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The nonlinear Schrödinger (NLS) equation describing the propagation of\udweakly rotational wave packets in an infinitely deep fluid in Lagrangian\udcoordinates has been derived. The vorticity is assumed to be an arbitrary\udfunction of Lagrangian coordinates and quadratic in the small parameter\udproportional to the wave steepness. The vorticity effects manifest\udthemselves in a shift of the wave number in the carrier wave and in\udvariation in the coefficient multiplying the nonlinear term. In the case of\udvorticity\uddependence on the vertical Lagrangian coordinate only (Gouyon waves), the shift of the wave number and the respective coefficient\udare constant. When the vorticity is dependent on both Lagrangian\udcoordinates, the shift of the wave number is horizontally inhomogeneous.\udThere are special cases (e.g., Gerstner waves) in which the vorticity is\udproportional to the squared wave amplitude and nonlinearity disappears, thus\udmaking the equations for wave packet dynamics linear. It is shown that the\udNLS solution for weakly rotational waves in the Eulerian variables may be\udobtained from the Lagrangian solution by simply changing the horizontal\udcoordinates

Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water