Hypergeometric Functions as Infinite-Soliton Tau Functions

It is known that resonant multisoliton solutions depend on higher times and a set of parameters (integrals of motion). We show that soliton tau functions of the Toda lattice (and of the multicomponent Toda lattice) are tau functions of a dual hierarchy, where the higher times and the parameters (integrals of motion) exchange roles. The multisoliton solutions turn out to be rational solutions of the dual hierarchy, and the infinite-soliton tau functions turn out to be hypergeometric-type tau functions of the dual hierarchy. The variables in the dual hierarchies exchange roles. Soliton momenta are related to the Frobenius coordinates of partitions in the decomposition of rational solutions with respect to Schur functions. As an example, we consider partition functions of matrix models: their perturbation series is, on one hand, a hypergeometric tau function and, on the other hand, can be interpreted as an infinite-soliton solution. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 222–250, February, 2006.     

Hamiltonian Structures of Fermionic Two-Dimensional Toda Lattice Hierarchies

By exhibiting the corresponding Lax-pair representations, we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies, which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL hierarchies as particular cases. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with involution generalizing the graded commutator in superalgebras, which allows describing these hierarchies in the framework of the Hamiltonian formalism and constructing their first two Hamiltonian structures. We obtain the first Hamiltonian structure for both bosonic and fermionic Lax operators and the second Hamiltonian structure only for bosonic Lax operators. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 90–102, January, 2006.     

Canonical and Grand Canonical Partition Functions of Dyson Gases as Tau-Functions of Integrable Hierarchies and Their Fermionic Realization

The partition function for a canonical ensemble of 2D Coulomb charges in a background potential (the Dyson gas) is realized as a vacuum expectation value of a group-like element constructed in terms of free fermionic operators. This representation provides an explicit identification of the partition function with a tau-function of the 2D Toda lattice hierarchy. Its dispersionless (quasiclassical) limit yields the tau-function for analytic curves encoding the integrable structure of the inverse potential problem and parametric conformal maps. A similar fermionic realization of partition functions for grand canonical ensembles of 2D Coulomb charges in the presence of an ideal conductor is also suggested. Their representation as Fredholm determinants is given and their relation to integrable hierarchies, growth problems and conformal maps is discussed.     

N=(1|1) Supersymmetric Dispersionless Toda Lattice Hierarchy

Generalizing the graded commutator in superalgebras, we propose a new bracket operation on the space of graded operators with an involution. We study properties of this operation and show that the Lax representation of the two-dimensional N=(1|1) supersymmetric Toda lattice hierarchy can be realized via the generalized bracket operation; this is important in constructing the semiclassical (continuum) limit of this hierarchy. We construct the continuum limit of the N=(1|1) Toda lattice hierarchy, the dispersionless N=(1|1) Toda hierarchy. In this limit, we obtain the Lax representation, with the generalized graded bracket becoming the corresponding Poisson bracket on the graded phase superspace. We find bosonic symmetries of the dispersionless N=(1|1) supersymmetric Toda equation.     

Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation u _t = Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α _k (n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator. We derive an explicit formula for α _k in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].      

Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation u _t = Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α _k (n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator. We derive an explicit formula for α _k in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].     

Regular Solution and Lattice Miura Transformation of Bigraded Toda Hierarchy

The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy （BTH）. As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the （1, 2）-BTH with a 3 × 3-sized Lax matrix, and discuss some geometric structures of the solution from which the difference between the （1, 2）- BTH and the original Toda hierarchy is shown. After this, the authors construct another kind of Lax representation of （N, 1）-BTH which does not use the fractional operator of Lax operator. Then the authors introduce the lattice Miura transformation of （N, 1）-BTH which leads to equations depending on one field, and meanwhile some specific examples which contain the Volterra lattice equation （a useful ecological competition model） are given.     

Singular sectors of the one-layer Benney and dispersionless Toda systems and their interrelations

We completely describe the singular sectors of the one-layer Benney system (classical long-wave equation) and dispersionless Toda system. The associated Euler-Poisson-Darboux equations E(1/2, 1/2) and E(−1/2,−1/2) are the main tool in the analysis. We give a complete list of solutions of the one-layer Benney system depending on two parameters and belonging to the singular sector. We discuss the relation between Euler-Poisson-Darboux equations E(ɛ, ɛ) with the opposite sign of ɛ.     

Hurwitz numbers and products of random matrices

We study multimatrix models, which may be viewed as integrals of products of tau functions depending on the eigenvalues of products of random matrices. We consider tau functions of the two-component Kadomtsev–Petviashvili (KP) hierarchy (semi-infinite relativistic Toda lattice) and of the B-type KP (BKP) hierarchy introduced by Kac and van de Leur. Such integrals are sometimes tau functions themselves. We consider models that generate Hurwitz numbers H^E,F, where E is the Euler characteristic of the base surface and F is the number of branch points. We show that in the case where the integrands contain the product of n > 2 matrices, the integral generates Hurwitz numbers with E ≤ 2 and F ≤ n+2. Both the numbers E and F depend both on n and on the order of the factors in the matrix product. The Euler characteristic E can be either an even or an odd number, i.e., it can match both orientable and nonorientable (Klein) base surfaces depending on the presence of the tau function of the BKP hierarchy in the integrand. We study two cases, the products of complex and the products of unitary matrices.     

