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e taken into account in these models.
A novel model (D-model) of wave turbulence is introduced based on the nonlinear
resonance analysis. It allows one to reproduce in a single frame various
nonlinear wave phenomena such as intermittency, formation and direction of energy
cascades, possible growth of nonlinearity due to direct energy cascades, etc.
depending on the initial state. No statistical assumptions are used in this D-model
– all effects are due to the behavior of distinct modes taking part in resonant interactions.
Classical energy spectra are obtained as a particular case of a more
general form of energy spectra. This generic model can be applied to the experimental
and theoretical study of numerous wave turbulent systems appearing in
hydrodynamics, nonlinear optics, electrodynamics, convection theory, etc. The
D-model is sketched in the monograph [10]; a detailed investigation of dynamical
cascades in specific wave systems has been given in a number of further papers,
e.g. [6, 12].
4 Didactic potential of the subject
Since 2005 the author has taught a one-semester advanced course “Nonlinear Resonance
Analysis” for undergraduates in pure and applied mathematics, and computer
science at the Johannes Kepler University, Linz. The material of the course
is dynamical and takes account of the new scientific results in the area. Some results
which were obtained in the framework of students’ projects have been later
developed and published as joint papers with the course attendees.
Some results from the theory of nonlinear resonances have also been incorporated
into the one-semester course “Algebraic and discrete methods in biology”, at Linz.
The general theory of nonlinear resonances and its application for specific wave
turbulent systems have been presented by the author in the form of an intensive
lecture course (organized by Prof. I. Procaccia, 2008) at the Weizmann Institute
for Sciences, Israel.
An important addendum for these courses has been the development of the interactive
web portal CENREC (Center for Nonlinear Resonance Computation,
http://www.dynamics-approx.jku.at/portal). Modern computer technologies are
available for transforming theoretical results into efficiently executable services
that may be easily accessed by users via the world wide web. Our ultimate goal
in the developing of this portal is to provide a “virtual laboratory” for the areas
where nonlinear resonances occur in order to demonstrate the practical applicability
of the theoretical results to students, researchers, and engineers. Such a laboratory
serves educational purposes by providing online-training facilities in graduate
and in post-graduate education. Furthermore, it may provide a framework for
researchers by allowing the performance online virtual experiments without the
necessity of having access to actual physical equipment.
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