This Demonstration shows the development of a discrete rogue wave according to the commonly used Ablowitz–Ladik equation: . As one application, such a wave can model light propagation in strongly interacting parallel optical waveguides. The amplitude, , is defined only at a discrete set of sites (), although the propagation variable is continuous.

A rogue wave is a unique event that occurs when a disturbance builds up on a constant background to form a high amplitude part, but later decreases into the background. This Demonstration illustrates the curious fact that rogue waves can exist in discrete systems, for example in an array of optical waveguides. Thus they are analogous to those in continuous systems, like the nonlinear Schrödinger equation (NLSE). Rogue waves arise as solutions of the Ablowitz–Ladik system, even though this is not a direct discretization of the NLSE.

Waves of the two lowest orders are shown here. You can switch from one to the other by clicking "basic" or "second order". You can also change the distance of propagation and observe the rise of the central part above the background when the distance, , approaches zero. The maximum can indeed be much higher than the other nearby sites. When the offset is 1/2 or -1/2, then the two central heights are equal.

For more details, see A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, "Discrete Rogue Waves of the Ablowitz-Ladik and Hirota Equations," Physical Review E82, 2010. DOI: 10.1103/PhysRevE.82.026602.