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At least 19 recordsLinked to original sources

arXiv2022

Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation

We present several algebraic and differential-geometric constructions of tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. In particular, we obtain a family of new (nonlinear) polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang--Baxter equation. Liouville integrability is established for some of these maps. Also, we show how to derive linear tetrahedron maps as linear approximations of nonlinear ones, using Lax representations and the differentials of nonlinear tetrahedron maps on manifolds. We apply this construction to two nonlinear maps: a tetrahedron map obtained in [arXiv:1708.05694] in a study of soliton solutions of vector KP equations and a tetrahedron map obtained in [arXiv:2005.13574] in a study of a matrix trifactorisation problem related to a Darboux matrix associated with a Lax operator for the NLS equation. We derive parametric families of new linear tetrahedron maps (with nonlinear dependence on parameters), which are linear approximations for these nonlinear ones. Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron map from Sergeev's classification [arXiv:solv-int/9709006]. These matrix generalisations can be regarded as tetrahedron maps in noncommutative variables. Besides, several tetrahedron maps on arbitrary groups are constructed.

nlin.SI↗

arXiv1993

On Pentagon And Tetrahedron Equations

We show that solutions of Pentagon equations lead to solutions of the Tetrahedron equation. The result is obtained in the spectral parameter dependent case.

hep-th↗

arXiv1996

The symmetry structure of the heavenly equation

We show that excitations of physical interest of the heavenly equation are generated by symmetry operators which yields two reduced equations with different characteristics. One equation is of the Liouville type and gives rise to gravitational instantons, including those found by Eguchi-Hanson and Gibbons-Hawking. The second equation appears for the first time in the theory of heavenly spaces and provides meron-like configurations endowed with a fractional topological charge. A link is also established between the heavenly equation and the socalled Schr{ö}der equation, which plays a crucial role in the bootstrap model and in the renormalization theory.

hep-th↗

arXiv1998

Bispectrality of KP Solitons

It is by now well known that the wave functions of rational solutions to the KP hierarchy (those which can be achieved as limits of the pure n-soliton solutions) satisfy an additional eigenvalue equation for ordinary differential operators in the spectral parameter. This property is known as ``bispectrality'' and has proved to be both interesting and useful. In this note, it is shown that certain (non-rational) soliton solutions of the KP hierarchy satisfy an eigenvalue equation for a non-local operator constructed by composing ordinary differential operators in the spectral parameter with translation operators in the spectral parameter, and therefore have a form of bispectrality as well. Considering the results relating ordinary bispectrality to the self-duality of the rational Calogero-Moser particle system, it seems likely that this new form of bispectrality should be related to the duality of the Ruijsenaars system.

math-ph↗

arXiv1998

Darboux Transformations of Bispectral Quantum Integrable Systems

We present an approach to higher dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the algebro-geometric definition of quantum integrability, we utilize the bispectral duality of quantum Hamiltonian systems to construct non-trivial Darboux transformations between completely integrable quantum systems. As an application, we are able to construct new quantum integrable systems as the Darboux transforms of trivial examples (such as symmetric products of one dimensional systems) or by Darboux transformation of well-known bispectral systems such as quantum Calogero-Moser.

math-ph↗

arXiv1998

On solutions of the Schlesinger Equations in Terms of $Θ$-Functions

In this paper we construct explicit solutions and calculate the corresponding $τ$-function to the system of Schlesinger equations describing isomonodromy deformations of $2\times 2$ matrix linear ordinary differential equation whose coefficients are rational functions with poles of the first order; in particular, in the case when the coefficients have four poles of the first order and the corresponding Schlesinger system reduces to the sixth Painlevé equation with the parameters $1/8, -1/8, 1/8, 3/8$, our construction leads to a new representation of the general solution to this Painlevé equation obtained earlier by K. Okamoto and N. Hitchin, in terms of elliptic theta-functions.

math-ph↗

arXiv1998

Linear r-Matrix Algebra for a Hierarchy of One-Dimensional Particle Systems Separable in Parabolic Coordinates

