At the beginning of this course, a decision concerning the language will be made: German or English. As a rule, if one participant has an insufficient knowledge of German, the course will be held in English.
Typically, physical systems are classically described by, in general nonlinear, (partial) differential equations. The nonlinearity may lead to surprising effects, like chaos (not addressed in this course) or some unexpected structure formation.
Mathematically the problem arises of getting information about the space of solutions and the behavior of solutions. In general, nonlinear partial differential equations are still a challenge of present day mathematics. In the presence of (enough) symmetries one can hope for some substantial analytical insight. (Numerical simulations are a different approach, but are typically very difficult in case of nonlinear partial differential equations.)
The first part of this course enters this subject in an elementary way, concentrating on various examples. In particular, we address the concept of a symmetry of a differential equation.
There is a special class of nonlinear partial differential equations which possess an infinite number of symmetries and which are, in some sense, completely solvable. This class includes the so-called soliton equations, which frequently appear as special cases, approximations, or in a certain limit of various physical systems. They include
Moreover, the course involves a bit about
The course will be kept on an elementary level. But it consists to a large extent of (for you) new mathematics. You should bring along some serious interest in mathematical physics (rather than phenomenological physics).
Last update: 27 September 2012