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A Nonlinear Schrödinger Equation for Gravity-Capillary Waves on Deep Water with Constant Vorticity

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Abstract

The surface gravity-capillary waves on deep water with constant vorticity in the region bounded by the free surface and the infinitely deep plane bottom are considered. A nonlinear Schrödinger equation is derived from a system of exact nonlinear integro-differential equations in conformal variables written in the implicit form taking into account surface tension. In deriving the nonlinear Schrödinger equation, the role of the mean flow is taken into account. The nonlinear Schrödinger equation is investigated for modulation instability. A soliton solution of the nonlinear Schrödinger equation that represents a soliton of the “ninth wave” type is obtained.

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ACKNOWLEDGMENTS

The author would like to express her deeply gratitude to Yu.I. Troitskaya for her constant attention to the present study and useful discussion of the results obtained.

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  1. Nizhny Novgorod Planetarium named after G.M. Grechko, Nizhny Novgorod, Russia

    M. I. Shishina

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  1. M. I. Shishina
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Correspondence to M. I. Shishina.

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Translated by E.A. Pushkar

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Shishina, M.I. A Nonlinear Schrödinger Equation for Gravity-Capillary Waves on Deep Water with Constant Vorticity. Fluid Dyn 58, 72–83 (2023). https://doi.org/10.1134/S0015462822601851

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