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Dynamics of gravity–capillary solitary waves in deep water

Published online by Cambridge University Press:  15 August 2012

Zhan Wang
Department of Mathematics, University of Wisconsin–Madison, Madison WI, 53706, USA
Paul A. Milewski*
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Email address for correspondence:


The dynamics of solitary gravity–capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time-dependent solutions, we simplify the full potential flow problem by using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonlinear focusing of wavepackets, quasi-elastic collisions, and the generation of propagating, spatially localized, time-periodic structures akin to breathers.

Copyright © Cambridge University Press 2012

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1. Ablowitz, M. J. & Segur, H. 1979 On the evolution of packets of water waves. J. Fluid Mech. 92, 691715.Google Scholar
2. Akers, B. & Milewski, P. A. 2009 A model equation for wavepacket solitary waves arising from capillary–gravity flows. Stud. Appl. Math. 122, 249274.Google Scholar
3. Akers, B. & Milewski, P. A. 2010 Dynamics of three-dimensional gravity–capillary solitary waves in deep water. SIAM J. Appl. Math. 70, 23902408.Google Scholar
4. Akylas, T. R. 1993 Envelope solitons with stationary crests. Phys. Fluids A 5, 789791.Google Scholar
5. Akylas, T. R. & Cho, Y. 2008 On the stability of lumps and wave collapse in water waves. Phil. Trans. Math. Phys. Engng Sci. 366, 27612774.Google Scholar
6. Alfimov, G. L., Eleonsky, V. M., Kulagin, N. E., Lerman, L. M. & Silin, V. P. 1990 On existence of non-trivial solutions for the equation . Physica D 44, 168177.Google Scholar
7. Calvo, D. C., Yang, T. S. & Akylas, T. R. 2002 Stability of steep gravity–capillary waves in deep water. J. Fluid Mech. 452, 123143.Google Scholar
8. Chiao, R. Y., Garmire, E. & Townes, C. 1964 Self-trapping of optical beams. Phys. Rev. Lett. 13, 479482.Google Scholar
9. Cho, Y., Diorio, J. D., Akylas, T. R. & Duncan, J. H. 2011a Resonantly forced gravity–capillary lumps on deep water. Part 2. Theoretical model. J. Fluid Mech. 672, 288306.Google Scholar
10. Cho, Y., Diorio, J. D., Duncan, J. H. & Akylas, T. R. 2011b Resonantly forced gravity–capillary lumps on deep water. Part 1. Experiments. J. Fluid Mech. 672, 268387.Google Scholar
11. Coifman, R. & Meyer, Y. 1985 Nonlinear harmonic analysis and analytic dependence. Proc. Symp. Pure Math. 43, 7178.Google Scholar
12. Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108, 7383.Google Scholar
13. Dias, F., Dyachenko, A. I. & Zakharov, V. E. 2008 Theory of weakly damped free-surface flow: a new formulation based on potential flow solutions. Phys. Lett. A 372, 12971302.Google Scholar
14. Falcon, L., Laroche, C. & Fauve, S. 2007 Observation of gravity–capillary wave turbulence. Phys. Rev. Lett. 98, 094503.Google Scholar
15. Iooss, G. & Kirrmann, P. 1996 Capillary gravity waves on the free surface of an inviscid fluid of infinite depth: existence of solitary waves. Arch. Rat. Mech. Anal. 136, 119.Google Scholar
16. Kim, B. & Akylas, T. R. 2005 On gravity–capillary lumps. J. Fluid Mech. 540, 337351.Google Scholar
17. Kim, B. & Akylas, T. R. 2007 Transverse instability of gravity–capillary solitary waves. J. Engng Math. 58, 167175.Google Scholar
18. Kim, B., Dias, F. & Milewski, P. A. 2012 On weakly nonlinear gravity–capillary solitary waves. Part 1. Bifurcation of solitary wavepackets. Wave Motion 49 (2), 221237.Google Scholar
19. Longuet-Higgins, M. S. 1989 Capillary–gravity waves of solitary type on deep water. J. Fluid Mech. 200, 451478.Google Scholar
20. Milewski, P. A. 2005 Fast communication: three-dimensional localized gravity–capillary waves. Commun. Math. Sci. 3, 8999.Google Scholar
21. Milewski, P. A. & Tabak, E. G. 1999 A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows. SIAM J. Sci. Comput. 21 (3), 11021114.Google Scholar
22. Milewski, P. A., Vanden-Broeck, J.-M. & Wang, Z. 2010 Dynamics of steep two-dimensional gravity–capillary solitary waves. J. Fluid Mech. 664, 466477.Google Scholar
23. Nicholls, D. P. 2007 Boundary perturbation methods for water waves. GAMM-Mitt 30 (1), 4474.Google Scholar
24. Părău, E. I., Vanden-Broeck, J.-M. & Cooker, M. J. 2005 Nonlinear three-dimensional gravity–capillary solitary waves. J. Fluid Mech. 536, 99105.Google Scholar
25. Rypdal, K. & Rasmussen, J. J. 1988 Stability of solitary structures in the nonlinear Schrödinger equations. Phys. Scr. 40, 192201.Google Scholar
26. Saffman, P. G. 1985 The superharmonic instability of finite-amplitude water waves. J. Fluid Mech. 159, 169174.Google Scholar
27. Sulem, C. & Sulem, P. L. 1999 The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences , vol. 139. Springer.Google Scholar
28. Vanden-Broeck, J.-M. & Dias, F. 1992 Gravity–capillary solitary waves in water of infinite depth and related free-surface flows. J. Fluid Mech. 240, 549557.Google Scholar
29. Wright, J. W. 1978 Detection of ocean waves by microwave radar: the modulation of short gravity–capillary waves. Boundary-Layer Meteorol. 13, 87105.Google Scholar
30. Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190194.Google Scholar
31. Zakharov, V. E. 1972 Collapse of Langmuir waves. Sov. Phys. JETP 35 (5), 908914.Google Scholar
32. Zhang, X. 1995 Capillary–gravity and capillary waves generated in a wind wave tank: observations and theory. J. Fluid Mech. 289, 5182.Google Scholar