In the first part, we study the existence and bifurcation results for quasi periodic traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions and with polynomial type potentials. We employ variational ana topological methods to prove the existence of traveling waves in nonlocal DNLS lattice. Next, we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential (nonlocal NLS). While in the case of cubic and quintic interactions of the same kind (e.g. both attractive or both repulsive), only a symmetry breaking bifurcation can be identified, a remarkable effect that emerges e.g. in the setting of repulsive cubic but attractive quintic interactions is a ``symmetry restoring'' bifurcation. Namely, in addition to the supercritical pitchfork that leads to a spontaneous symmetry breaking of the anti-symmetric state, there is a subcritical pitchfork that eventually reunites the asymmetric daughter branch with the anti-symmetric parent one. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting. The model is argued to be of physical relevance, especially so in the context of optical thermal media.
2019/04/26 Fri 14:00 - 17:00
Room 609, Building No.6
Aristotle University of Thessaloniki