Skip to main content Accessibility help
Hostname: page-component-5d6d958fb5-x2fsp Total loading time: 0.715 Render date: 2022-11-26T12:28:30.913Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Dromions of flexural-gravity waves

Published online by Cambridge University Press:  19 February 2013

Mohammad-Reza Alam*
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
Email address for correspondence:


Here we show that weakly nonlinear flexural-gravity wave packets, such as those propagating on the surface of ice-covered waters, admit three-dimensional fully localized solutions that travel with a constant speed without dispersion or dissipation. These solutions, that are formed at the intersection of line-soliton mean-flow tracks, have exponentially decaying tails in all directions and are called dromions in contrast to lumps that decay only algebraically. We derive, by asymptotic expansion and assuming multiple scales for spatial and temporal variations, the three-dimensional weakly nonlinear governing equations that describe the coupled motion of the wavepacket envelope and the underlying mean current. We show that in the limit of long waves and strong flexural rigidity these equations reduce to a system of nonlinear elliptic–hyperbolic partial differential equations similar to the Davey–Stewartson I (DSI) equation, but with major differences in the coefficients. Specifically, and contrary to DSI equations, the elliptic and hyperbolic operators in the flexural-gravity equations are not canonical resulting in complications in analytical considerations. Furthermore, standard computational techniques encounter difficulties in obtaining the dromion solution to these equations owing to the presence of a spatial hyperbolic operator whose solution does not decay at infinity. Here, we present a direct (iterative) numerical scheme that uses pseudo-spectral expansion and pseudo-time integration to find the dromion solution to the flexural-gravity wave equation. Details of this direct simulation technique are discussed and properties of the solution are elaborated through an illustrative case study. Dromions may play an important role in transporting energy over the ice cover in the Arctic, resulting in the ice breaking far away from the ice edge, and also posing danger to icebreaker ships. In fact we found that, contrary to DSI dromions that only exist in water depths of less than 5 mm, flexural-gravity dromions exist for a broad range of ice thicknesses and water depths including values that may be realized in polar oceans.

©2013 Cambridge University Press

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.)


