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| Titel |
On the nonlinear stage of modulation instability |
| VerfasserIn |
Andrey Gelash, 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 |
250072507
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| Zusammenfassung |
| There is a point of common agreement that freak (or rogue) waves appear as a result of
modulation instability of quasimonochromatic nonlinear waves. What is a nonlinear stage of
modulation instability? In dimension D = 2,3, the answer is known - modulation instability
leads to formation of finite time singularities - collapses. In dimension D = 1 collapses are
forbidden. However in this case development of modulation instability leads to formation of
extreme waves where energy density exceeds the mean level in order of magnitude. As a
result the study of long-time consequences of modulation instability is a problem of big
practical importance, crucial for creation of a freak wave theory in ocean. In the
recent time the simplest and most universal model for description of these waves
is the focusing NLSE (Nonlinear Schrödinger Equation). In terms of the NLSE
model we should study instability of the "condensate". The focusing NLSE is the
model of the first approximation. For the surface of fluid this model describes the
essentially weakly nonlinear quasimonochromatic wave trains. Nowadays a lot
of models generalizing the NLSE are developed. Also, freak waves in the ocean
were studied by numerical modeling of exact Euler equations for potential flow
with free boundary. The behavior of freak waves studied by NLSE and by more
sophisticated models shows considerable quantitative difference. Nevertheless, advanced
improvement of NLSE does not lead to any qualitatively new effects. That means that a
careful and detailed study of NLSE solutions is still very important problem. NLSE
is a completely integrable system, having many exact solutions. It is natural to
hope that the nonlinear development of the modulation instability is described by
one of such solutions. We are interested only in instability growing from small
spatially localized perturbation of condensate. Historically first such solution was
found by Peregrine in 1983. Now "multi-Peregrine" are known. These solutions
have a weak point - they are small perturbations of condensate only in the limit
t ---. These solutions are homoclinic - they describe freak waves appearing "from
nowhere" and completely disappearing in future. We find another class of exact
solutions of NLSE which are small perturbation of condensate not at t -- but in the
initial moment of time t = 0. This is the special class of 2N-solitonic solutions of
NLSE in presence of condensate. They grow exponentially, saturate and turn to a
system of localized solitons propagating with fast group velocities in both directions.
In general these solutions are not symmetric with respect to reflection of spatial
coordinate. It is important that these solutions do not change a condensate phase.
They describe the nonlinear stage of the modulation instability of a condensate. |
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