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arXiv1999

Chern-Simons Solitons in Quantum Potential

The self-dual Chern-Simons solitons under the influence of the quantum potential are considered. The single-valuedness condition for an arbitrary integer number $N \ge 0$ of solitons leads to quantization of Chern-Simons coupling constant $κ= m {e^{2} \over g}$, and the integer strength of quantum potential $s = 1 - m^{2}$. As we show, the Jackiw-Pi model corresponds to the first member (m = 1) of our hierarchy of the Chern-Simons gauged nonlinear Schrödinger models, admitting self-dual solitons. New type of exponentially localized Chern-Simons solitons for the Bloch electrons near the hyperbolic energy band boundary are found.

hep-th↗
arXiv1998

On the relation between orthogonal, symplectic and unitary matrix ensembles

For the unitary ensembles of $N\times N$ Hermitian matrices associated with a weight function $w$ there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are $2\times2$ matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever $w'/w$ is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of $w'/w$. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations.

solv-int↗
arXiv1999

Determinants of Airy Operators and Applications to Random Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener-Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum.

math.FA↗
arXiv1997

Generating function of correlators in the sl_2 Gaudin model

For the sl_2 Gaudin model (degenerated quantum integrable XXX spin chain) an exponential generating function of correlators is calculated explicitely. The calculation relies on the Gauss decomposition for the SL_2 loop group. From the generating function a new explicit expression for the correlators is derived from which the determinant formulas for the norms of Bethe eigenfunctions due to Richardson and Gaudin are obtained.

solv-int↗
arXiv1999

Classical skew orthogonal polynomials and random matrices

Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the cases that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre and Jacobi cases.

solv-int↗
arXiv1997

Lax pair tensors and integrable spacetimes

The use of Lax pair tensors as a unifying framework for Killing tensors of arbitrary rank is discussed. Some properties of the tensorial Lax pair formulation are stated. A mechanical system with a well-known Lax representation -- the three-particle open Toda lattice -- is geometrized by a suitable canonical transformation. In this way the Toda lattice is realized as the geodesic system of a certain Riemannian geometry. By using different canonical transformations we obtain two inequivalent geometries which both represent the original system. Adding a timelike dimension gives four-dimensional spacetimes which admit two Killing vector fields and are completely integrable.

gr-qc↗
arXiv1999

Exact Solution of the Quantum Calogero-Gaudin System and of its q-Deformation

A complete set of commuting observables for the Calogero-Gaudin system is diagonalized, and the explicit form of the corresponding eigenvalues and eigenfunctions is derived. We use a purely algebraic procedure exploiting the co-algebra invariance of the model; with the proper technical modifications this procedure can be applied to the $q-$deformed version of the model, which is then also exactly solved.

solv-int↗
arXiv1993

Quantum Groups

These notes correspond rather accurately to the translation of the lectures given at the Fifth Mexican School of Particles and Fields, held in Guanajuato, Gto., in December~1992. They constitute a brief and elementary introduction to quantum symmetries from a physical point of view, along the lines of the forthcoming book by C. Gómez, G. Sierra and myself.

hep-th↗
arXiv1993

Casorati Determinant Solution for the Relativistic Toda Lattice Equation

The relativistic Toda lattice equation is decomposed into three Toda systems, the Toda lattice itself, Bäcklund transformation of Toda lattice and discrete time Toda lattice. It is shown that the solutions of the equation are given in terms of the Casorati determinant. By using the Casoratian technique, the bilinear equations of Toda systems are reduced to the Laplace expansion form for determinants. The $N$-soliton solution is explicitly constructed in the form of the Casorati determinant.

solv-int↗
arXiv1992

Constant Solutions of Reflection Equations and Quantum Groups

To the Yang-Baxter equation an additional relation can be added. This is the reflection equation which appears in various places, with or without spectral parameter. For example, in factorizable scattering on a half-line, integrable lattice models with non-periodic boundary conditions, non-commutative differential geometry on quantum groups, etc. We study two forms of spectral parameter independent reflection equations, chosen by the requirement that their solutions be comodules with respect to the quantum group coaction leaving invariant the reflection equations. For a variety of known solutions of the Yang-Baxter equation we give the constant solutions of the reflection equations. Various quadratic algebras defined by the reflection equations are also given explicitly.

