On the trigonometric polynomial Ansatz for gravity water waves

Abstract

We show that Gerstner’s wave exhausts all solutions to the gravity water wave problem that have particle paths prescribed by trigonometric polynomials of degree one. We further show that any Ansatz of degree two reduces to Gerstner’s wave.

Keywords:

• Gerstner’s wave
• trigonometric polynomial Ansatz
• water waves with vorticity

AMS Subject Classifications:

• 35Q31
• 76B15

Acknowledgements

The financial support of the ERC Advanced Grant “Nonlinear studies of water flows with vorticity” is acknowledged. The author would like to thank the referees for several useful comments and suggestions.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by the European Research Council [NWFV 267116].