Integrable systems in infinite dimension refer to an area in mathematical physics which is devoted to the study of a certain group of partial differential equations, many of them soliton equations like the classic Korteweg-de Vries equation and the nonlinear Schrodinger equation. One of the striking features is the existence of solutions with particle character, called solitons, remarkable in view of nonlinearity of the governing equations. Methodologically, integrable systems are a meeting point (melting pan) for methods from very diverse parts of mathematics. The main idea of this mini course is to highlight interactions of some of the main approaches to integrable systems, the inverse scattering method and an operator theoretic approach in the first place, and symmetry methods like Backlund transformations, recursion operators and hierarchies to a minor extent.
Throughout we will emphasise the recent topic ofnon-commutative integrable systems, like vector- and matrix soliton equations, where many fundamental questions are still open. Notably, the construction of solutions is not interesting only under the mathematical viewpoint, but also under the physical one. Indeed, very important applications of soliton equations are in nonlinear optics, for instance.
The lectures are going to be reasonably self-contained. Some familiarity with PDE's and functional analysis is certainly helpful, but not required. An overview on the basic notions used during the course are provided when needed.
The course is organised in six lectures (a two hours lecture each week as indicated below).
Needed material as well as references are provided by the Lecturer.
AULA 1B1 PAL RM002
Lectures 1-2 March 7 time 12-14
Lectures 3-4 March 14 time 12-14
Lectures 5-6 March 21 time 12-14