Exact dynamics for fully connected nonlinear networks

TL;DR

This paper derives exact solutions for the nonlinear Schrödinger equation on fully connected networks, revealing two dynamical transitions influenced by nonlinearity, including a regime where the network mimics linear behavior with a shifted frequency.

Contribution

It provides the first exact analytical solutions for the nonlinear Schrödinger equation in fully connected networks, identifying dynamical transitions and the conditions for linear-like behavior.

Findings

Identification of hyperbolic and trigonometric dynamical transitions.

Existence of a regime where the nonlinear network behaves like a linear one with a renormalized frequency.

Exact solutions elucidate the impact of nonlinearity on network dynamics.

Abstract

We investigate the dynamics of the discrete nonlinear Schr\"{o}dinger equation in fully connected networks. For a localized initial condition the exact solution shows the existence of two dynamical transitions as a function of the nonlinearity parameter, a hyperbolic and a trigonometric one. In the latter the network behaves exactly as the corresponding linear one but with a renormalized frequency.