We consider a hierarchy of many-particle systems on the line with polynomial potentials separable in parabolic coordinates. The first non-trivial member of this hierarchy is a generalization of an integrable case of the Hénon-Heiles system. We give a Lax representation in terms of $2\times 2$ matrices for the whole hierarchy and construct the associated linear r-matrix algebra with the r-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is proposed. Classical integration in a particular case is carried out and quantization of the system is discussed with the help of separation variables. This paper was published in the rary issues: Sfb 288 Preprint No. 110, Berlin and Nonlinear Mathematical Physics, {\bf 1(3)}, 275-294 (1994)

solv-int↗

arXiv1999

Self-similar solutions of NLS-type dynamical systems

We study self-similar solutions of NLS-type dynamical systems. Lagrangian approach is used to show that they can be reduced to three canonical forms, which are related by Miura transformations. The fourth Painleve equation (PIV) is central in our consideration - it connects Heisenberg model, Volterra model and Toda model to each other. The connection between the rational solutions of PIV and Coulomb gas in a parabolic potential is established. We discuss also the possibility to obtain an exact solution for optical soliton i.e. of the NLS equation with time-dependent dispersion.

solv-int↗

arXiv1999

Self-similarity in Spectral Problems and q-special Functions

Similarity symmetries of the factorization chains for one-dimensional differential and finite-difference Schrödinger equations are discussed. Properties of the potentials defined by self-similar reductions of these chains are reviewed. In particular, their algebraic structure, relations to $q$-special functions, infinite soliton systems, supersymmetry, coherent states, orthogonal polynomials, one-dimensional Ising chains and random matrices are outlined.

solv-int↗

arXiv1999

Resonant Bifurcations

We consider dynamical systems depending on one or more real parameters, and assuming that, for some ``critical'' value of the parameters, the eigenvalues of the linear part are resonant, we discuss the existence -- under suitable hypotheses -- of a general class of bifurcating solutions in correspondence to this resonance. These bifurcating solutions include, as particular cases, the usual stationary and Hopf bifurcations. The main idea is to transform the given dynamical system into normal form (in the sense of Poincaré-Dulac), and to impose that the normalizing transformation is convergent, using the convergence conditions in the form given by A. Bruno. Some specially interesting situations, including the cases of multiple-periodic solutions, and of degenerate eigenvalues in the presence of symmetry, are also discussed with some detail.

solv-int↗

arXiv1999

The Pfaff lattice, Matrix integrals and a map from Toda to Pfaff

We study the Pfaff lattice, introduced by us in the context of a Lie algebra splitting of gl(infinity) into sp(infinity) and lower-triangular matrices. We establish a set of bilinear identities, which we show to be equivalent to the Pfaff Lattice. In the semi-infinite case, the tau-functions are Pfaffians; interesting examples are the matrix integrals over symmetric matrices (symmetric matrix integrals) and matrix integrals over self-dual quaternionic Hermitean matrices (symplectic matrix integrals). There is a striking parallel of the Pfaff lattice with the Toda lattice, and more so, there is a map from one to the other. In particular, we exhibit two maps, dual to each other, (i) from the the Hermitean matrix integrals to the symmetric matrix integrals, and (ii) from the Hermitean matrix integrals to the symplectic matrix integrals. The map is given by the skew-Borel decomposition of a skew-symmetric operator. We give explicit examples for the classical weights.

solv-int↗

arXiv1997

Hidden Symmetries of the Principal Chiral Model Unveiled

By relating the two-dimensional U(N) Principal Chiral Model to a simple linear system we obtain a free-field parametrisation of solutions. Obvious symmetry transformations on the free-field data give symmetries of the model. In this way all known `hidden symmetries' and Bäcklund transformations, as well as a host of new symmetries, arise.

hep-th↗

arXiv1994

Unified Approach to Thermodynamic Bethe Ansatz and Finite Size Corrections for Lattice Models and Field Theories

We present a unified approach to the Thermodynamic Bethe Ansatz (TBA) for magnetic chains and field theories that includes the finite size (and zero temperature) calculations for lattice BA models. In all cases, the free energy follows by quadratures from the solution of a {\bf single} non-linear integral equation (NLIE). [A system of NLIE appears for nested BA]. We derive the NLIE for: a) the six-vertex model with twisted boundary conditions; b) the XXZ chain in an external magnetic field $h_z$ and c) the sine-Gordon-massive Thirring model (sG-mT) in a periodic box of size $\b \equiv 1/T $ using the light-cone approach. This NLIE is solved by iteration in one regime (high $T$ in the XXZ chain and low $T$ in the sG-mT model). In the opposite (conformal) regime, the leading behaviors are obtained in closed form. Higher corrections can be derived from the Riemann-Hilbert form of the NLIE that we present.