Hypergeometric Solutions of Soliton Equations

We consider multivariable hypergeometric functions related to Schur functions and show that these hypergeometric functions are tau functions of the KP hierarchy and are simultaneously the ratios of Toda lattice tau functions evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts the parameters of the hypergeometric functions. We construct the determinant representation and the integral representation of a special type for the KP tau functions. We write a system of linear differential and difference equations on these tau functions, which play the role of string equations.     

Renormalization-Group Transformation in a 2n-Component Fermionic Hierarchical Model

We study the 2N-component fermionic model on a hierarchical lattice and give explicit formulas for the renormalization-group transformation in the space of coefficients that determine a Grassmann-valued density of the free measure. We evaluate the inverse renormalization-group transformation. The de.nition of the renormalization-group fixed points reduces to a solution of a system of algebraic equations. We investigate solutions of this system for N = 1, 2, 3. For α = 1, we prove an analogue of the central limit theorem for fermionic 2N-component fields. We discover an interesting relation between renormalization-group transformations in bosonic and fermionic hierarchical models and show that one of these transformations is obtained from the other by replacing N with -N. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 251–266, February, 2006.     

On the Toda and Kac–van Moerbeke hierarchies

We provide a comprehensive treatment of the single and double commutation method as a tool for constructing soliton solutions of the Toda and Kac–van Moerbeke hierarchy on arbitrary background. In addition, we present a novel construction based on the single commutation method. As an illustration we compute the N-soliton solution of the Toda and Kac–van Moerbeke hierarchy. Received November 20, 1997; in final form July 7, 1998     

Elliptic solutions of the Toda lattice hierarchy and the elliptic Ruijsenaars–Schneider model

Theoretical and Mathematical Physics - We consider solutions of the 2D Toda lattice hierarchy that are elliptic functions of the “zeroth” time $$t_0=x$$ . It is known that their poles...     

Modeling Adiabatic <Emphasis Type="Italic">N</Emphasis>-Soliton Interactions and Perturbations

We analyze a perturbed version of the complex Toda chain (CTC) in an attempt to describe the adiabatic N-soliton train interactions of the perturbed nonlinear Schrodinger equation. We study perturbations with weak quadratic and periodic external potentials analytically and numerically. The perturbed CTC adequately models the N-soliton train dynamics for both types of potentials. As an application of the developed theory, we consider the dynamics of a train of matter-wave solitons confined in a parabolic trap and an optical lattice. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 302–312, August, 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.     

The binary nonlinearization of generalized Toda hierarchy by a special choice of parameters

A new discrete matrix spectral problem with two arbitrary constants is introduced. The corresponding 2-parameter hierarchy of integrable lattice equations, which can be reduced to the hierarchy of Toda lattice, is obtained by discrete zero curvature representation. Moreover, the Hamiltonian structure and a hereditary operators are deduced by applying the discrete trace identity. Finally, an integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonlinearization for the resulting hierarchy by a special choice of parameters.     

Pfaff $\tau$-functions

Consider the two-dimensional Toda lattice, with certain skew-symmetric initial condition, which is preserved along the locus of the space of time variables. Restricting the solution to , we obtain another hierarchy called Pfaff lattice, which has its own tau function, being equal to the square root of the restriction of 2D-Toda tau function. We study its bilinear and Fay identities, W and Virasoro symmetries, relation to symmetric and symplectic matrix integrals and quasiperiodic solutions. Received: 20 September 1999 / Published online: 1 February 2002     

Vacuum solutions in the Neveu-Schwarz field theory of a fermionic string and the zero-curvature representation in graded spaces

We demonstrate that vacuum solutions in the Neveu-Schwarz field theory of a fermionic string incorporating the GSO(−) sector are naturally related to the zero-curvature representation in a graded space of special form. We use the same representation to describe the equivalence of the cubic and nonpolynomial theories with the GSO(−) sector also taken into account. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 131–141, April, 2009.     

Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy

Following the approach of Carlet et al. (2011) [9], we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type A defined by Dubrovin and Zhang.     

Nonlocal lattice fermionic models on a two-dimensional torus

Abelian fermionic models described by the SLAC action on a two-dimensional finite lattice are considered. In vector U(1) models modified by introducing additional Pauli-Villars regularization, nonlocal effects are suppressed, and the results are in good agreement with the continuous-theory results. For chiral fermions, the lattice determinant phase differs from the determinant phase in the continuous theory. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 93–105, April, 1998