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On the evolution of packets of water waves

Published online by Cambridge University Press:  19 April 2006

Mark J. Ablowitz
Applied Mathematics Program, Princeton University, Princeton, New Jersey 08540 Permanent address: Mathematics Department, Clarkson College, Potsdam, New York.
Harvey Segur
Aeronautical Research Associates of Princeton, Inc., 50 Washington Road, P.O. Box 2229, Princeton, New Jersey 08540


We consider the evolution of packets of water waves that travel predominantly in one direction, but in which the wave amplitudes are modulated slowly in both horizontal directions. Two separate models are discussed, depending on whether or not the waves are long in comparison with the fluid depth. These models are two-dimensional generalizations of the Korteweg-de Vries equation (for long waves) and the cubic nonlinear Schrödinger equation (for short waves). In either case, we find that the two-dimensional evolution of the wave packets depends fundamentally on the dimensionless surface tension and fluid depth. In particular, for the long waves, one-dimensional (KdV) solitons become unstable with respect to even longer transverse perturbations when the surface-tension parameter becomes large enough, i.e. in very thin sheets of water. Two-dimensional long waves (‘lumps’) that decay algebraically in all horizontal directions and interact like solitons exist only when the one-dimensional solitons are found to be unstable.

The most dramatic consequence of surface tension and depth, however, occurs for capillary-type waves in sufficiently deep water. Here a packet of waves that are everywhere small (but not infinitesimal) and modulated in both horizontal dimensions can ‘focus’ in a finite time, producing a region in which the wave amplitudes are finite. This nonlinear instability should be stronger and more apparent than the linear instabilities examined to date; it should be readily observable.

Another feature of the evolution of short wave packets in two dimensions is that all one-dimensional solitons are unstable with respect to long transverse perturbations. Finally, we identify some exact similarity solutions to the evolution equations.

Research Article
© 1979 Cambridge University Press

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Ablowitz, M. J. & Haberman, R. 1975 Phys. Rev. Lett. 35, 1185.
Ablowitz, M. J., Kaup, D. J., Newell, A. C. & Segur, H. 1974 Stud. Appl. Math. 53, 249.
Ablowitz, M. J., Ramani, A. & Segur, H. 1978 Lett. Nuovo cimento, 23, 333.
Ablowitz, M. J. & Satsuma, J. 1978 J. Math. Phys. 19, 2180.
Ablowitz, M. J. & Segur, H. 1977a Stud. Appl. Math. 57, 13.
Ablowitz, M. J. & Segur, H. 1977b Phys. Rev. Lett. 38, 1103.
Anker, D. & Freeman, N. E. 1978 Proc. Roy. Soc. A 360, 529.
Benney, D. J. & Roskes, G. J. 1969 Stud. Appl. Math. 48, 377.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford.
Chen, H. 1975 J. Math. Phys. 16, 2382.
Chiao, R. Y., Garmire, E. & Townes, C. H. 1964 Phys. Rev. Lett. 13, 479.
Davey, A. & Stewartson, K. 1974 Proc. Roy. Soc. A 388, 191.
Djordjevic, V. D. & Redekopp, L. G. 1977 J. Fluid Mech. 79, 703.
Dryuma, V. 1974 Sov. Phys. J. Exp. Theor. Phys. 19, 387.
Freeman, N. C. & Davey, A. 1975 Proc. Roy. Soc. A 344, 427.
Hammack, J. Preprint.
Hammack, J. & Segur, H. 1974 J. Fluid Mech. 65, 209.
Hammack, J. & Segur, H. 1978 J. Fluid Mech. 84, 337.
Hasimoto, H. & Ono, H. 1972 J. Phys. Soc. Japan 33, 805.
Hayes, W. D. 1973 Proc. Roy. Soc. A 332, 199.
Ince, E. L. 1944 Ordinary Differential Equations. Dover.
Kadomtsev, B. & Petviashvili, V. 1970 Sov. Phys. Dokl. 15, 539.
Lin, J. E. & Strauss, W. A. 1978 J. Funct. Anal. 30, 245.
Manakov, S. V. 1974 Sov. Phys. J. Exp. Theor. Phys. 38, 693.
Saffman, P. G. & Yuen, H. C. 1978 Phys. Fluids 21, 1450.
Satsuma, J. 1976 J. Phys. Soc. Japan 40, 286.
Segur, H. 1976 J. Math. Phys. 17, 714.
Segur, H. & Ablowitz, M. J. 1976 J. Math. Phys. 17, 710.
Talanov, V. I. 1967 Radiophys. 9, 260.
Ursell, F. 1953 Proc. Camb. Phil. Soc. 49, 685.
Vlasov, V. N., Petrishchev, I. A. & Talanov, V. I. 1974 Quant. Electron., Radiophys. 14, 1062.
Yuen, H. C. & Lake, B. M. 1975 Phys. Fluids 18, 956.
Zakharov, V. E. 1968 Sov. Phys. J. Appl. Mech. Tech. Phys. 4, 86.
Zakharov, V. E. & Rubenchik, A. M. 1974 Sov. Phys. J. Exp. Theor. Phys. 38, 494.
Zakharov, V. E. & Shabat, A. B. 1974 Func. Anal. Appl. 8, 226.
Zakharov, V. E. & Synakh, V. S. 1976 Sov. Phys. J. Exp. Theor. Phys. 41, 465.