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arXiv1998

From 2D Integrable Systems to Self-Dual Gravity

We explain how to construct solutions to the self-dual Einstein vacuum equations from solutions of various two-dimensional integrable systems by exploiting the fact that the Lax formulations of both systems can be embedded in that of the self-dual Yang--Mills equations. We illustrate this by constructing explicit self-dual vacuum metrics on $\R^2\times Σ$, where $Σ$ is a homogeneous space for a real subgroup of $SL(2, \C)$ associated with the two-dimensional system.

solv-int↗
arXiv1997

Functional representation of the Ablowitz-Ladik hierarchy

The Ablowitz-Ladik hierarchy (ALH) is considered in the framework of the inverse scattering approach. After establishing the structure of solutions of the auxiliary linear problems, the ALH, which has been originally introduced as an infinite system of difference-differential equations is presented as a finite system of difference-functional equations. The representation obtained, when rewritten in terms of Hirota's bilinear formalism, is used to demonstrate relations between the ALH and some other integrable systems, the Kadomtsev-Petviashvili hierarchy in particular.

solv-int↗
arXiv1998

Explicit solution of the quantum three-body Calogero-Sutherland model

Quantum integrable systems generalizing Calogero-Sutherland systems were introduced by Olshanetsky and Perelomov (1977). Recently, it was proved that for systems with trigonometric potential, the series in the product of two wave functions is a deformation of the Clebsch-Gordan series. This yields recursion relations for the wave functions of those systems. In this note, this approach is used to compute the explicit expressions for the three-body Calogero-Sutherland wave functions, which are the Jack polynomials. We conjecture that similar results are also valid for the more general two-parameters deformation introduced by Macdonald.

hep-th↗
arXiv1998

Integrable Top Equations associated with Projective Geometry over Z_2

We give a series of integrable top equations associated with the projective geometry over Z_2 as a (2^n-1)-dimensional generalisation of the 3D Euler top equations. The general solution of the (2^n-1)D top is shown to be given by an integration over a Riemann surface with genus (2^{n-1}-1)^2.

math-ph↗
arXiv1998

Dynamical boundary conditions for integrable lattices

Some special solutions to the reflection equation are considered. These boundary matrices are defined on the common quantum space with the other operators in the chain. The relations with the Drinfeld twist are discussed.

solv-int↗
arXiv1998

The Complex Toda Chains and the Simple Lie Algebras - Solutions and Large Time Asymptotics

The asymptotic regimes of the N-site complex Toda chain (CTC) with fixed ends related to the classical series of simple Lie algebras are classified. It is shown that the CTC models have much richer variety of asymptotic regimes than the real Toda chain (RTC). Besides asymptotically free propagation (the only possible regime for the RTC), CTC allow bound state regimes, various intermediate regimes when one (or several) group(s) of particles form bound state(s), singular and degenerate solutions. These results can be used e.g., in describing the soliton interactions of the nonlinear Schroedinger equation. Explicit expressions for the solutions in terms of minimal sets of scattering data are proposed for all classical series B_r - D_r.

solv-int↗
arXiv1997

Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy

The algebraic matrix hierarchy approach based on affine Lie $sl (n)$ algebras leads to a variety of 1+1 soliton equations. By varying the rank of the underlying $sl (n)$ algebra as well as its gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy. The soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine $sl (n)$ algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial dependence on the KdV (odd) time flows and KP (odd and even) time flows which distinguishes them from the conventional structure of the Darboux-Bäcklund Wronskian solutions of the constrained KP hierarchy.

solv-int↗
arXiv1998

N-Soliton Solutions to a New (2 + 1) Dimensional Integrable Equation

We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, $ϕ_{xt} + ϕ_{xxxz}/4 + ϕ_x ϕ_{xz} + ϕ_{xx} ϕ_z/2 + \partial_x^{-1} ϕ_{zzz}/4 = 0$. This equation is obtained by unifying two directional generalization of the KdV equation, composing the closed ring with the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura transformation which yields the same ring in the corresponding modified equations.

