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  • Print publication year: 2016
  • Online publication date: February 2016

7 - Validity and Non-Validity of the Nonlinear Schrödinger Equation as a Model for Water Waves

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[1] V. E., Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. Sov. Phys. J. Appl. Mech. Tech. Phys, 4:190-194, 1968.
[2] M., Shinbrot. The initial value problem for surface waves under gravity. I. The simplest case. Indiana Univ. Math. J., 25(3):281-300, 1976.
[3] T., Kano and T., Nishida. Sur les ondes de surface de l'eau avec une justification mathématique des équations des ondes en eau peu profonde. J. Math. Kyoto Univ., 19(2):335-370, 1979.
[4] D., Lannes. Well-posedness of the water-waves equations. J. Amer. Math. Soc., 18(3):605-654 (electronic), 2005.
[5] V. I., Nalimov. The Cauchy-Poisson problem. Dinamika Splošn. Sredy, 254 (Vyp. 18 Dinamika Zidkost. so Svobod. Granicami):104-210, 1974.
[6] H., Yosihara. Gravity waves on the free surface of an incompressible perfect fluid of finite despth. Publ. Res. Inst. Math. Sci., 18:49-96, 1982.
[7] H., Yosihara. Capillary-gravity waves for an incompressible ideal fluid. J. Math. Kyoto Univ., 23:649-694, 1983.
[8] W., Craig. An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Partial Differential Equations, 10(8):787-1003, 1985.
[9] S., Wu. Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math., 130(1):39-72, 1997.
[10] S., Wu. Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc., 12(2):445-495, 1999.
[11] G., Schneider and C. E., Wayne. 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.
[12] T., Iguchi. Well-posedness of the initial value problem for capillary-gravity waves. Funkcial. Ekvac., 44(2):219-241, 2001.
[13] G., Schneider and C. E., Wayne. The rigorous approximation of long-wavelength capillary-gravity waves. Arch. Rat. Mech. Anal., 162:247-285, 2002.
[14] D. M., Ambrose and N., Masmoudi. The zero surface tension limit of two-dimensional water waves. Comm. Pure Appl. Math., 58(10): 1287–1315, 2005.
[15] S., Wu. Almost global wellposedness of the 2-D full water wave problem. Invent. Math., 177(1):45-135, 2009.
[16] S., Wu. Global wellposedness of the 3-D full water wave problem. Invent. Math., 184(1):125-220, 2011.
[17] P., Germain, N., Masmoudi, and J., Shatah. Global solutions for the gravity water waves equation in dimension 3. Ann. of Math. (2), 175(2):691-754, 2012.
[18] N., Totz and S., Wu. A rigorous justification of the modulation approximation to the 2D full water wave problem. Commun. Math. Phys., 310(3):817-883, 2012.
[19] W.-P., Düll, G., Schneider, and C. E., Wayne. Justification of the Nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth. Arch. Rat. Mech. Anal., published online 2015.
[20] W., Craig, C., Sulem, and P.-L., Sulem. Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity, 5(2):497-522, 1992.
[21] P., Kirrmann, G., Schneider, and A., Mielke. The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A, 122(1-2):85-91, 1992.
[22] L. A., Kalyakin. Asymptotic decay of a one-dimensional wavepacket in a nonlinear dispersive medium. Math. USSR Sbornik, 60(2):457-483, 1988.
[23] J., Shatah. Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. PureAppl. Math., 38(5):685-696, 1985.
[24] W.-P., Düll and G., Schneider. Justification of the nonlinear Schrodinger equation for a resonant Boussinesq model. Indiana Univ. Math. J., 55(6):1813-1834, 2006.
[25] G., Schneider. Bounds for the nonlinear Schrodinger approximation of the Fermi-Pasta-Ulam system. Appl. Anal., 89(9):1523-1539, 2010.
[26] G., Schneider, D. A., Sunny, and D., Zimmermann. The NLS approximation makes wrong predictions for the water wave problem in case of small surface tension and spatially periodic boundary conditions. J. Dyn. Diff. Eq., published online 2014.
[27] G., Schneider and C. E., Wayne. Estimates for the three-wave interaction of surface water waves. European J. Appl. Math., 14(5):547-570, 2003.
[28] G., Schneider and O., Zink. Justification of the equations for the resonant four wave interaction. In EQUADIFF 2003, pages 213-218. World Sci. Publ., Hackensack, NJ, 2005.
[29] G., Schneider. Justification and failure of the nonlinear Schrodinger equation in case of non-trivial quadratic resonances. J. Diff. Eq., 216(2):354-386, 2005.
[30] W.-P., Dull, A., Hermann, G., Schneider, and D., Zimmermann. Justification of the 2D NLS equation for a fourth order nonlinear wave equation - quadratic resonances do not matter much in case of analytic initial conditions -. Preprint, Universität Stuttgart:19p., 2014.
[31] G., Schneider. Approximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation. J. Diff. Eq., 147:333-354, 1998.
[32] N., Masmoudi and K., Nakanishi. Multifrequency NLS scaling for a model equation of gravity-capillary waves. Commun. Pure Appl. Math., 66(8): 1202-1240, 2013.
[33] B., Deconinck and K., Oliveras. The instability of periodic surface gravity waves. J. Fluid Mech., 675:141-167, 2011.