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 Novgorod, 603950, Russia 9 10 The nonlinear Schrödinger (NLS) equation describing the propagation of weakly 11 rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has 12 been derived. The vorticity is assumed to be an arbitrary function of Lagrangian 13 coordinates and quadratic in the small parameter proportional to the wave 14 steepness. The vorticity effects manifest themselves in a shift of the wavenumber 15 in the carrier wave as well as in variation of the coefficient multiplying the 16 nonlinear term. In the case of the dependence of vorticity on the vertical 17 Lagrangian coordinate only (the Gouyon waves), the shift of the wavenumber and 18 the respective coefficient are constant. When the vorticity is dependent on both 19 Lagrangian coordinates, the shift of the wavenumber is horizontally 20 inhomogeneous. There are special cases (e.g., Gerstner waves) when the vorticity 21 is proportional to squared wave amplitude and non-linearity disappears, thus 22 making the equations for wave packet dynamics linear. It is shown that the NLS 23 solution for weakly rotational waves in the Eulerian variables may be obtained 24 from the Lagrangian solution by simply changing the horizontal coordinates. 25 26 27