Research Aspect

The pluridisciplinarity character of the project makes up its main originality. We intend to promote the development of nonlinear science through a focussed training of Young Researchers ("fellows"). These fellows are expected to play a role as future leaders of the field in Europe.

The pluridisciplinarity character of the field of nonlinear physics results essentially from the fact that all scientific disciplines have to deal with nonlinear ordinary and partial differential equations. (ODEs and PDEs) Universality of the routes to chaos, for example, can be understood as generic behaviors of low-dimensional dynamical systems. It is quite instructive to remark that the understanding of this question progressed a lot in the years 1970-80 by works implementing simple ideas from phase transitions (renormalisation group). This progress has been done by physicists (not by mathematicians). The situation today is similar with respect to spatio-temporal phenomena observed in many different areas of science (from physics to biology), which seems also to possess a great degree of universality. The lack of a mathematical framework for PDEs, similar to the qualitative theory of ODEs (Poincare) makes the question very difficult. We quote Feynman, Lectures II/2 (41): ├Čthe next great era of awakening of human intellect may well produce a method of understanding the content of equations. Today we cannot. Today we cannot see that the water flow equations contain such things as the barber pole structure of turbulence that one sees between rotating cylinders. Today we cannot see whether Schroedingers equation contains frogs, musical composers, or morality -- or whether not.├«. The development of a qualitative theory of partial differential equation is the decisive open problem, which probably needs new mathematical concepts.

There is actually a great difficulty for learning the subject, which is connected with its ambiguous relation to mathematics. The qualitative theory of ODEs, for example, is not present in the curriculum of most physicists, because it is typically considered part of mathematics. The situation is quite serious. Students in physics often start a PhD thesis, without even knowing the Poincare representation of the dynamics (phase portrait) of the system that they may consider.

Webmaster bernd.schlesier@uni-bayreuth.de Last modified