BIRS Workshop Lecture Videos
Construction of unstable KAM tori for a system of coupled NLS equations. Vilaça da Rocha, Victor
The systems of coupled NLS equations occur in some physical problems, in particular in nonlinear optics (coupling between two optical waveguides, pulses or polarized components...). From the mathematical point of view, the coupling effects can lead to truly nonlinear behaviors, such as the beating effect (solutions with Fourier modes exchanging energy) of GrÃ Â©bert, Paturel and Thomann (2013). In this talk, I will use the coupling between two NLS equations on the 1D torus to construct a family of linearly unstable tori, and therefore unstable quasi-periodic solutions. The idea is to take profit of the Hamiltonian structure of the system via the construction of a Birkhoff normal form and the application of a KAM theorem. In particular, we will see of this surprising behavior (this is the first example of unstable tori for a 1D PDE) is strongly related to the existence of beating solutions. This is a work in collaboration with BenoÃ Â®t GrÃ Â©bert (UniversitÃ Â© de Nantes).
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