Ablowitz, M. J. & Segur, H. 1979 On the evolution of packets of water waves. J. Fluid Mech. 92, 691715.CrossRefGoogle Scholar
Akylas, T. R., Dias, F. & Grimshaw, R. H. J. 1998 The effect of the induced mean flow on solitary waves in deep water. J. Fluid Mech. 355, 317328.CrossRefGoogle Scholar
Benney, D. J. & Luke, J. C. 1964 On the interactions of permanent waves of finite amplitude. J. Math. Phys. 43, 309313.CrossRefGoogle Scholar
Berger, K. M. & Milewsky, P. A. 2000 The generation and evolution of lump solitary waves in surface-tension-dominated flows. SIAM J. Appl. Maths 61 (3), 731750.CrossRefGoogle Scholar
Besse, C., Mauser, N. J. & Stimming, H. P. 2004 Numerical study of the Davey–Stewartson system. ESIAM; Math. Model. Numer. Anal. 38 (6), 10351054.CrossRefGoogle Scholar
Boiti, M., Leon, J. J.-P., Martina, L. & Pempinelli, F. 1988 Scattering of localized solitons in the plane. Phys. Lett. A 132 (8,9), 432439.CrossRefGoogle Scholar
Champagne, B. & Winternitz, P. 1988 On the infinite-dimensional symmetry group of the Davey–Stewartson equations. J. Math. Phys. 29 (1), 18.CrossRefGoogle Scholar
Chen, X. J., Jensen, J. J., Cui, W. C. & Fu, S. X. 2003 Hydroelasticity of a floating plate in multidirectional waves. Ocean Engng 30 (15), 19972017.CrossRefGoogle Scholar
Clarkson, P. A. & Hood, S. 1994 New symmetry reductions and exact solutions of the Davey–Stewartson system. I. Reductions to ordinary differential equations. J. Math. Phys. 35 (1), 255283.CrossRefGoogle Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338 (1613), 101110.CrossRefGoogle Scholar
Djordjevic, V. D. & Redekopp, L. G. 1977 On two-dimensional packets of capillary-gravity waves. J. Fluid Mech. 79 (4), 703714.CrossRefGoogle Scholar
Duan, W.-S. 2003 Weakly two-dimensional modulated wave packet in dusty plasmas. Phys. Plasmas 10 (7), 30223055.CrossRefGoogle Scholar
Fokas, A. S. & Santini, P. M. 1990 Dromions and a boundary value problem for the Davey–Stewartson 1 equation. Physica D 44 (1–2), 99130.CrossRefGoogle Scholar
Forbes, L. K. 1986 Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution. J. Fluid Mech. 169, 409428.CrossRefGoogle Scholar
Freeman, N. C. & Davey, A. 1975 On the evolution of packets of long surface waves. Proc. R. Soc. Lond. A 344 (1638), 427433.CrossRefGoogle Scholar
Gilson, C. R. & Nimmo, J. J. C. 1991 A direct method for dromion solutions of the Davey–Stewartson equations and their asymptotic properties. Proc. R. Soc. Lond. A 435 (1894), 339357.CrossRefGoogle Scholar
Groves, M. D. & Sun, S.-M. 2008 Fully localised solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem. Arch. Rat. Mech. Anal. 188, 191.CrossRefGoogle Scholar
Hayashi, N. & Hirata, H. 1996 Global existence and asymptotic behaviour in time of small solutions to the elliptic hyperbolic Davey Stewartson system. Nonlinearity 9 (6), 13871409.CrossRefGoogle Scholar
Hietarinta, J. 1990 One-dromion solutions for genetic classes of equations. Phys. Lett. A 149 (2–3), 113118.CrossRefGoogle Scholar
Hietarinta, J. & Hirota, R. 1990 Multidromion solutions to the Davey–Stewartson equation. Phys. Lett. A 145 (5), 237244.CrossRefGoogle Scholar
Hogan, S. J. 1985 The fourth-order evolution equation for deep-water gravity-capillary waves. Proc. R. Soc. Lond. A 402 (1823), 359372.CrossRefGoogle Scholar
Hǎrǎguş-Courcelle, M. & Il’ichev, A. 1998 Three-dimensional solitary waves in the presence of additional surface effects. Eur. J. Mech. (B Fluids) 17 (5), 739768.CrossRefGoogle Scholar
Hizel, E., Turgay, N. C. & Guldogan, B. 2009 The symmetry reductions and new exact solutions of the generalized Davey–Stewartson equation. Intl J. Contemp. Math. Sci. 4 (18), 883894.Google Scholar
Johnson, R. S. 1997 A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press.CrossRefGoogle Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15 (6), 539541.Google Scholar
Kim, B. & Akylas, T. R. 2005 On gravity capillary lumps. J. Fluid Mech. 540, 337351.CrossRefGoogle Scholar
Korobkin, A., Parau, E. I. & Vanden-Broeck, J.-M. 2011 The mathematical challenges and modelling of hydroelasticity. Proc. R. Soc. Lond. A 369 (1947), 28032812.Google ScholarPubMed
Kuznetsov, E. A., Rubenchik, A. M. & Zakharov, V. E. 1986 Soliton stability in plasmas and hydrodynamics. Phys. Rep. 142 (3), 103165.CrossRefGoogle Scholar
Leblond, H. 1999 Electromagnetic waves in ferromagnets: a Davey–Stewartson-type model. J. Phys. A 32 (45), 79077932.CrossRefGoogle Scholar
Leblond, H. 2001 The Davey–Stewartson model in quadratic media: a way to control pulses. Soliton-driven Photonics 31 (4), 215218.CrossRefGoogle Scholar
Liu, A. K. & Mollo-Christensen, E. 1988 Wave propagation in a solid ice pack. J. Phys. Oceanogr. 18 (11), 17021712.2.0.CO;2>CrossRefGoogle Scholar
Marko, J. R. 2003 Observations and analyses of an intense waves-in-ice event in the Sea of Okhotsk. J. Geophys. Res. 108 (C9, 3296).CrossRefGoogle Scholar
Miles, J & Sneyd, A. D. 2003 The response of a floating ice sheet to an accelerating line load. J. Fluid Mech. 497, 435439.CrossRefGoogle Scholar
Milewski, P. A., Vanden-Broeck, J.-M. & Wang, Z. 2011 Hydroelastic solitary waves in deep water. J. Fluid Mech. 679, 628640.CrossRefGoogle Scholar
Nishinari, K. & Yajima, T. 1994 Numerical studies on stability of Dromion and its collisions. J. Phys. Soc. Japan 63 (10), 35383541.CrossRefGoogle Scholar
Osborne, A. R. 2009 Nonlinear Ocean Waves and the Inverse Scattering Transform. Academic.Google Scholar
Parau, E. I. & Vanden-Broeck, J.-M. 2011 Three-dimensional waves beneath an ice sheet due to a steadily moving pressure. Proc. R. Soc. Lond. A 369 (1947), 29732988.Google ScholarPubMed
Radha, R. & Lakshmanan, M. 1994 Singularity analysis equations and localized coherent structures in $(2+ 1)$ dimensional generalized Korteweg-de Vries. J. Math. Phys. 35 (9), 47464756.CrossRefGoogle Scholar
Schultz, C. L., Ablowitz, M. J. & Bar Yaacov, D. 1987 Davey–Stewartson I system: a quantum $(2+ 1)$ -dimensional integrable system. Phys. Rev. Lett. 59 (25), 28252828.CrossRefGoogle Scholar
Strathdee, J., Robinson, W. H. & Haines, E. M. 1991 Moving loads on ice plates of finite thickness. J. Fluid Mech. 226, 3761.CrossRefGoogle Scholar
Suzuki, H. 2005 Overview of megafloat: concept, design criteria, analysis, and design. Mar. Struct. 18 (2), 111132.CrossRefGoogle Scholar
White, P. W. & Weideman, J. A. C. 1994 Numerical simulation of solitons and dromions in the Davey–Stewartson system. Math. Comput. Simul. 37 (4–5), 469479.CrossRefGoogle Scholar
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure 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 or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ 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.

Dromions of flexural-gravity waves
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.

Dromions of flexural-gravity waves
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.

Dromions of flexural-gravity waves
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? *