The Lagrange form of the nonlinear Schrödinger equation for low - 1 vorticity waves in deep water : rogue wave aspect
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3 Anatoly Abrashkin and Efim Pelinovsky 4 5 a National Research University Higher School of Economics (HSE), Nizhny Novgorod 6 603155, Russia 7 b Institute of Applied Physics, 603950, 46 Ulyanov str., Nizhny Novgorod, Russia 8 c Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia 9 10 11 Abstract: 12 The nonlinear Schrödinger equation (NLS equation) describing weakly 13 rotational wave packets in an infinity-depth fluid in the Lagrangian coordinates is 14 derived. The vorticity is assumed to be an arbitrary function of the Lagrangian 15 coordinates and quadratic in the small parameter proportional to the wave 16 steepness. It is proved that the modulation instability criteria of the low-vorticity 17 waves and deep water potential waves coincide. All the known solutions of the 18 NLS equation for rogue waves are applicable to the low-vorticity waves. The effect 19 of vorticity is manifested in a shift of the wave number in the carrier wave. In case 20 of vorticity dependence on the vertical Lagrangian coordinate only (the Gouyon 21 waves) this shift is constant. In a more general case, where the vorticity is 22 dependent on both Lagrangian coordinates, the shift of the wave number is 23 horizontally heterogeneous. There is a special case with the Gerstner waves where 24 the vorticity is proportional to the square of the wave amplitude, and the resulting 25 non-linearity disappears, thus making the equations of the dynamics of the 26 Gerstner wave packet linear. It is shown that the NLS solution for weakly 27 rotational waves in the Eulerian variables can be obtained from the Lagrangian 28 solution by the ordinary change of the horizontal coordinates. 29 30 31
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Lagrange form of the nonlinear Schrödinger equation for 1 low - vorticity waves in deep water
1 low-vorticity waves in deep water 2 3 Anatoly Abrashkin 1 and Efim Pelinovsky 2,3 4 1 National Research University Higher School of Economics (HSE), 5 25/12 Bol'shaya Pecherskaya str., Nizhny Novgorod, 603155, Russia 6 2 Institute of Applied Physics RAS, 46 Ulyanov str., Nizhny Novgorod, 603950, Russia 7 3 Nizhny Novgorod State Technical University n.a. R. Alekseev, 24 Minin str., Nizhny 8 No...
متن کاملLagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water
The nonlinear Schrödinger (NLS) equation describing the propagation of weakly rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. The vorticity effects manifest themselves in a shift of the wave number...
متن کاملInteractive comment on “The Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water: rogue wave aspect” by Anatoly Abrashkin and Efim Pelinovsky
The paper describes a new derivation of the NLS equation, based on a Lagrangian coordinates approach, in the presence of weak vorticity. First, an introduction presents several previously existing derivations of the NLS equation, and offers an interesting review of recent developments designed to take vorticity into account. Then, the Lagrange coordinates, and associated general equations are p...
متن کاملInteractive comment on “The Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water: rogue wave aspect” by
The paper describes a new derivation of the NLS equation, based on a Lagrangian coordinates approach, in the presence of weak vorticity. First, an introduction presents several previously existing derivations of the NLS equation, and offers an interesting review of recent developments designed to take vorticity into account. Then, the Lagrange coordinates, and associated general equations are p...
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