Benjamin-Feir instability in nonlinear dispersive waves. (English) Zbl 1268.35113

Summary: The authors extended the derivation to the nonlinear Schrödinger equation in two-dimensions, modified by the effect of non-uniformity. The authors derived several classes of soliton solutions in \(2+1\) dimensions. When the solution is assumed to depend on space and time only through a single argument of the function, they showed that the two-dimensional nonlinear Schrödinger equation is reduced either to the sine-Gordon for the hyperbolic case or sinh-Gordon equations for the elliptic case. Moreover, the authors extended this method to obtain analytical solutions to the nonlinear Schrödinger equation in two space dimensions plus time. This contains some interesting solutions such as the plane solitons, the N multiple solitons, the propagating breathers and quadratic solitons. The authors displayed graphically the obtained solutions by using the software Mathematica 5.


35Q55 NLS equations (nonlinear Schrödinger equations)
35C08 Soliton solutions


Full Text: DOI


[1] Benjamin, T. B.; Feir, J. E., The disintegration of wavetrains on deep water. Part 1. Theory, J. Fluid Mech., 27, 417 (1967) · Zbl 0144.47101
[2] Benjamin, T. B., The stability of solitary waves, Proc. R. Soc. Lond. Ser. A, 328, 153 (1972)
[3] Hasimoto, H.; Ono, H., Nonlinear modulation of gravity waves, J. Phys. Soc. Japan, 33, 805 (1972)
[4] Stuart, J. T.; DiPrima, R. C., The Eckhaus and Benjamin-Feir resonance mechanisms, Proc. R. Soc. Lond. Ser. A, 362, 27 (1978)
[5] Yuen, H. C.; Lake, B. M., Nonlinear deep water waves: theory and experiment, Phys. Fluids, 18, 956 (1975) · Zbl 0326.76018
[6] Debnath, L., Nonlinear Water Waves (1994), Academic Press, Inc.: Academic Press, Inc. New York · Zbl 0793.76001
[7] Infeld, E.; Rowlands, G., Nonlinear Waves Solitons and Chaos (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0726.76018
[8] Lake, B. M.; Yuen, H. C.; Rundgaldier, H.; Ferguson, W. E., Nonlinear deep water waves: theory and experiment. Part 2. Evolution of a continuous wave train, J. Fluid Mech., 83, 49 (1977)
[9] Benny, D. J., Nonlinear gravity wave interactions, J. Fluid Mech., 14, 577 (1962) · Zbl 0117.43605
[10] Benjamin, T. B.; Bridges, T. J., Reappraisal of the Kelvin-Helmholtz problem. Part 2. Introduction of the Kelvin-Helmholtz, Superharmonic and Benjamin-Feir instabilities, J. Fluid Mech., 333, 327 (1997) · Zbl 0892.76027
[11] Bridges, T. J.; Mielke, A., A proof of the Benjamin-Feir instability, Arch. Ration. Mech. Anal., 133, 145 (1995) · Zbl 0845.76029
[12] Grimshaw, R. H.J.; Pullin, D. I., Stability of finite-amplitude interfacial waves. Part 1. Modulational instability for small-amplitude waves, J. Fluid Mech., 160, 297 (1985) · Zbl 0614.76020
[13] Pullin, D. I.; Grimshaw, R. H.J., Stability of finite-amplitude interfacial waves. Part 2. Numerical results, J. Fluid Mech., 160, 317 (1985) · Zbl 0614.76020
[14] Yuen, H. C., Nonlinear dynamics of interfacial waves, Physica D, 12, 71 (1984) · Zbl 0575.76026
[15] Van Duin, C. A., The effect of non-uniformity of modulated wave packets on the mechanism of Benjamin-Feir instability, J. Fluid Mech., 399, 237 (1999) · Zbl 0960.76034
[16] Johnson, R. S., A Modern Introduction to the Mathematical Technique of Water Waves (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0892.76001
[17] Whitham, G. B., Linear and Nonlinear Waves (1974), Wiley: Wiley New York · Zbl 0373.76001
[18] Nayfeh, A. H., Introduction to Perturbation Techniques (1981), Wiley: Wiley New York · Zbl 0449.34001
[19] Berkshire, F. H.; Gibbon, J. D., Collapse in the \(n\)-dimensional nonlinear Schrödinger equation a parallel with Sundman’s results in the N-body problem, Stud. Appl. Math., 69, 229 (1983) · Zbl 0536.35063
[20] Gibbon, J. D.; McGuinness, M. J., Nonlinear focusing and the Kelvin-Helmholtz instability, Phys. Lett. A, 77, 118 (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.