This course will provide an introduction to the topic of discrete integrable systems. The course material will be based around two important types of integrable difference equations. The first are the multidimensionally consistent partial difference equations classified by Adler, Bobenko, and Suris (ABS), and the second are the discrete Painlevé equations associated to the classification of rational surfaces of Sakai.

Planned topics include:

1) Techniques to derive integrable difference equations: Bäcklund transformations, singularity confinement, integrable mappings and deautonomization

2) Integrability as multidimensional consistency, derivation of Lax pairs, discrete soliton solutions

3) Space of initial conditions, discrete Painlevé equation as translations in affine root lattice, configurations of eight points and characterisation via affine Weyl groups

4) Elliptic Painlevé equation

5) Special function solutions of discrete Painlevé equations, tau functions and Hirota bilinear equations.

References:

H. Sakai, "Rational surfaces associated with affine root systems and geometry of the Painlevé equations", Commun. Math. Phys. 220, 165-229 (2001)

V.E. Adler, A.I. Bobenko, Yu.B. Suris, "Classification of integrable equations on quad-graphs. The consistency approach", Commun. Math. Phys. 233, 513-543 (2003)

K. Kajiwara, M. Noumi, Y. Yamada, "Geometric aspects of Painlevé equations", J. Phys. A: Math. Theor. 50, 073001 (2017)

J. Hietarinta, N. Joshi, F. Nijhoff, "Discrete systems and integrability", Cambridge University Press (2016)

## Discrete Integrable Systems

External Lecturer:

Andrew Kels

Course Type:

PhD Course

Academic Year:

2020-2021

Period:

October - November

Duration:

20 h

Description:

Research Group:

Location:

A-136