Selection of standing waves at small energy for NLS with a trapping potential in 1 D

PDE Seminar
Tuesday, September 5, 2023 - 3:30pm for 1 hour (actually 50 minutes)
Scipio Cuccagna – Universita` di Trieste – scuccagna@units.it
Gong Chen

Due to linear superposition, solutions of a Linear Schrodinger Equation with a trapping potential,  produce a discrete  quasiperiodic part. When  a nonlinear perturbation is turned on,  it is known in principle, and proved in various situations,  that at small energies there is a phenomenon of standing wave selection where, up to radiation,  quasiperiodicity breaks down and there is convergence to a periodic wave.  We will discuss  this phenomenon in 1 D, where cubic nonlinearities are long range perturbations of the linear equations. Our aim is to show that a very effective framework to see these phenomena is provided by   a combination of the dispersion theory of  Kowalczyk, Martel and Munoz  along with  Maeda's  notion of Refined Profile.