Skip to main content Accessibility help
×
Home
Hostname: page-component-7f7b94f6bd-gszfc Total loading time: 0.333 Render date: 2022-07-01T05:27:33.999Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Water waves, nonlinear Schrödinger equations and their solutions

Published online by Cambridge University Press:  17 February 2009

Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrödinger (NLS) equation are enumerated.

Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].

In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Ablowitz, M. J. and Segur, H., “On the evolution of packets of water waves”, J. Fluid Mech. 92 (1979), 691715.CrossRefGoogle Scholar
[2]Ablowitz, M. J. and Segur, H., Solitons and the inverse scattering transform (SIAM, Philadelphia, 1981).CrossRefGoogle Scholar
[3]Broer, L. J. F., “Approximate equations for long water waves”, Appl. Sci. Res. 31 (1975), 377395.CrossRefGoogle Scholar
[4]Bryant, P. J., “Nonlinear wave groups in deep water”, Stud. Appl. Math. 61 (1979), 130.CrossRefGoogle Scholar
[5]Davey, A. and Stewartson, K., “On three-dimensional packets of surface waves”, Proc. Roy. Soc. London Ser. A 338 (1974), 101110.CrossRefGoogle Scholar
[6]Dysthe, K. B., “Note on a modification to the nonlinear Schrödinger equation for application to deep water waves”, Proc. Roy. Soc. London Ser. A 369 (1979), 105114.CrossRefGoogle Scholar
[7]Hui, W. H. and Hamilton, J., “Exact solutions of a three-dimensional nonlinear Schrödinger equation applied to gravity waves’, J. Fluid Mech. 93 (1979), 117134.CrossRefGoogle Scholar
[8]Johnson, R. S., “On the modulation of water waves in the neighbourhood of kh = 1.373”, Proc. Roy. Soc. London Ser. A 357 (1977), 131141.CrossRefGoogle Scholar
[9]Karpman, V. I. and Solov'ev, V. V., ‘A perturbational approach to the two-soliton systems’, Physica 3D (1981), 487502.Google Scholar
[10]Lake, B. M., Yuen, H. C., Rundgaldier, H. and Ferguson, W. E., “Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train”, J. Fluid Mech. 83 (1977), 4974.CrossRefGoogle Scholar
[11]Larsen, L. H., “Surface waves and low frequency noise in the deep ocean”, Geophys. Res. Letters 5 (1978), 499501.CrossRefGoogle Scholar
[12]Larsen, L. H., “An instability of packets of short gravity waves in waters of finite depth”, J. Phys. Oceanog. 9 (1979), 11391143.2.0.CO;2>CrossRefGoogle Scholar
[13]Ma, Y.-C., “The perturbed plane-wave solution of the cubic Schrödinger equation”, Stud. Appl. Math. 60 (1979), 4358.CrossRefGoogle Scholar
[14]Miles, J. W., ‘The Korteweg-deVnes equation: an historical essay”, J. Fluid Mech. 106 (1981), 103147.CrossRefGoogle Scholar
[15]Miles, J. W., “An envelope soliton problem”, SIAM J. Appl. Math. 41 (1981), 227230.CrossRefGoogle Scholar
[16]Mollo-Christensen, E. and Ramamonjiarisoa, A., “Modeling the presence of wave groups in a random field”, J. Geophys. Res. 83 (1978), 41174122.CrossRefGoogle Scholar
[17]Peregrine, D. H., “Equations for water waves and the approximations behind them”, Waves on beaches (ed. Meyer, R.), (Academic Press, New York, 1972).Google Scholar
[18]Peregrine, D. H., “Wave jumps and caustics in the refraction of finite-amplitude water waves” (submitted for publication).Google Scholar
[19]Satsuma, J. and Yajima, N., “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media”, Progr. Theoret. Phys. Suppl. 56 (1974), 284306.CrossRefGoogle Scholar
[20]Whitham, G. B., Linear and non-linear waves (Wiley-Interscience, New York, 1974).Google Scholar
[21]Yue, D. K. P. and Mei, C. C., “Forward diffraction of Stokes waves by a thin wedge”, J. Fluid Mech. 99 (1980), 3352.CrossRefGoogle Scholar
[22]Yuen, H. C. and Lake, B. M., “Nonlinear dynamics of deep-water gravity waves”, Adv. Appl. Mech. 22 (1982), 67229.CrossRefGoogle Scholar
[23]Zakharov, V. E. and Shabat, A. B., “Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media”, Soviet Phys. JETP 34 (1972), 6269Google Scholar
(transl. of Zh. Eksp. Teor. Fiz. 61, 118134).Google Scholar
[24]Zakharov, V. E. and Shabat, A. B., “Interaction between solitons in a stable medium”, Soviet Phys. JETP 37 (1973), 823828Google Scholar
(transl. of Zh. Eksp. Teor. Fiz. 64, 16271639).Google Scholar
You have Access
1103
Cited by

Save article to Kindle

To save this article 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.

Water waves, nonlinear Schrödinger equations and their solutions
Available formats
×

Save article to Dropbox

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

Water waves, nonlinear Schrödinger equations and their solutions
Available formats
×

Save article to Google Drive

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

Water waves, nonlinear Schrödinger equations and their solutions
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *