Boundary driven waveguide arrays: Supratransmission and saddle-node bifurcation

Abstract

In this report, we consider a semi-infinite discrete nonlinear Schrödinger 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.

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References in corpus (8)

• Energy transmission in the forbidden bandgap of a nonlinear chain
• Nonlinear supratransmission and bistability in the Fermi-Pasta-Ulam model
• Radiationless Travelling Waves In Saturable Nonlinear Schrödinger Lattices
• Nonlinear Band Gap Transmission in Optical Waveguide Arrays
• Nonlinear supratransmission as a fundamental instability
• On the propagation of binary signals in damped mechanical systems of oscillators
• Calculated threshold of supratransmission phenomena in waveguide arrays with saturable nonlinearity
• Bistable light detectors with nonlinear waveguide arrays

Cited by in corpus (10)

• Supratransmission in a Disordered Nonlinear Periodic Structure
• Nonlinear supratransmission in multicomponent systems
• Nonlinear magnetoinductive transmission lines
• Surge of power transmission in flat and nearly flat band lattices
• Nonlinear bandgap transmission by discrete rogue waves induced in a pendulum chain
• Exploiting bifurcations in waveguide arrays for light detectors
• In-band supratransmission in nonlinear flat band lattices
• Supratransmission of rogue wave-like breathing solitons
• Pearl supratransmission in a boundary-driven two-dimensional nonlinear Schrödinger equation with a hole
• Supratransmission in Lattices with Purely Nonlinear Coupling