Abstract
We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data of the form \({\epsilon B(\epsilon \alpha)e^{ik \alpha}}\) for k > 0, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrödinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times of order \({O(\epsilon^{-2})}\) provided the initial data differs from the wave packet by at most \({O(\epsilon^{3/2})}\) in Sobolev spaces. These results are obtained by directly applying modulational analysis to the evolution equation with no quadratic nonlinearity constructed in Wu (Invent Math 117(1):45–135, 2009) and by the energy method.
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References
Alvarez-Samaniego B., Lannes D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171(3), 485–541 (2008)
Cazenave, T.: Semilinear Schrödinger Equations. Volume 10 of Courant Lecture Notes. Providence, RI: Amer. Math. Soc., 2003
Coifman R.R., David G., Meyer Y.: La solution des conjectures de Calderón. Adv. Math. 48, 144–148 (1983)
Coifman R.R., McIntosh A., Meyer Y.: L’integral de cauchy definit un operateur borne sur L 2 pour les courbes lipschitziennes. Ann. of Math., 2nd Series 116(2), 361–387 (1982)
Craig W., Sulem C., Sulem P.-L.: Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5(2), 497–522 (1992)
Craig W.: An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Part. Diff. Eqs. 10(8), 787–1003 (1985)
Hashimoto H., Ono H.: Nonlinear modulation of gravity waves. J. Phys. Soc. Jpn. 33, 805–811 (1972)
Johnson, R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press, 1st edition, 2001
Kirrmann P., Schneider G., Mielke A.: The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 122(1-2), 85–91 (1992)
Miller, P.: Applied Asymptotic Analysis. Volume 75 of Graduate Studies in Mathematics. Providence, RI: Amer. Math. Soc., 2006
Schneider G., Wayne C.E.: The long-wave limit for the water wave problem. I. The case of zero surface tension. Comm. Pure Appl. Math. 53(12), 1475–1535 (2000)
Schneider G., Wayne C.E.: Justification of the NLS approximation for a quasilinear water wave model. J. Diff. Eqs. 251(2), 238–269 (2011)
Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Volume 139 of Applied Mathematical Sciences. Berlin-Heidelberg-New York: Springer, 1999
Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130(1), 39–72 (1997)
Wu S.: Almost global wellposedness of the 2-D full water wave problem. Invent. Math. 177(1), 45–135 (2009)
Wu S.: Global wellposedness of the 3-D full water wave problem. Invent. Math. 184(1), 125–220 (2011)
Zakharov V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zhurnal Prikladnoi Mekhaniki i Teckhnicheskoi Fiziki 9(2), 86–94 (1969)
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Communicated by P. Constantin
The authors are supported in part by NSF grant DMS-0800194.
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Totz, N., Wu, S. A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem. Commun. Math. Phys. 310, 817–883 (2012). https://doi.org/10.1007/s00220-012-1422-2
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DOI: https://doi.org/10.1007/s00220-012-1422-2