solv-int↗
arXiv1997

Complete Nondiagonal Reflection Matrices of RSOS/SOS and Hard Hexagon Models

In this paper we compute the most general nondiagonal reflection matrices of the RSOS/SOS models and hard hexagon model using the boundary Yang-Baxter equations. We find new one-parameter family of reflection matrices for the RSOS model in addition to the previous result without any parameter. We also find three classes of reflection matrices for the SOS model, which has one or two parameters. For the hard hexagon model which can be mapped to RSOS(5) model by folding four RSOS heights into two, the solutions can be obtained similarly with a main difference in the boundary unitarity conditions. Due to this, the reflection matrices can have two free parameters. We show that these extra terms can be identified with the `decorated' solutions. We also generalize the hard hexagon model by `folding' the RSOS heights of the general RSOS(p) model and show that they satisfy the integrability conditions such as the Yang- Baxter and boundary Yang-Baxter equations. These models can be solved using the results for the RSOS models.

solv-int↗
arXiv1999

Critical and off-critical studies of the Baxter-Wu model with general toroidal boundary conditions

The operator content of the Baxter-Wu model with general toroidal boundary conditions is calculated analytically and numerically. These calculations were done by relating the partition function of the model with the generating function of a site-colouring problem in a hexagonal lattice. Extending the original Bethe-ansatz solution of the related colouring problem we are able to calculate the eigenspectra of both models by solving the associated Bethe-ansatz equations. We have also calculated, by exploring the conformal invariance at the critical point, the mass ratios of the underlying massive theory governing the Baxter-Wu model in the vicinity of its critical point.

cond-mat.stat-mech↗
arXiv1999

Relationship Between the Energy Eigenstates of Calogero-Sutherland Models With Oscillator and Coulomb-like Potentials

We establish a simple algebraic relationship between the energy eigenstates of the rational Calogero-Sutherland model with harmonic oscillator and Coulomb-like potentials. We show that there is an underlying SU(1,1) algebra in both of these models which plays a crucial role in such an identification. Further, we show that our analysis is in fact valid for any many-particle system in arbitrary dimensions whose potential term (apart from the oscillator or the Coulomb-like potential) is a homogeneous function of coordinates of degree -2. The explicit coordinate transformation which maps the Coulomb-like problem to the oscillator one has also been determined in some specific cases.

solv-int↗
arXiv1999

Multiscale Analysis of Discrete Nonlinear Evolution Equations

The method of multiscale analysis is constructed for dicrete systems of evolution equations for which the problem is that of the far behavior of an input boundary datum. Discrete slow space variables are introduced in a general setting and the related finite differences are constructed. The method is applied to a series of representative examples: the Toda lattice, the nonlinear Klein-Gordon chain, the Takeno system and a discrete version of the Benjamin-Bona-Mahoney equation. Among the resulting limit models we find a discrete nonlinear Schroedinger equation (with reversed space-time), a 3-wave resonant interaction system and a discrete modified Volterra model.

solv-int↗
arXiv1999

Dynamical correlation functions of the spin Calogero-Sutherland model

The thermodynamic limit of the dynamical density and spin-density two-point correlation functions for the spin Calogero-Sutherland model are derived from Uglov's finite-size results. The resultant formula for the density two-point correlation function is consistent with the previous conjecture on the basis of the minimal number of elementary excitations.

cond-mat.str-el↗
arXiv1999

Lax pair tensors in arbitrary dimensions

A recipe is presented for obtaining Lax tensors for any n-dimensional Hamiltonian system admitting a Lax representation of dimension n. Our approach is to use the Jacobi geometry and coupling-constant metamorphosis to obtain a geometric Lax formulation. We also exploit the results to construct integrable spacetimes, satisfying the weak energy condition.

solv-int↗
arXiv1999

Finite genus solutions for the Ablowitz-Ladik hierarchy

The question of constructing the finite genus quasiperiodic solutions for the Ablowitz-Ladik hierarchy (ALH) is considered by establishing relations between the ALH and the Fay's identity for the theta-functions. It is shown that using a limiting procedure one can derive from the latter an infinite number of differential identities which can be arranged as an infinite set of differential-difference equations coinciding with the equations of the ALH, and that the original Fay's identity can be rewritten in a form similar to the functional equations representing the ALH which have been derived in the previous works of the author. This provides an algorithm for obtaining some class of quasiperiodic solutions for the ALH, which can be viewed as an alternative to the inverse scattering transform or the algebro-geometrical approach.

solv-int↗