Boundary Driven Waveguide Arrays: Supratransmission and Saddle-Node Bifurcation

In this paper, we consider a semi-infinite discrete nonlinear Schrodinger 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 c...

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