We consider the glN -invariant Calogero-Sutherland Models with N = 1, 2, 3, . . . in a unified framework, which is the framework of Symmetric Polynomials. By the framework we mean an isomorphism between the space of states of the glN -invariant Calogero-Sutherland Model and the space of Symmetric Laurent Polynomials. In this framework it becomes apparent that all the glN -invariant Calogero-Sut...

We introduce and study a system of coupled nonlocal nonlinear Schrödinger equations that interpolates between the mixed, focusing–defocusing Manakov on one hand limiting case intermediate equation other. show this new system, which we call mixed (IMM) admits multi-soliton solutions governed by complexification hyperbolic Calogero–Moser (CM) system. Furthermore, spatially periodic version IMM fo...

Following Shastry and Sutherland I construct the super Lax operators for the Calogero model in the oscillator potential. These operators can be used for the derivation of the eigenfunctions and integrals of motion of the Calogero model and its supersymmetric version. They allow to infer several relations involving the Lax matrices for this model in a fast way. It is shown that the super Lax ope...

We study the rational potentials V (x), with sextic growth at infinity , such that the corresponding one-dimensional Schrödinger equation has no monodromy in the complex domain for all values of the spectral parameter. We investigate in detail the subclass of such potentials which can be constructed by the Darboux transformations from the well-known class of quasi-exactly solvable potentials V ...

We study new interactions between degrees of freedom for Calogero, Sutherland and confined Calogero spin models. These interactions are encoded by the generators of the Lie algebra so(N) or sp(N). We find the symmetry algebras of these new models: the half-loop algebra based on so(N) or sp(N) for the Calogero models and the Yangian of so(N) or sp(N) for the two types of other models. MSC: 70H06...

We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a τ -function. We investigate the potential using the non-local ∂̄ dressing method of Manakov and Zakharov, and we show that it can be interpreted as the Fredholm determinant of an integral equation which naturally appears within that approach. Finally...