Peregrine soliton

The Peregrine soliton (or Peregrine breather) is an analytic solution of the nonlinear Schrödinger equation. This soliton or breather solution was proposed in 1983 by Howell Peregrine, researcher at the mathematics department of the University of Bristol.

Main properties Contrary to the usual fundamental soliton that can maintain its profile unchanged during propagation, the Peregrine soliton presents a double spatio-temporal localization. Therefore, starting from a weak oscillation on a continuous background, the Peregrine soliton develops undergoing a progressive increase of its amplitude and a narrowing of its temporal duration. At the point of maximum compression, the amplitude is three times the level of the continuous background (and if one considers the intensity as it is relevant in optics, there is a factor 9 between the peak intensity and the surrounding background). After this point of maximal compression, the wave's amplitude decreases and its width increases. These features of the Peregrine soliton are fully consistent with the quantitative criteria usually used in order to qualify a wave as a rogue wave. Therefore, the Peregrine soliton is an attractive hypothesis to explain the formation of those waves which have a high amplitude and may appear from nowhere and disappear without a trace.

Mathematical expression

In the spatio-temporal domain

The Peregrine soliton is a solution of the one-dimensional nonlinear Schrödinger equation that can be written in normalized units as follows:

i ∂ ψ ∂ τ + 1 2 ∂ 2 ψ ∂ ξ 2 + | ψ | 2 ψ = 0 {\displaystyle i{\frac {\partial \psi }{\partial \tau }}+{\frac {1}{2}}{\frac {\partial ^{2}\psi }{\partial \xi ^{2}}}+|\psi |^{2}\psi =0}

with ξ {\displaystyle \xi } the spatial coordinate and τ {\displaystyle \tau } the temporal coordinate. ψ ( ξ , τ ) {\displaystyle \psi (\xi ,\tau )} being the envelope of a surface wave in deep water. The dispersion is anomalous and the nonlinearity is self-focusing (note that similar results could be obtained for a normally dispersive medium combined with a defocusing nonlinearity). The Peregrine analytical expression is:

ψ ( ξ , τ ) = [ 1 − 4 ( 1 + 2 i τ ) 1 + 4 ξ 2 + 4 τ 2 ] e i τ {\displaystyle \psi (\xi ,\tau )=\left[1-{\frac {4(1+2i\tau )}{1+4\xi ^{2}+4\tau ^{2}}}\right]e^{i\tau }}

so that the temporal and spatial maxima are obtained for ξ = 0 {\displaystyle \xi =0} and τ = 0 {\displaystyle \tau =0} .

In the spectral domain

It is also possible to mathematically express the Peregrine soliton according to the spatial frequency η {\displaystyle \eta } :

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