We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relate the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These basis can be related to the eigenfunctions of Cal...

In a recent paper Calogero and Alcántara [Kinet. Relat. Models, 4 (2011), pp. 401-426] derived a Lorentz-invariant Fokker-Planck equation, which corresponds to the evolution of a particle distribution associated with relativistic Brownian Motion. We study the “one and one-half” dimensional version of this problem with nonlinear electromagnetic interactions the relativistic Vlasov-Maxwell-Fokker...

In this article we study multiple SLE?, for ??(0,4], driven by Dyson Brownian motions. This model was introduced in the unit disk Cardy, (2003) connection with Calogero–Sutherland model. We prove Carathéodory convergence of perturbed Loewner chains under different initial conditions and diffusivity ??(0,4] case N=2 driving forces. Our proofs use analysis Bessel processes estimates on differenti...

Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the von Mises transformation, with the unknown function taken as the new independent variable and an appropriate partial derivative taken as the new dependent variable. RF-pairs and associated Bäcklund transformations are constructed for evolution equations of general form (speci...

We construct the r-matrix for the generalization of the Calogero-Moser system introduced by Gibbons and Hermsen. By reduction procedures we obtain the r-matrix for the O(N) Euler-Calogero-Moser model and for the standard AN Calogero-Moser model. PAR LPTHE 93-55 L.P.T.H.E. Université Paris VI (CNRS UA 280), Box 126, Tour 16, 1 étage, 4 place Jussieu, F-75252 PARIS CEDEX 05

It is by now well known that the wave functions of rational solutions to the KP hierarchy which can be achieved as limits of the pure n-soliton solutions satisfy an eigenvalue equation for ordinary differential operators in the spectral parameter. This property is known as “bispectrality” and has proved to be both interesting and useful. In this note, it is shown that certain (non-rational) sol...

We establish a connection between the hyperbolic relativistic Calogero-Moser systems and a class of soliton solutions to the Tzitzeica equation (aka the DoddBullough-Zhiber-Shabat-Mikhailov equation). In the 6N -dimensional phase space Ω of the relativistic systems with 2N particles and N antiparticles, there exists a 2N dimensional Poincaré-invariant submanifold ΩP corresponding to N free part...

We present a class of periodic solutions the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation, which was recently introduced by authors together with Langmann as classical, continuum limit an Inozemtsev-type spin chain. These exact analytic are constructed via spin-pole ansatz written in terms certain elliptic functions. The dynamical parameters our solve Calogero-Moser (CM) syst...