Solutions of the Generic Non-Compact Weyl Equation

In this paper, solutions of the generic non-compact Weyl equation are obtained. In particular, by identifying a suitable similarity transformation and introducing a non-trivial change of variables we are able to implement azimuthal dependence on the solutions of the diagonal non-compact Weyl equation. We also discuss some open questions related to the construction of infinite BPS monopole configurations.Comment: 12 pages, Latex. Few extra comments and a reference adde

Non-diagonal reflection for the non-critical XXZ model

The most general physical boundary $S$-matrix for the open XXZ spin chain in the non-critical regime ($\cosh (\eta)>1$) is derived starting from the bare Bethe ansazt equations. The boundary $S$-matrix as expected is expressed in terms of $\Gamma_q$-functions. In the isotropic limit corresponding results for the open XXX chain are also reproduced.Comment: 8 pages Late

Direct Calculation of Breather S Matrices

We formulate a systematic Bethe-Ansatz approach for computing bound-state (``breather'') S matrices for integrable quantum spin chains. We use this approach to calculate the breather boundary S matrix for the open XXZ spin chain with diagonal boundary fields. We also compute the soliton boundary S matrix in the critical regime.Comment: 23 pages, LaTeX, 1 eps figur

Fusion and Analytical Bethe Ansatz for the A_{\n-1}^{(1)} Open Spin Chain

We generalise the fusion procedure for the A_{\n-1}^{(1)} open spin chain (\n>2) and we show that the transfer matrix satisfies a crossing property. We use these results to solve the A_{\n-1}^{(1)} open spin chain with U_{q} (SU(\n)) symmetry by means of the analytical Bethe ansatz method. Our results coincide with the known ones obtained by means of the nested Bethe ansatz

New reflection matrices for the U_q(gl(m|n)) case

We examine super symmetric representations of the B-type Hecke algebra. We exploit such representations to obtain new non-diagonal solutions of the reflection equation associated to the super algebra U_q(gl(m|n)). The boundary super algebra is briefly discussed and it is shown to be central to the super symmetric realization of the B-type Hecke algebraComment: 13 pages, Latex. A few alterations regarding the representations. A reference adde

On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary

Motivated by a study of the crossing symmetry of the `gemini' representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang--Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley--Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the Boundary Yang--Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin's commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary -- despite differing from this model in having reference states.) Using this multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.Comment: 43 pages, multiple figure

Selected Topics in Classical Integrability

Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete and continuum classical integrable models is introduced. Using this framework the associated classical integrals of motion and the corresponding Lax pair are extracted based on algebraic considerations. Our attention is restricted to classical discrete and continuum integrable systems with periodic boundary conditions. Typical examples of discrete (Toda chain, discrete NLS model) and continuum integrable models (NLS, sine-Gordon models and affine Toda field theories) are also discussed.Comment: 40 pages, Latex. A few typos correcte

Partial differential systems with nonlocal nonlinearities: Generation and solutions

We develop a method for generating solutions to large classes of evolutionary partial differential systems with nonlocal nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples including reaction-diffusion systems with nonlocal quadratic nonlinearities and the nonlinear Schrodinger equation with a nonlocal cubic nonlinearity. In each case we demonstrate our approach with numerical simulations. We discuss the effectiveness of our approach and how it might be extended.Comment: 4 figure

Introduction to Quantum Integrability

In this article we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang-Baxter and boundary Yang-Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.Comment: 56 pages, Latex. A few typos correcte