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| Titel |
Statistics of extreme waves in the framework of one-dimensional Nonlinear Schrodinger Equation |
| VerfasserIn |
Dmitry Agafontsev, Vladimir Zakharov |
| Konferenz |
EGU General Assembly 2013
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 15 (2013) |
| Datensatznummer |
250072207
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| Zusammenfassung |
| We examine the statistics of extreme waves for one-dimensional classical focusing Nonlinear
Schrodinger (NLS) equation,
iΨt + Ψxx + |Ψ |2Ψ = 0,
(1)
as well as the influence of the first nonlinear term beyond Eq. (1) - the six-wave interactions -
on the statistics of waves in the framework of generalized NLS equation accounting for
six-wave interactions, dumping (linear dissipation, two- and three-photon absorption) and
pumping terms,
2 4
iΨt +(1 - idl)Ψxx + (1+ id2p)|Ψ| Ψ+ (α + id3p)|Ψ |Ψ = ipΨ, (2)
α,dl,d2p,d3p,p -ª 1, d3p -ª α.
We solve these equations numerically in the box with periodically boundary conditions
starting from the initial data Ψt=0 = F(x) + ε(x), where F(x) is an exact modulationally
unstable solution of Eq. (1) seeded by stochastic noise ε(x) with fixed statistical
properties. We examine two types of initial conditions F(x): (a) condensate state
F(x) = 1 for Eq. (1)-(2) and (b) cnoidal wave for Eq. (1). The development of
modulation instability in Eq. (1)-(2) leads to formation of one-dimensional wave
turbulence. In the integrable case the turbulence is called integrable and relaxes to
one of infinite possible stationary states. Addition of six-wave interactions term
leads to appearance of collapses that eventually are regularized by the dumping
terms. The energy lost during regularization of collapses in (2) is restored by the
pumping term. In the latter case the system does not demonstrate relaxation-like
behavior.
We measure evolution of spectra Ik =< |Ψk|2 >, spatial correlation functions and the
PDFs for waves amplitudes |Ψ|, concentrating special attention on formation of "fat tails" on
the PDFs. For the classical integrable NLS equation (1) with condensate initial condition
we observe Rayleigh tails for extremely large waves and a "breathing region" for
middle waves with oscillations of the frequency of waves appearance with time,
while nonintegrable NLS equation with dumping and pumping terms (2) with the
absence of six-wave interactions α = 0 demonstrates perfectly Rayleigh PDFs
without any oscillations with time. In case of the cnoidal wave initial condition we
observe severely non-Rayleigh PDFs for the classical NLS equation (1) with the
regions corresponding to 2-, 3- and so on soliton collisions clearly seen of the PDFs.
Addition of six-wave interactions in Eq. (2) for condensate initial condition results
in appearance of non-Rayleigh addition to the PDFs that increase with six-wave
interaction constant α and disappears with the absence of six-wave interactions
α = 0.
References:
[1] D.S. Agafontsev, V.E. Zakharov, Rogue waves statistics in the framework of
one-dimensional Generalized Nonlinear Schrodinger Equation, arXiv:1202.5763v3. |
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