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It has been recently (2014) discovered that rogue waves arise not only in the context of deep sea waves, but also in that of fiber optics. To be precise, consider a single-mode fiber, which its slowly varying electrical field envelope adheres the NLS dynamics $$i\psi _z + \psi _{t t} + |\psi|^2\psi =0 \, ,$$ where $t$ is retatded time/moving frame time, and $z$ is the propogation distance. There is a small but positive probability to have extremely large amplitudes $|\psi|^2$ with a small disturbance in the initial condition.

My questions: Other then an analogy to deep-sea waves and a new phenomenon, is this discovery supposed to have any implications on fiber optics?

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Yes indeed there are applications of these special solitary waves or solitons. In fiber optics communications over very long distances there is a problem with spreading or dispersion of the pulses. These solitons solutions have a cancellation mechanism that balances the effects of nonlinarity against dispersion so they can propagate over large distances free of dispersion.

BTW, this is not a recent discovery, but has been known for many years.

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  • $\begingroup$ Hey Lewis, thanks. Couple of questions, though (1) I can see how sech like solitons can be used for long distance pulse transmission. However, inasmuch as I know, the solitons associated with rogue waves, Peregrine's solitons, are usually of finite background, i.e. of infinite energy (L^2) norm. How does that works physically? (b) I think it has been known that NLS can manifest rogue waves, but observed in fibers for the first time in 2014, and published in Nature. $\endgroup$
    – Amir Sagiv
    Commented Dec 2, 2016 at 14:53
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    $\begingroup$ @AmirSagiv Wow, I had no idea this had become such an active field of research in the last 10 years. Thanks for directing me toward this new literature which started around 2007. My answer will need to be updated in light of this new research. It will take me a while to digest these papers, but my general impression is that rogue wave solutions combine many different nonlinear phenomena including solitons, chaos theory, stochastic resonance, and maybe even self-organized criticality. A huge number of PDEs exhibit these behaviors, so there are probably many things still to be learned. $\endgroup$
    – Lewis Miller
    Commented Dec 15, 2016 at 14:23

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