hep-th↗
arXiv1993

Two-Dimensional Integrable Systems and Self-Dual Yang-Mills Equations

The relation between two--dimensional integrable systems and four--dimen\-sional self--dual Yang--Mills equations is considered. Within the twistor description and the zero--curvature representation a method is given to associate self--dual Yang-Mills connections with integrable systems of the Korteweg--de Vries and non--linear Schrödinger type or principal chiral models. Examples of self--dual connections are constructed that as points in the moduli do not have two independent conformal symmetries.

hep-th↗
arXiv1994

Local Geometric Invariants of Integrable Evolution Equations

The integrable hierarchy of commuting vector fields for the localized induction equation of 3D hydrodynamics, and its associated recursion operator, are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept-out surface.

solv-int↗
arXiv1995

Quantum chains with a Catalan tree pattern of conserved charges: the $Δ= -1$ XXZ model and the isotropic octonionic chain

A class of quantum chains possessing a family of local conserved charges with a Catalan tree pattern is studied. Recently, we have identified such a structure in the integrable $SU(N)$-invariant chains. In the present work we find sufficient conditions for the existence of a family of charges with this structure in terms of the underlying algebra. Two additional systems with a Catalan tree structure of conserved charges are found. One is the spin 1/2 XXZ model with $Δ=-1$. The other is a new octonionic isotropic chain, generalizing the Heisenberg model. This system provides an interesting example of an infinite family of noncommuting local conserved quantities.

hep-th↗
arXiv1994

The Korteweg-de Vries Hierarchy and Long Water-Waves

By using the multiple scale method with the simultaneous introduction of multiple times, we study the propagation of long surface-waves in a shallow inviscid fluid. As a consequence of the requirements of scale invariance and absence of secular terms in each order of the perturbative expansion, we show that the Korteweg-de Vries hierarchy equations do appear in the description of such waves. Finally, we show that this procedure of eliminating secularities is closely related to the renormalization technique introduced by Kodama and Taniuti.

patt-sol↗
arXiv1995

Infinite-Dimensional Estabrook-Wahlquist Prolongations for the sine-Gordon Equation

We are looking for the universal covering algebra for all symmetries of a given pde, using the sine-Gordon equation as a typical example for a non-evolution equation. For non-evolution equations, Estabrook-Wahlquist prolongation structures for non-local symmetries depend on the choice of a specific sub-ideal, of the contact module, to define the pde. For each inequivalent such choice we determine the most general solution of the prolongation equations, as sub-algebras of the (infinite-dimensional) algebra of all vector fields over the space of non-local variables associated with the pde, in the style of Vinogradov covering spaces. We show explicitly how previously-known prolongation structures, known to lie within the Kac-Moody algebra, $A_1^{(1)}$, are special cases of these general solutions, although we are unable to identify the most general solutions with previously-studied algebras. We show the existence of gauge transformations between prolongation structures, viewed as determining connections over the solution space, and use these to relate (otherwise) distinct algebras. Faithful realizations of the universal algebra allow integral representations of the prolongation structure, opening up interesting connections with algebras of Toeplitz operators over Banach spaces, an area that has only begun to be explored.

solv-int↗
arXiv1994

$R$--Matrix Construction of Electromagnetic Models for the Painlevé Transcendents

The Painlevé transcendents $P_{\rom{I}}$--$P_{\rom{V}}$ and their representations as isomonodromic deformation equations are derived as nonautonomous Hamiltonian systems from the classical $R$--matrix Poisson bracket structure on the dual space $\wt{\frak{sl}}_R^*(2)$ of the loop algebra $\wt{\frak{sl}}_R(2)$. The Hamiltonians are obtained by composing elements of the Poisson commuting ring of spectral invariant functions on $\wt{\frak{sl}}_R^*(2)$ with a time--dependent family of Poisson maps whose images are $4$--dimensional rational coadjoint orbits in $\wt{\frak{sl}}_R^*(2)$. Each system may be interpreted as describing a particle moving on a surface of zero curvature in the presence of a time--varying electromagnetic field. The Painlevé equations follow from reduction of these systems by the Hamiltonian flow generated by a second commuting element in the ring of spectral invariants.

hep-th↗
arXiv1996

Nonlinear Discrete Systems with Nonanalytic Dispersion Relations

A discrete system of coupled waves (with nonanalytic dispersion relation) is derived in the context of the spectral transform theory for the Ablowitz Ladik spectral problem (discrete version of the Zakharov-Shabat system). This 3-wave evolution problem is a discrete version of the stimulated Raman scattering equations, and it is shown to be solvable for arbitrary boundary value of the two radiation fields and initial value of the medium state. The spectral transform is constructed on the basis of the D-bar approach.

solv-int↗