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Discrete Integrable Systems

External Lecturer: 
Andrew Kels
Course Type: 
PhD Course
Academic Year: 
October - November
20 h

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.

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)


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