# Exactly Solvable And Integrable Systems

###### Featured Researches

## 'Universality' of the Ablowitz-Ladik hierarchy

The aim of this paper is to summarize some recently obtained relations between the Ablowitz-Ladik hierarchy (ALH) and other integrable equations. It has been shown that solutions of finite subsystems of the ALH can be used to derive a wide range of solutions for, e.g., the 2D Toda lattice, nonlinear Schrödinger, Davey-Stewartson, Kadomtsev-Petviashvili (KP) and some other equations. Similar approach has been used to construct new integrable models: O(3,1) and multi-field sigma models. Such 'universality' of the ALH becomes more transparent in the framework of the Hirota's bilinear method. The ALH, which is usually considered as an infinite set of differential-difference equations, has been presented as a finite system of functional-difference equations, which can be viewed as a generalization of the famous bilinear identities for the KP tau-functions.

Read more## 3D symplectic map

Quantum 3D R-matrix in the classical (i.e. functional) limit gives a symplectic map of dynamical variables. The corresponding 3D evolution model is considered. An auxiliary problem for it is a system of linear equations playing the role of the monodromy matrix in 2D models. A generating function for the integrals of motion is constructed as a determinant of the auxiliary system.

Read more## A (1) n Toda Solitons: a Relation between Dressing transformations and Vertex Operators

Affine Toda equations based on simple Lie algebras arise by imposing zero curvature condition on a Lax connection which belongs to the corresponding loop Lie algebra in the principal gradation. In the particular case of A (1) n Toda models, we exploit the symmetry of the underlying linear problem to calculate the dressing group element which generates arbitrary N -soliton solution from the vacuum. Starting from this result we recover the vertex operator representation of the soliton tau functions.

Read more## A (2+1) dimensional integrable spin model: Geometrical and gauge equivalent counterpart, solitons and localized coherent structures

A non-isospectral (2+1) dimensional integrable spin equation is investigated. It is shown that its geometrical and gauge equivalent counterparts is the (2+1) dimensional nonlinear Schrödinger equation introduced by Zakharov and studied recently by Strachan. Using a Hirota bilinearised form, line and curved soliton solutions are obtained. Using certain freedom (arbitrariness) in the solutions of the bilinearised equation, exponentially localized dromion-like solutions for the potential is found. Also, breaking soliton solutions (for the spin variables) of the shock wave type and algebraically localized nature are constructed.

Read more## A Bethe ansatz solution for the closed U q [sl(2)] Temperley-Lieb quantum spin chains

We solve the spectrum pf the closed Temperley-Lieb quantum spin chains using the coordinate Bethe ansatz. These Hamiltonians are invariante under the quantum group U q [sl(2)]

Read more## A Bilinear Approach to Discrete Miura Transformations

We present a systematic approach to the construction of Miura transformations for discrete Painlevé equations. Our method is based on the bilinear formalism and we start with the expression of the nonlinear discrete equation in terms of τ -functions. Elimination of τ -functions from the resulting system leads to another nonlinear equation, which is a ``modified'' version of the original equation. The procedure therefore yields Miura transformations. In this letter, we illustrate this approach by reproducing previously known Miura transformations and constructing new ones.

Read more## A Characterization of All Elliptic Solutions of the AKNS Hierarchy

An explicit characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy is presented. Our approach is based on (an extension of) a classical theorem of Picard, which guarantees the existence of solutions which are elliptic of the second kind for n-th order ordinary differential equations with elliptic coefficients associated with a common period lattice. As by-products we offer a detailed Floquet analysis of Dirac-type differential expressions with periodic coefficients, specifically emphasizing algebro-geometric coefficients, and a constructive reduction of singular hyperelliptic curves and their Baker-Akhiezer functions to the nonsingular case.

Read more## A Class of Coupled KdV systems and Their Bi-Hamiltonian Formulations

A Hamiltonian pair with arbitrary constants is proposed and thus a sort of hereditary operators is resulted. All the corresponding systems of evolution equations possess local bi-Hamiltonian formulation and a special choice of the systems leads to the KdV hierarchy. Illustrative examples are given.

Read more## A Coupled AKNS-Kaup-Newell Soliton Hierarchy

A coupled AKNS-Kaup-Newell hierarchy of systems of soliton equations is proposed in terms of hereditary symmetry operators resulted from Hamiltonian pairs. Zero curvature representations and tri-Hamiltonian structures are established for all coupled AKNS-Kaup-Newell systems in the hierarchy. Therefore all systems have infinitely many commuting symmetries and conservation laws. Two reductions of the systems lead to the AKNS hierarchy and the Kaup-Newell hierarchy, and thus those two soliton hierarchies also possess tri-Hamiltonian structures.

Read more## A Dirac Sea and thermodynamic equilibrium for the quantized three-wave interaction

The classical version of the three wave interaction models the creation and destruction of waves; the quantized version models the creation and destruction of particles. The quantum three wave interaction is described and the Bethe Ansatz for the eigenfunctions is given in closed form. The Bethe equations are derived in a rigorous fashion and are shown to have a thermodynamic limit. The Dirac sea of negative energy states is obtained as the infinite density limit. Finite particle/hole excitations are determined and the asymptotic relation of energy and momentum is obtained. The Yang-Yang functional for the relative free energy of finite density excitations is constructed and is shown to be convex and bounded below. The equations of thermal equilibrium are obtained.

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