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