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A. Zabrodin

Publications and source records attributed to A. Zabrodin.

At least 109 records · Page 6Linked to original sources
arXiv2002

Viscous fingering and a shape of an electronic droplet in the Quantum Hall regime

We show that the semiclassical dynamics of an electronic droplet confined in the plane in a quantizing inhomogeneous magnetic field in the regime when the electrostatic interaction is negligible is similar to viscous (Saffman-Taylor) fingering on the interface between two fluids with different viscosities confined in a Hele-Shaw cell. Both phenomena are described by the same equations with scales differing by a factor of up to $10^{-9}$. We also report the quasiclassical wave function of the droplet in an inhomogeneous magnetic field.

cond-mat↗
arXiv2001

Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions

We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green function under deformations of the domain reveals an integrable structure. The independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain. The solution to the Dirichlet boundary problem is expressed through the tau-function of the dispersionless Toda hierarchy. We also discuss a degenerate case of the Dirichlet problem on the plane with a gap. In this case the tau-function is identical to the partition function of the planar large $N$ limit of the Hermitean one-matrix model.

hep-th↗
arXiv2001

On Associativity Equations in Dispersionless Integrable Hierarchies

We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda hierarchies. We show, therefore, that any tau-function of dispersionless KP or Toda hierarchy provides a solution to associativity equations. In general, they depend on infinitely many variables. We also discuss the particular solution to the dispersionless Toda hierarchy that describes conformal mappings and construct a family of new solutions to the WDVV equations depending on finite number of variables.

hep-th↗
arXiv2001

Dispersionless limit of Hirota equations in some problems of complex analysis

The integrable structure, recently revealed in some classical problems of the theory of functions in one complex variable, is discussed. Given a simply connected domain in the complex plane, bounded by a simple analytic curve, we consider the conformal mapping problem, the Dirichlet boundary problem, and to the 2D inverse potential problem associated with the domain. A remarkable family of real-valued functionals on the space of such domains is constructed. Regarded as a function of infinitely many variables, which are properly defined moments of the domain, any functional from the family gives a formal solution to the problems listed above. These functions are shown to obey an infinite set of dispersionless Hirota equations. This means that they are $τ$-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda lattice. In addition to our previous studies, we show that, with a more general definition of the moments, this connection is not specific for any particular solution to the Hirota equations but reflects the structure of the hierarchy itself.

math.CV↗
arXiv2000

$τ$-function for analytic curves

We review the concept of $τ$-function for simple analytic curves. The $τ$-function gives a formal solution to the 2D inverse potential problem and appears as the $τ$-function of the integrable hierarchy which describes conformal maps of simply-connected domains bounded by analytic curves to the unit disk. The $τ$-function also emerges in the context of topological gravity and enjoys an interpretation as a large $N$ limit of the normal matrix model.

hep-th↗
arXiv2000

Conformal maps and dispersionless integrable hierarchies

We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy singled out by the conditions known as "string equations". The same hierarchy locally solves the 2D inverse potential problem, i.e. reconstruction of the domain out of a set of its harmonic moments. This is the same solution which is known to describe 2D gravity coupled to c=1 matter. We also introduce a concept of the $τ$-function for analytic curves.

hep-th↗
arXiv1999

Finite-gap difference opeators with elliptic coefficients and their spectral curves

We review recent results on the finite-gap properties of difference operators with elliptic coefficients and give explicit characterization of spectral curves for difference analogues of the higher Lamé operators. This curve parametrizes double-Bloch solutions to the difference Lamé equation. The curve depends on a positive integer number \ell, related to its genus g by g=2\ell, and two continuous parameters: the lattice spacing ηand the modular parameter τ. Isospectral deformations of the difference Lamé operator under Volterra flows are also discussed.

math.QA↗
arXiv1998

On the spectral curve of the difference Lamé operator

We give two "complementary" descriptions of the curve $Γ$ parametrizing double-Bloch solutions to the difference analogue of the Lamé equation. The curve depends on a positive integer number $\ell$ and two continuous parameters: the "lattice spacing" $η$ and the modular parameter $τ$. Apart from being a covering of the elliptic curve with the modular parameter $τ$, $Γ$ is a hyperelliptic curve of genus $2\ell$. We also point out connections between the spectral curve and representations of the Sklyanin algebra.

math.QA↗
arXiv1998

Tau-function for discrete sine-Gordon equation and quantum R-matrix

We prove that the tau-function of the integrable discrete sine-Gordon model apart from the "standard" bilinar identities obeys a number of "non-standard" ones. They can be combined into a bivector 3-dimensional difference equation which is shown to contain Hirota's difference analogue of the sine-Gordon equation and both auxiliary linear problems for it. We observe that this equation is most naturally written in terms of the quantum R-matrix for the XXZ spin chain and looks then like a relation of the "vertex-face correspondence" type.

