Organizers
Andrew N. W. Hone, University of Kent
Frank W. Nijhoff, University of Leeds
Yang Shi, Flinders University
Da-jun Zhang(张大军), Shanghai University
Abstract
Elliptic discrete integrable systems are among the richest of the whole class of integrable equations, both continuous as well as discrete. Their solutions in terms of special functions involve novel features, such as bi-elliptic addition formulae and elliptic generalisations of classical functions and orthogonal polynomials. A lot of progress have been made on various aspects of the theory, but some important elliptic models have received not as much attention as they deserve. One of the aims of the workshop is to repair this inbalance. Thus, the workshop will bring together experts who have worked on various aspects of integrability and elliptic function theory, and thus aim at bringing progress in dealing with these challenging but rewarding model systems.
Description of the aim
Elliptic integrable systems are, in principle, exactly solvable model equations defined through ordinary or partial difference equations and that contain parameters associated with an elliptic curve. In the continuous case the outstanding examples are the Krichever-Novikov equation and the fully anisotropic Landau-Lifschitz equation, where in the former case the equation contains an arbitrary quartic polynomial in the dependent variable, to which one can associate an elliptic curve, while in the latter case the anisotropy parameters in general position are associated with the moduli of an elliptic curve. The discovery of discrete analogues of these equations and their significance within the framework of integrable systems forms the main motivation for this proposal. In addition, classes of ordinary difference equations of discrete-time many-body systems exhibit features which are radically different and richer than those of the corresponding continuous-time many-body systems. (Recently a novel example of such an elliptic many-body system arising as reduction of the BKP equation was found by Zabrodin). Furthermore, there is an entire classification of difference equations of Painlevé type (i.e. equations whose general solutions are meromorphic w.r.t. the movable singularities) at the top of which stands the elliptic Painlevé equation discovered in 1999 by H. Sakai. About a year earlier V. Adler discovered a discrete analogue of the Krichever-Novikov equation, which stands at the top of the well-known Adler-Bobenko-Suris (ABS) classification of quadrilateral lattice equations, which are four-point partial difference equations integrable in the sense of being multidimensionally consistent (i.e., consistently embeddable in a higher-dimensional lattice). Regarding the Landau-Lifschitz equations, there are three independent versions known: one proposed by Nijhoff and Papageorgiou in 1989, another constructed by Adler and Yamilov in 1996, and yet a third proposed by Adler himself in 2000. Almost nothing is known to date about these three models, not even if/how they are interrelated. Nonetheless, it would be of great interest to know their solution structures and their connection with other integrable systems. Finally, there is also a three-dimensional system of equations, namely an elliptic version of the Kadomtsev-Petviashvili equation, which was first proposed by Date, Jimbo and Miwa in 1983 and further studied recently by Fu and Nijhoff.
Apart from elliptic models, there are also classes of elliptic solutions of integrable lattice equations that exhibit novel features, such as a new concept of elliptic Nth root of unity, appearing in elliptic solutions of higher rank lattice equations of Gel’fand-Dickey type.
There are many outstanding problems in the theory of elliptic discrete systems, and resolving them has become urgent, as they stand at the top of the tree of integrable systems: knowing their resolution would imply the resolution of similar questions for the degenerate cases, which comprises almost all integrable equations.
In recent years, various novel mathematical techniques and methods have been developed to study these discrete systems and their solutions, bringing together ideas stemming from several branches of mathematics and physics, that are usually distinct, now come together: asymptotic analysis, algebraic geometry, representation theory, spectral/isomonodromy analysis, random matrix theory, exactly solvable models, theory of special functions and combinatorial geometry. These will form the unifying themes of the workshop, which will comprise the following topics:
--Elliptic solutions of higher-rank and multidimensional lattice equations;
--The construction of higher-rank elliptic lattice systems;
--Bi-elliptic addition formulae (as appearing in the soliton solutions of the ABS Q4 equation);
--Elliptic discrete time many-body systems;
--The solution structure of the variants of the lattice Landau-Lifschitz equations;
--Elliptic Lax pairs and isomonodromic deformation problems;
--Elliptic orthogonal polynomials and corresponding elliptic discrete integrable systems;
--Reductions to elliptic autonomous and non-autonomous ordinary difference equations, such as the QRT map and elliptic Painlevé equation;
--Algebro-geometric solutions of integrable lattice equations, and higher-genus lattice systems;
--Elliptic solutions of recurrence relations for integer (Somos) sequences;
--Connections with models of quantum theory and statistical mechanics.
