During the recent history of mathematics the theory of difference equations (∆Es)
has been lagging behind the analogous theory of differential equations (DEs). In the last two
decades, however, a considerable amount of progress has been made in understanding the structures behind certain specific classes of difference equations which we call integrable
. This course provides an overview of these modern developments, highlighting the connections with various other branches of mathematics.
Integrable systems form a special class of mathematical models and equations that allow for exact and rigorous methods for their solution. They come in all kinds of forms and shapes, such as nonlinear evolution equations (PDEs), Hamiltonian many-body systems, special types of nonlinear ODEs and certain quantum mechanical models. They possess remarkable properties, such as the existence of (multi-)soliton solutions, infinite number of conservation laws, higher and generalised symmetries, underlying infinite-dimensional group structures, etc. Their study has led to the development of new mathematical techniques, such as the inverse scattering transform method, finite-gap integration techniques and the application of Riemann-Hilbert problems.
A remarkable feature is that most integrable systems possess natural discrete analogues, described by difference equations rather than differential equations. Obviously, one can discretize a given differential equation in many ways, but to find a discretization that preserves the essential integrability features of an integrable differential equation is a far from trivial enterprise. Nonetheless, such discretizations have been found and constructed, and the resulting difference equations not only possess all the hall marks of integrability, but in fact turn out to be richer and more transparent than their continuous counterparts. Through their study a major boost has been given to the theory of difference equations in general, leading to the introduction of new mathematical notions and phenomena.
The proposed course is meant to be an introduction to this relatively new and exciting area of research, which draws together many facets of modern pure and applied mathematics, such as "discrete differential geometry", special function theory, geometric numerical integration, algebraic geometry and analysis. Nevertheless, the course will be given on a rather elementary level, without assuming any specific prerequisites beyond standard undergraduate mathematics. It will emphasise the interconnection between the various models and their emergence from basic principles.
Topics to be covered
1. Elementary theory of difference equations and difference calculus;
1. Bäcklund and Darboux transformations (BTs) and the tansition from continuous to discrete eqs;
2. Integrability and classication of lattice equations of KdV type;
3. Continuum limits: differential-difference equations and nonlinear evolution equations;
4. Soliton solutions on the lattice;
5. Reductions to finite-dimensional dynamical maps;
6. Symmetries of P∆Es and similarity reductions;
7. Special functions: Hypergeometric functions and contiguity relations;
8. Orthogonal polynomials and Padé approximants;
9. Integrability of mappings and the discrete Painlevé property;
10. Analytic difference equations and isomonodromy theory;
11. Elliptic functions and addition formulae;
12. Difference geometry;
13 Ultra-discrete systems and tropical geometry.
(Some of these topics are optional).