Abstract
This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schrödinger equation (NLS) on the natural cubic time scale.
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References
Alvarez-Samaniego B, Lannes D. Large time existence for 3D water-waves and asymptotics. Invent Math, 2008, 171: 485–541
Chirilus-Bruckner M, Düll W-P, Schneider G. NLS approximation of time oscillatory long waves for equations with quasilinear quadratic terms. Math Nachr, 2015, 288: 158–166
Dull W-P, Heβ M. Existence of long time solutions and validity of the nonlinear Schrödinger approximation for a quasilinear dispersive equation. J Differential Equations, 2018, 264: 2598–2632
Düll W-P, Schneider G, Wayne C E. Justification of the nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth. Arch Ration Mech Anal, 2016, 220: 543–602
Dyachenko A I, Kuznetsov E A, Spector M D, et al. Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys Lett A, 1996, 221: 73–79
Hasimoto H, Ono H. Nonlinear modulation of gravity waves. J Phys Soc Japan, 1972, 33: 805–811
Hunter J K, Ifrim M, Tataru D. Two dimensional water waves in holomorphic coordinates. Comm Math Phys, 2016, 346: 483–552
Ifrim M, Tataru D. Two dimensional water waves in holomorphic coordinates II: Global solutions. Bull Soc Math France, 2016, 144: 369–394
Ifrim M, Tataru D. The lifespan of small data solutions in two dimensional capillary water waves. Arch Ration Mech Anal, 2017, 225: 1279–1346
Ifrim M, Tataru D. Two-dimensional gravity water waves with constant vorticity I: Cubic lifespan. Anal PDE, 2019, 12: 903–967
Koch H, Tataru D. Conserved energies for the cubic nonlinear Schröodinger equation in one dimension. Duke Math J, 2018, 167: 3207–3313
Ovsjannikov L V. To the shallow water theory foundation. Arch Mech (Arch Mech Stos), 1974, 26: 407–422
Schneider G. Validity and non-validity of the nonlinear Schrödinger equation as a model for water waves. In: Lectures on the Theory of Water Waves. London Mathematical Society Lecture Note Series, vol. 426. Cambridge: Cambridge University Press, 2016, 121–139
Schneider G, Wayne C E. Justification of the NLS approximation for a quasilinear water wave model. J Differential Equations, 2011, 251: 238–269
Schneider G, Wayne C E. Corrigendum: The long-wave limit for the water wave problem I. The case of zero surface tension. Comm Pure Appl Math, 2012, 65: 587–591
Totz N, Wu S J. A rigorous justification of the modulation approximation to the 2D full water wave problem. Comm Math Phys, 2012, 310: 817–883
Wu S J. Almost global wellposedness of the 2-D full water wave problem. Invent Math, 2009, 177: 45–135
Zakharov V E. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J Appl Mech Tech Phys, 1968, 9: 190–194
Acknowledgements
The first author was supported by a Clare Boothe Luce Professorship. The second author was supported by the National Science Foundation of USA (Grant No. DMS-1800294) and a Simons Investigator grant from the Simons Foundation.
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Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday
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Ifrim, M., Tataru, D. The NLS approximation for two dimensional deep gravity waves. Sci. China Math. 62, 1101–1120 (2019). https://doi.org/10.1007/s11425-018-9501-y
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DOI: https://doi.org/10.1007/s11425-018-9501-y