## Seminario

minicorso

**INTEGRABLE SYSTEMS - METHODS OF MATHEMATICAL PHYSICS IN INTERACTION**

**PROF. Cornelia SCHIEBOLD**

Mid Sweden University, Sundsvall, Sweden

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**