hep-th↗

arXiv1998

Spontaneous magnetization of the XXZ Heisenberg spin-1/2 chain

Determinant representations of form factors are used to represent the spontaneous magnetization of the Heisenberg XXZ chain (Delta >1) on the finite lattice as the ratio of two determinants. In the thermodynamic limit (the lattice of infinite length), the Baxter formula is reproduced in the framework of Algebraic Bethe Ansatz. It is shown that the finite size corrections to the Baxter formula are exponentially small.

solv-int↗

arXiv1998

Form factors of the XXZ Heisenberg spin-1/2 finite chain

Form factors for local spin operators of the XXZ Heisenberg spin-1/2 finite chain are computed. Representation theory of Drinfel'd twists for the sl2 quantum affine algebra in finite dimensional modules is used to calculate scalar products of Bethe states (leading to Gaudin formula) and to solve the quantum inverse problem for local spin operators in the finite XXZ chain. Hence, we obtain the representation of the n-spin correlation functions in terms of expectation values(in ferromagnetic reference state) of the operator entries of the quantum monodromy matrix satisfying Yang-Baxter algebra. This leads to the direct calculation of the form factors of the XXZ Heisenberg spin-1/2 finite chain as determinants of usual functions of the parameters of the model. A two-point correlation function for adjacent sites is also derived using similar techniques.

math-ph↗

arXiv2023

Evolution equations for quark-gluon distributions in multi-color QCD and open spin chains

We study the scale dependence of the twist-3 quark-gluon parton distributions using the observation that in the multi-color limit the corresponding QCD evolution equations possess an additional integral of motion and turn out to be effectively equivalent to the Schrodinger equation for integrable open Heisenberg spin chain model. We identify the integral of motion of the spin chain as a new quantum number that separates different components of the twist-3 parton distributions. Each component evolves independently and its scale dependence is governed by anomalous dimension given by the energy of the spin magnet. To find the spectrum of the QCD induced open Heisenberg spin magnet we develop the Bethe Ansatz technique based on the Baxter equation. The solutions to the Baxter equation are constructed using different asymptotic methods and their properties are studied in detail. We demonstrate that the obtained solutions provide a good qualitative description of the spectrum of the anomalous dimensions and reveal a number of interesting properties. We show that the few lowest anomalous dimensions are separated from the rest of the spectrum by a finite mass gap and estimate its value.

hep-ph↗

arXiv1997

Normal Ordering in the Theory of Correlation Functions of Exactly Solvable Models

We study models of quantum statistical mechanics which can be solved by the algebraic Bethe ansatz. The general method of calculation of correlation functions is based on the method of determinant representations. The auxiliary Fock space and auxiliary Bose fields are introduced in order to remove the two body scattering and represent correlation functions as a mean value of a determinant of a Fredholm integral operator; the representation has a simple form for large space and time separations. In this paper we explain how to calculate the mean value in the auxiliary Fock space of asymptotic expression of the Fredholm determinant. It is necessary for the evaluation of the asymptotic form of the physical correlation functions.

hep-th↗

arXiv1999

Boundary breathers in the sinh-Gordon model

We present an investigation of the boundary breather states of the sinh-Gordon model restricted to a half-line. The classical boundary breathers are presented for a two parameter family of integrable boundary conditions. Restricting to the case of boundary conditions which preserve the ϕ--> -ϕsymmetry of the bulk theory, the energy spectrum of the boundary states is computed in two ways: firstly, by using the bootstrap technique and subsequently, by using a WKB approximation. Requiring that the two descriptions of the spectrum agree with each other allows a determination of the relationship between the boundary parameter, the bulk coupling constant, and the parameter appearing in the reflection factor derived by Ghoshal to describe the scattering of the sinh-Gordon particle from the boundary.

hep-th↗