solv-int↗
arXiv1998

Vacuum curves of elliptic L-operators and representations of Sklyanin algebra

An algebro-geometric approach to representations of Sklyanin algebra is proposed. To each 2 \times 2 quantum L-operator an algebraic curve parametrizing its possible vacuum states is associated. This curve is called the vacuum curve of the L-operator. An explicit description of the vacuum curve for quantum L-operators of the integrable spin chain of XYZ type with arbitrary spin $\ell$ is given. The curve is highly reducible. For half-integer $\ell$ it splits into $\ell +{1/2}$ components isomorphic to an elliptic curve. For integer $\ell$ it splits into $\ell$ elliptic components and one rational component. The action of elements of the L-operator to functions on the vacuum curve leads to a new realization of the Sklyanin algebra by difference operators in two variables restricted to an invariant functional subspace.

solv-int↗
arXiv1998

Hidden quantum R-matrix in discrete time classical Heisenberg magnet

We construct local M-operators for an integrable discrete time version of the classical Heisenberg magnet by convolution of the twisted quantum trigonometric 4$\times$4 R-matrix with certain vectors in its "quantum" space. Components of the vectors are identified with $τ$-functions of the model. Hirota's bilinear formalism is extensively used. The construction generalizes the known representation of M-operators in continuous time models in terms of Lax operators and classical $r$-matrix.

solv-int↗
arXiv1997

Zero curvature representation for classical lattice sine-Gordon equation via quantum R-matrix

Local M-operators for the classical sine-Gordon model in discrete space-time are constructed by convolution of the quantum trigonometric 4$\times$4 R-matrix with certain vectors in its "quantum" space. Components of the vectors are identified with $τ$-functions of the model. This construction generalizes the known representation of M-operators in continuous time models in terms of Lax operators and classical $r$-matrix.

hep-th↗
arXiv1997

On trigonometric intertwining vectors and non-dynamical R-matrix for the Ruijsenaars model

We elaborate the trigonometric version of intertwining vectors and factorized L-operators. The starting point is the corresponding elliptic construction with Belavin's R-matrix. The naive trigonometric limit is singular and a careful analysis is needed. It is shown that the construction admits several different trigonometric degenerations. As a by-product, a quantum Lax operator for the trigonometric Ruijsenaars model intertwined by a non-dynamical R-matrix is obtained. The latter differs from the standard trigonometric R-matrix of $A_n$ type. A connection with the dynamical R-matrix approach is discussed.

hep-th↗
arXiv1997

Fusion rules for Quantum Transfer Matrices as a Dynamical System on Grassmann Manifolds

We show that the set of transfer matrices of an arbitrary fusion type for an integrable quantum model obey these bilinear functional relations, which are identified with an integrable dynamical system on a Grassmann manifold (higher Hirota equation). The bilinear relations were previously known for a particular class of transfer matrices corresponding to rectangular Young diagrams. We extend this result for general Young diagrams. A general solution of the bilinear equations is presented.

solv-int↗
arXiv1997

Elliptic solutions to difference non-linear equations and related many-body problems

We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for $τ$-functions. Starting from a given algebraic curve, we express the $τ$-function and the Baker-Akhiezer function in terms of the Riemann theta function. We show that the elliptic solutions, when the $τ$-function is an elliptic polynomial, form a subclass of the general algebro-geometric solutions. We construct the algebraic curves of the elliptic solutions. The evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the Ruijsenaars-Schneider many body system. The zeros obey equations which have the form of nested Bethe-Ansatz equations, known from integrable quantum field theories. We discuss the Lax representation and the action-angle-type variables for the many body system. We also discuss elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D Toda equations and describe the loci of the zeros.

hep-th↗
arXiv1997

A survey of Hirota's difference equations

A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations. Similarly to the continuous theory, HBDE is a member of an infinite hierarchy. The central point of our exposition is a discrete version of the zero curvature condition explicitly written in the form of discrete Zakharov-Shabat equations for M-operators realized as difference or pseudo-difference operators. A unified approach to various types of M-operators and zero curvature representations is suggested. Different reductions of HBDE to 2-dimensional equations are considered. Among them discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical examples are discussed in detail.

solv-int↗
arXiv1996

Discrete Hirota's equation in quantum integrable models

The recent progress in revealing classical integrable structures in quantum models solved by Bethe ansatz is reviewed. Fusion relations for eigenvalues of quantum transfer matrices can be written in the form of classical Hirota's bilinear difference equation. This equation is also known as the completely discretized version of the 2D Toda lattice. We explain how one obtains the specific quantum results by solving the classical equation. The auxiliary linear problem for the Hirota equation is shown to generalize Baxter's T-Q relation.

hep-th↗