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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union

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doi:10.5194/npg-2016-71
© Author(s) 2016. This work is distributed
under the Creative Commons Attribution 3.0 License.
Research article
14 Dec 2016
Review status
A revision of this discussion paper is under review for the journal Nonlinear Processes in Geophysics (NPG).
The Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water: rogue wave aspect
Anatoly Abrashkin1 and Efim Pelinovsky2,3 1National Research University Higher School of Economics (HSE), Nizhny Novgorod 603155, Russia
2Institute of Applied Physics, 603950, 46 Ulyanov str., Nizhny Novgorod, Russia
3Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia
Abstract. The nonlinear Schrödinger equation (NLS equation) describing weakly rotational wave packets in an infinity-depth fluid in the Lagrangian coordinates is derived. The vorticity is assumed to be an arbitrary function of the Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is proved that the modulation instability criteria of the low-vorticity waves and deep water potential waves coincide. All the known solutions of the NLS equation for rogue waves are applicable to the low-vorticity waves. The effect of vorticity is manifested in a shift of the wave number in the carrier wave. In case of vorticity dependence on the vertical Lagrangian coordinate only (the Gouyon waves) this shift is constant. In a more general case, where the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally heterogeneous. There is a special case with the Gerstner waves where the vorticity is proportional to the square of the wave amplitude, and the resulting non-linearity disappears, thus making the equations of the dynamics of the Gerstner wave packet linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables can be obtained from the Lagrangian solution by the ordinary change of the horizontal coordinates.

Citation: Abrashkin, A. and Pelinovsky, E.: The Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water: rogue wave aspect, Nonlin. Processes Geophys. Discuss., doi:10.5194/npg-2016-71, in review, 2016.
Anatoly Abrashkin and Efim Pelinovsky
Anatoly Abrashkin and Efim Pelinovsky
Anatoly Abrashkin and Efim Pelinovsky

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Short summary
The nonlinear Schrödinger equation describing weakly rotational wave packets in a fluid in the Lagrangian coordinates is derived. The rogue effects are possible in the low-vorticity waves. The effect of vorticity is manifested in a shift of the wave number in the carrier wave. Special attention is paid to the Gouyon and Gerstner waves. It is shown that this equation in the Eulerian variables can be obtained from the Lagrangian solution by the ordinary change of the horizontal coordinates.
The nonlinear Schrödinger equation describing weakly rotational wave packets in a fluid in the...
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