q-breathers in Discrete Nonlinear Schroedinger lattices

TL;DR

This paper investigates $q$-breathers in discrete nonlinear Schrödinger lattices, demonstrating their existence, localization properties, and transition to delocalization, with implications for nonlinear optical and atomic systems.

Contribution

It proves the existence of $q$-breathers in 1D, 2D, and 3D DNLS lattices, analyzes their localization and delocalization transitions, and introduces $q$-breather vortices in 3D.

Findings

$q$-breathers exist in weakly nonlinear DNLS lattices.

Localization depends on a parameter involving norm density and nonlinearity.

Delocalization occurs at a critical parameter value, especially at spectrum edges.

Abstract

$q$-breathers 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 $q$-breathers in one- two- and three-dimensional discrete nonlinear Sch\"{o}dinger (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 $q$-breathers 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 $q$-breathers delocalize via resonances, signaling a breakdown of the normal mode picture and a transition into 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 $q$-breathers supplements these findings. For three-dimensional lattices, we find $q$-breather vortices, which violate time reversal symmetry and generate a vortex ring flow of energy in normal mode space.