Handbook

This course is designed for Honours and postgraduate students with lectures used to introduce students to the theoretical content.

Scope of the course: By means of specific examples, this course serves as a gentle introduction to the world of integrable systems. Topics may include: Classical Liouville-Arnold integrability of ODEs based on Hamiltonian/Lagrangian mechanics, symmetries, conservation laws (Noether’s theorem). Integrability of infinite-dimensional systems: PDEs, soliton theory, Lax pairs, Bäcklund transformations, (differential) geometric aspects. Basic theory of (linear) difference equations. Discrete (nonlinear) integrable systems: (algebraic) geometry and features of integrable maps and partial difference equations, discrete Painlevé equations, and integrability detectors. Applications in mathematics and physics.

Historical context: The discovery of the Inverse Scattering Transform (IST) method and its application to the celebrated Korteweg-de Vries (KdV) equation in 1967 by Gardner, Green, Kruskal and Miura is regarded as one of the most important developments in mathematical physics in the past 50 years. It followed the numerical discovery in 1965 by Kruskal and Zabusky of a groundbreaking nonlinear phenomenon, namely the soliton interaction. The IST method is a nonlinear analogue of the Fourier transform method for privileged systems of nonlinear partial differential equations which are known as soliton equations or integrable systems. The latter term refers to the fact that these systems have properties that generalise those of classical integrable systems in the sense of classical Hamiltonian mechanics.

Significance in mathematics and physics: Remarkably, in the past 50 years, it has been demonstrated that the applicability of integrable systems is not confined to the original physical setting of the important Fermi-Pasta-Ulam problem. Indeed, the ubiquitous nature of integrable systems is reflected in their (apparent and disguised) presence in a wide range of both mathematical fields (e.g. partial differential and difference equations, differential and algebraic geometry, Galois theory, representation theory, special functions, quantum and cluster algebras, number theory, Nevanlinna theory in complex analysis) and physical fields (e.g. general relativity, twistor theory, field theory, (continuum) mechanics, nonlinear optics, Josephson junctions, Bose-Einstein condensates, biophysics, surface and water waves, plasma physics).