Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-06T12:12:57.253Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  05 February 2016

Guido Schneider
Affiliation:
Universität Stuttgart
Thomas J. Bridges
Affiliation:
University of Surrey
Mark D. Groves
Affiliation:
Universität des Saarlandes, Saarbrücken, Germany
David P. Nicholls
Affiliation:
University of Illinois, Chicago
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2016

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[4] D., Lannes. Well-posedness of the water-waves equations. J. Amer. Math. Soc., 18(3):605-654 (electronic), 2005.Google Scholar
[5] V. I., Nalimov. The Cauchy-Poisson problem. Dinamika Splošn. Sredy, 254 (Vyp. 18 Dinamika Zidkost. so Svobod. Granicami):104-210, 1974.Google Scholar
[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.Google Scholar
[7] H., Yosihara. Capillary-gravity waves for an incompressible ideal fluid. J. Math. Kyoto Univ., 23:649-694, 1983.Google Scholar
[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.Google Scholar
[9] S., Wu. Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math., 130(1):39-72, 1997.Google Scholar
[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.Google Scholar
[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.Google Scholar
[12] T., Iguchi. Well-posedness of the initial value problem for capillary-gravity waves. Funkcial. Ekvac., 44(2):219-241, 2001.Google Scholar
[13] G., Schneider and C. E., Wayne. The rigorous approximation of long-wavelength capillary-gravity waves. Arch. Rat. Mech. Anal., 162:247-285, 2002.Google Scholar
[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.Google Scholar
[15] S., Wu. Almost global wellposedness of the 2-D full water wave problem. Invent. Math., 177(1):45-135, 2009.Google Scholar
[16] S., Wu. Global wellposedness of the 3-D full water wave problem. Invent. Math., 184(1):125-220, 2011.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[22] L. A., Kalyakin. Asymptotic decay of a one-dimensional wavepacket in a nonlinear dispersive medium. Math. USSR Sbornik, 60(2):457-483, 1988.Google Scholar
[23] J., Shatah. Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. PureAppl. Math., 38(5):685-696, 1985.Google Scholar
[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.Google Scholar
[25] G., Schneider. Bounds for the nonlinear Schrodinger approximation of the Fermi-Pasta-Ulam system. Appl. Anal., 89(9):1523-1539, 2010.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[31] G., Schneider. Approximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation. J. Diff. Eq., 147:333-354, 1998.Google Scholar
[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.Google Scholar
[33] B., Deconinck and K., Oliveras. The instability of periodic surface gravity waves. J. Fluid Mech., 675:141-167, 2011.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×