q-breathers in discrete nonlinear Schrödinger lattices

q-breathers (QBs) are exact time-periodic solutions of extended nonlinear systems continued from the normal modes of the corresponding linearized system. They are localized in the space of normal modes. The existence of these solutions in a weakly anharmonic atomic chain explained essential features of the Fermi–Pasta–Ulam (FPU) paradox. We study QBs in one-, two- and three-dimensional discrete nonlinear Schrödinger (DNLS) lattices—theoretical playgrounds for light propagation in nonlinear optical waveguide networks, and the dynamics of cold atoms in optical lattices. We prove the existence of these solutions for weak nonlinearity. We find that the localization of QBs is controlled by a single parameter which depends on the norm density, nonlinearity strength and seed wave vector. At a critical value of that parameter QBs delocalize via resonances, signaling a breakdown of the normal mode picture and a transition into the strong mode–mode interaction regime. In particular, this breakdown takes place at one of the edges of the normal mode spectrum, and in a singular way also in the center of that spectrum. A stability analysis of QBs supplements these findings. We relate our findings to previous studies of time-periodic multibreather standing waves. For three-dimensional lattices, we find QB vortices which violate time reversal symmetry and generate a vortex ring flow of energy in normal mode space.