lunedi 18 Aprile, ore 14:30, aula 9 il Dr. Johannes Giannoulis della Technische Universität München terra’ un seminario dal titolo
“Macroscopic modulation of waves in weakly nonlinear lattices”
We present results concerning the macroscopic dynamics in multidimensional nonlinear atomic lattices. In one dimension, the most famous representative of such chains is the Fermi-Pasta-Ulam (FPU) chain. If we impose initial data in the form of small, macroscopically amplitude-modulated plane waves of the linearized problem, we ask for the (macroscopic) dynamics of the amplitudes. For time scales of first order in the scaling parameter, we see that the amplitudes are transported with the group velocity of the underlying carrier wave, while for time scales of second order in time and in a space frame moving with this velocity, we obtain that the amplitude is deformed according to the nonlinear Schrödinger equation. We show how such formally well known results can be justified in a mathematical rigorous way, and will discuss the relevance of nonresonance and resonance conditions for the underlying carrier waves, which manifest themselves in the macroscopic equations in the case of several interacting modulated waves. The latter results are shown in the cases of multidimensional lattices of scalar displacement and in the case of vector-valued (more precisely: diatomic) chains.