Non Linear Propagation in Reissner Beams: an integrable system?
Résumé
In the seventies, Arnold has a geometric approach by considering
a dynamical system as a map taking values in an abstract Lie
group. As such, he was able to highlight fundamental equivalencies between
rigid body motion and
uids dynamic depending on the specic
Lie group chosen (group of rotations in the former and group of dieomorphisms
in the latter). Following his idea, nonlinear propagation of
waves can also be formalized in their intrinsic qualities by adding space
variables independent to time. For a simple one-dimensional acoustical
system, it gives rise to the Reissner beam model for which the motion
of each dierent section, labelled by the arc length s, is encoding in the
Special Euclidean Lie group SE(3) - a natural choice to describe motion
in our 3-dimensional space. It turns out that, fortunately as a map over
spacetime, this multi-symplectic approach can be related to the study of
harmonic maps for which two dimensional cases can be solved exactly.
It allows us to identify, among the family of problems, a particular case
where the system is completely integrable. Among almost explicit solutions
of this fully nonlinear problem, it is tempting to identify solitons,
and to test the known numerical methods on these solutions.