# Symmetries and Nonlinear Differential Equations in Physics (Symmetrien und Nichtlineare Differenzialgleichungen in der Physik)

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

• Korteweg-deVries (KdV) equation
• Nonlinear Schrödinger (NLS) equation
• Sine-Gordon equation
• Ernst equation of General Relativity
The occurence of these equations in a physical context will be described and some of their features (exact solutions) will be derived. This includes, for example, a section about the so-called inverse scattering method (here applied to the KdV equation). The first three of the above examples possess (exponentially) localized wave solutions, called solitons with surprising properties, including a kind of nonlinear superposition principle. The latter is also shared by mathematically similar solutions of the remaining two equations.

Moreover, the course involves a bit about

• finite-dimensional integrable Hamiltonian systems (classical mechanics)
• the so-called inverse scattering method (applied to the KdV equation), using the formalism of quantum mechanics
• gauge theory (in particular the self-dual Yang-Mills equation)
• basic differential geometry

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