Abstract. Let W be an integrable positive Hermitian q × q-matrix valued function on the dual group of a discrete abelian group G such that W is integrable. Generalizing results of T. Nakazi and ofA. G. Miamee [N] andM. Pourahmadi [MiP] for q = 1 we establish a correspondencebetween trigonometric approximation problems in L(W ) and certain approximation problems in L(W). The result is applied to...

The spectral statistics of the circular billiard with a point-scatterer is investigated. In the semiclassical limit, the spectrum is demonstrated to be composed of two uncorrelated level sequences. The first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the sec...

For the integrable N -particle Calogero-Moser system with elliptic potential it is shown that the Lax operator found by Krichever possesses a classical r-matrix structure. The r-matrix is a natural generalisation of the matrix found recently by Avan and Talon (hep-th/9210128) for the trigonometric potential. The r-matrix depends on the spectral parameter and only half of the dynamical variables...

Abstract: Based on the work by Smirnov and Zeitlin, we study a simple realization of the matrix construction of the affine Jacobi varieties. We find that the realization is given by a classical integrable model, the extended Lotka-Volterra lattice. We investigate the integrable structure of the representative for the gauge equivalence class of matrices, which is isomorphic to the affine Jacobi ...

This paper gives an integrable hierarchy of nonlinear evolution equations. In this hierarchy there are the following representative equations: ut = ∂ 5 xu − 2 3 , ut = ∂ 5 x (u 1 3 )xx − 2(u 1 6 )x u ; uxxt + 3uxxux + uxxxu = 0. The first two are in the positive order hierarchy while the 3rd one is in the negative order hierarchy. The whole hierarchy is shown integrable through solving a key 3×...

We propose a spinon basis for the integrable highest weight modules of ŝl2 at levels k ≥ 1, and discuss the underlying Yangian symmetry. Evaluating the characters in this spinon basis provides new quasiparticle type expressions for the characters of these integrable modules, and explicitly exhibits the structure of an RSOS times a Yangian part, known e.g. from S-matrix results. We briefly discu...

In these notes we review the method to construct integrable deformations of the compactified c = 1 bosonic string theory by primary fields (momentum or winding modes), developed recently in collaboration with S. Alexandrov and V. Kazakov. The method is based on the formulation of the string theory as a matrix model. The flows generated by either momentum or winding modes (but not both) are inte...

We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of nonlinear PDEs linearizable by the matrix Hopf-Cole substitution (the Bürgers hierarchy). We derive examples of four-dimensional nonlinear matrix PDEs together wi...

This paper contains the generalization of the Feigin-Stoyanovsky construction to all integrable sl r+1-modules. We give formulas for the q-characters of any highest-weight integrable module of sl r+1 as a linear combination of the fermionic q-characters of the fusion products of a special set of integrable modules. The coefficients in the sum are the entries of the inverse matrix of generalized...

We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework for various multidimensional spinning tops, as well as Beauville systems, we prove that if the spectral curve is nodal, then all singularities on the correspo...