Mathematics

Maybe the most famous problem in the theory of non-linear Schrödinger equations is encapsulated in the so-called "Soliton Resolution Conjecture" which, loosely speaking, states that a generic finite energy solution u(t) of the NLS eventually splits into a certain number of mutually independent ground states and a dispersive component. This conjecture is completely open. In the talk I will review the related problem of the asymptotic stability of the ground states, which essentially is the soliton resolution conjecture in the very particular case when u(t) starts close to a ground state. I will focus in particular on the so called non-linear Fermi golden rule, which is a dissipation mechanism that often mathematical physicists introduce phenomenologically in their models, but that for the NLS (which is a Hamiltonian system) can be proved rigorously.