University of Leeds
Workshop on Integrable Systems, 9th/10th May, 2008

Available Abstracts

Thanasis Fokas (DAMTP, Cambridge)

Inversion of Certain Integrals and Integrability in 4+2 and 3+1 Dimensions


I will first review a method for deriving linear and nonlinear transform pairs, which is based on the spectral analysis of an eigenvalue equation and on the formulation of a d-bar problem.

Then, I will present two applications of this method:

  1. The derivation of a certain linear transform pair in one dimension, which characterizes the Dirichlet to Neumann map of the Laplace equation in the interior of a convex two-dimensional curvilinear domain.
  2. The derivation of a nonlinear Fourier transform pair in four dimensions, which can be used for the solution of the Cauchy problem of an integrable generalization of the Kadomtsev-Petviashvilli equation in 4+2, i.e. in four spatial and two temporal dimensions. The question of reducing this equation form 4+2 to 3+1 dimensions will also be discussed.

Alexander Mikhailov (Leeds)

Solutions of two-dimensional periodic Volterra system


Not Available

Frank Nijhoff (Leeds)

On soliton solutions of integrable lattice equations


In recent years integrable lattice equations, i.e. partial difference equations analogues of soliton PDEs, have gained a lot of interest. Whereas lattice equations of KdV type, and their soliton solutions have been known since the early 1980s, the more recent classification result of scalar quadrilateral lattices by Adler, Bobenko and Suris, which are integrable in the sense of "multidimensionally consistency", has provided a number of novel examples. In this talk an overview will be given of some of the old as well as novel results for soliton solutions for these equations.

Victor Buchstaber (Manchester)

Solutions of two-dimensional Lame and heat type equations


We will describe the solutions of some important differential equations in the term of two-dimensional sigma-functions.

Jon Nimmo (Glasgow)

Darboux transformations, noncommutative integrable systems and quasideterminants


Darboux transformations offer a standard technique to construct solutions of commutative integrable systems, leading to formulae involving determinants. Their use for noncommutative integrable also leads naturally expressions in terms of the quasideterminants of Gelfand and Retakh. This talk will review some recent results in this area.

Simon Ruijsenaars (Leeds)

A new approach to joint eigenfunctions of commuting Hamiltonians of elliptic Calogero-Moser type


In the literature one only finds fragmentary results concerning joint eigenfunctions of Calogero-Moser type Hamiltonians in the elliptic regime. For the relativistic Calogero-Moser case, even the existence of an orthogonal base of joint Hilbert space eigenfunctions has only been shown in special rank-1 cases. We present a quite unexpected new perspective for the general Hilbert space joint eigenfunction problem, which holds promise of a complete solution for both the A_N and BC_N cases.

Complete results for N=1 have already been obtained and will be sketched. The new approach leads to novel spectral symmetry results even for the well-studied nonrelativistic BC_1 case (aka the Heun equation). In the relativistic BC_N case an E_8 spectral symmetry emerges from our strategy, whose key ingredient consists of special Hilbert-Schmidt operators with kernels expressed solely in terms of the elliptic gamma function.

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Last updated 11th July, 2008