety associated with a hyperelliptic Riemann surface of genus g=N–1. We also study the real structure of the divisor and provide conjectures on the topology of the affine part of the real Jacobian and on the gluing rule over the divisor to compactify the manifold based on the sign representation of the Weyl group of .     

On the shapes of two-dimensional soliton perturbations in simple lattices

The Toda lattice and the discrete Korteweg-de Vries equation generalized to two dimensions are studied numerically. The interactions are assumed to be identical in both directions. It is shown that the equations have solutions in the form of plane linear and localized solitons. In contrast to equations integrable by the inverse scattering method, the parameters of solitons change in the course of their interaction and additional wave structures are formed. The basic types of solutions characterizing these processes are presented.     

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

The Allen-Cahn equation ? Δu = u ? u ³ in ?² has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem ?u″ =u ? u ³. In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in ?³.     

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.     

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [−1,1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann–Hilbert (RH) problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the RH problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order 1/n terms in the expansions. A critical step in the analysis of the RH problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order. In addition, the RH approach gives us also strong asymptotics of the orthogonal polynomials near the algebraic singularities in terms of Bessel functions.     

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.