Boundary driven waveguide arrays: Supratransmission and saddle-node bifurcation

TL;DR

This paper analyzes the nonlinear dynamics of waveguide arrays driven at one edge, revealing how supratransmission occurs via saddle-node bifurcation of static solitons when the forcing frequency is in the upper forbidden band.

Contribution

It provides a rigorous mathematical analysis linking supratransmission to saddle-node bifurcation of static solitons in a nonlinear Schrödinger model of waveguide arrays.

Findings

Supratransmission occurs at a critical amplitude in the upper forbidden band.

Two static solitons emerge and disappear in a saddle-node bifurcation.

One soliton is stable, the other unstable, within their existence regions.

Abstract

In this report, we consider a semi-infinite discrete nonlinear Schr\"odinger equation driven at one edge by a driving force. The equation models the dynamics of coupled waveguide arrays. When the frequency of the forcing is in the allowed-band of the system, there will be a linear transmission of energy through the lattice. Yet, if the frequency is in the upper forbidden band, then there is a critical driving amplitude for a nonlinear tunneling, which is called supratransmission, of energy to occur. Here, we discuss mathematically the mechanism and the source of the supratransmission. By analyzing the existence and the stability of the rapidly decaying static discrete solitons of the system, we show rigorously that two of the static solitons emerge and disappear in a saddle-node bifurcation at a critical driving amplitude. One of the emerging solitons is always stable in its existence region and the other is always unstable. We argue that the critical amplitude for supratransmission is then the same as the critical driving amplitude of the saddle-node bifurcation. We consider as well the case of the forcing frequency in the lower forbidden band. It is discussed briefly that there is no supratransmission because in this case there is only one rapidly decaying static soliton that exists and is stable for any driving amplitude.