Goldfishing by gauge theory

A new solvable many-body problem of goldfish type is identified and used to revisit the connection among two different approaches to solvable dynamical systems. An isochronous variant of this model is identified and investigated. Alternative versions of these models are presented. The behavior of the alternative isochronous model near its equilibrium configurations is investigated, and a remarkable Diophantine result, as well as related Diophantine conjectures, are thereby obtained.Comment: 22 page

On generalisations of Calogero-Moser-Sutherland quantum problem and WDVV equations

It is proved that if the Schr\"odinger equation $L \psi = \lambda \psi$ of Calogero-Moser-Sutherland type with $$L = -\Delta + \sum\limits_{\alpha\in{\cal A}_{+}} \frac{m_{\alpha}(m_{\alpha}+1) (\alpha,\alpha)}{\sin^{2}(\alpha,x)}$$ has a solution of the product form $\psi_0 = \prod_{\alpha \in {\cal {A}_+}} \sin^{-m_{\alpha}}(\alpha,x),$ then the function $F(x) =\sum\limits_{\alpha \in \cal {A}_{+}} m_{\alpha} (\alpha,x)^2 {\rm log} (\alpha,x)^2$ satisfies the generalised WDVV equations.Comment: 10 page

On the classification of scalar evolutionary integrable equations in 2+12+1 dimensions

We consider evolutionary equations of the form $u_t=F(u, w)$ where $w=D_x^{-1}D_yu$ is the nonlocality, and the right hand side $F$ is polynomial in the derivatives of $u$ and $w$. The recent paper \cite{FMN} provides a complete list of integrable third order equations of this kind. Here we extend the classification to fifth order equations. Besides the known examples of Kadomtsev-Petviashvili (KP), Veselov-Novikov (VN) and Harry Dym (HD) equations, as well as fifth order analogues and modifications thereof, our list contains a number of equations which are apparently new. We conjecture that our examples exhaust the list of scalar polynomial integrable equations with the nonlocality $w$. The classification procedure consists of two steps. First, we classify quasilinear systems which may (potentially) occur as dispersionless limits of integrable scalar evolutionary equations. After that we reconstruct dispersive terms based on the requirement of the inheritance of hydrodynamic reductions of the dispersionless limit by the full dispersive equation

Yang-Baxter maps and multi-field integrable lattice equations

A variety of Yang-Baxter maps are obtained from integrable multi-field equations on quad-graphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice equations introduced by Adler and Yamilov and which are related to the nonlinear superposition formulae for the B\"acklund transformations of the nonlinear Schr\"odinger system and specific ferromagnetic models.Comment: 16 pages, 4 figures, corrected versio

Topology and confinement at T \neq 0 : calorons with non-trivial holonomy

In this talk, relying on experience with various lattice filter techniques, we argue that the semiclassical structure of finite temperature gauge fields for T < T_c is dominated by calorons with non-trivial holonomy. By simulating a dilute gas of calorons with identical holonomy, superposed in the algebraic gauge, we are able to reproduce the confining properties below T_c up to distances r = O(4 fm} >> \rho (the caloron size). We compute Polyakov loop correlators as well as space-like Wilson loops for the fundamental and adjoint representation. The model parameters, including the holonomy, can be inferred from lattice results as functions of the temperature.Comment: Talk by M. M\"uller-Preussker at "Quark Confinement and Hadron Structure VII", Ponta Delgada, Azores, Portugal, September 2 - 7, 2006, 4 pages, 2 figures, to appear in the Proceeding

Backlund transformations for the sl(2) Gaudin magnet

Elementary, one- and two-point, Backlund transformations are constructed for the generic case of the sl(2) Gaudin magnet. The spectrality property is used to construct these explicitly given, Poisson integrable maps which are time-discretizations of the continuous flows with any Hamiltonian from the spectral curve of the 2x2 Lax matrix.Comment: 17 pages, LaTeX, refs adde

Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies

Analytic-bilinear approach for construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Generalized hierarchy incorporates basic hierarchy, modified hierarchy, singularity manifold equation hierarchy and corresponding linear problems. Different levels of generalized hierarchy are connected via invariants of Combescure symmetry transformation. Resolution of functional equations also leads to the $\tau$-function and addition formulae to it.Comment: 43 pages, Late

Integrable Schr\"odinger operators with magnetic fields: factorisation method on curved surfaces

The factorisation method for Schr\"odinger operators with magnetic fields on a two-dimensional surface $M^2$ with non-trivial metric is investigated. This leads to the new integrable examples of such operators and brings a new look at some classical problems such as Dirac magnetic monopole and Landau problem. The global geometric aspects and related spectral properties of the operators from the factorisation chains are discussed in details. We also consider the Laplace transformations on a curved surface and extend the class of Schr\"odinger operators with two integrable levels introduced in the flat case by S.P.Novikov and one of the authors.Comment: 20 page

On classical string configurations

Equations which define classical configurations of strings in $R^3$ are presented in a simple form. General properties as well as particular classes of solutions of these equations are considered.Comment: 10 pages, Latex, no figures, trivial corrections, submitted to Modern Physics Letters