Non linear propagation in reissner beams: An integrable system?

Abstract

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 fluids dynamic depending on the specific Lie group chosen (group of rotations in the former and group of diffeomorphisms 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 different 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.