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On periodic and solitary wavelike solutions of the intermediate long-wave equation

Published online by Cambridge University Press:  26 April 2006

Touvia Miloh
Affiliation:
School of Engineering, University of Tel-Aviv, Israel

Abstract

The intermediate long-wave (ILW) equation is a weakly nonlinear integrodifferential equation which governs the evolution of long internal waves in a stratified fluid of finite depth. It reduces to the Korteweg–de Vries (KdV) and to the Benjamin–Ono (BO) equations for shallow and large depths respectively. Solitary wave solutions of the ILW equation are well known, however analytic expressions for periodic solutions of the same equation do not seem to exist. Such expressions are derived in this paper and a remarkable property discovered for these periodic waves is that they can be represented as an infinite sum of spatially repeated solitons. Thus, nonlinear periodic solutions of the ILW equation are obtained by linear superposition of solitons.

Type
Research Article
Copyright
© 1990 Cambridge University Press

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References

Ablowitz, M. J., Fokas, A. S., Satsuma, J. & Segur, H. 1982 On the periodic intermediate long wave equation. J. Phys. A: Math. Gen. 15, 781.Google Scholar
Bateman, H. 1954 Tables of Integral Transforms. McGraw-Hill.
Benjamin, T. B. 1966 Internal waves of finite amplitude and permanent form. J. Fluid Mech. 25, 241.Google Scholar
Benjamin, T. B. 1967 Internal waves of permanent form in fluid of great depth. J. Fluid Mech. 29, 559.Google Scholar
Benney, C. J. 1966 Long nonlinear waves in fluid flows. J. Math. Phys. 45, 52.Google Scholar
Boyd, J. P. 1984 Cnoidal waves as exact sums of repeated solitary waves: new series for elliptic functions. SIAM J. Appl. Maths 44, 952.Google Scholar
Bromwich, T. J. 1952 An Introduction to the Theory of Infinite Series, 2nd edn. Macmillan.
Chen, H. H. & Lee, Y. C. 1979 Internal wave solitons of fluids with finite depth. Phys. Rev. Lett. 43, 264.Google Scholar
Christie, D. R., Muirhead, K. & Hales, A. 1978 On solitary waves in the atmosphere. J. Atmos. Sci. 35, 805.Google Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593.Google Scholar
Ekman, V. W. 1904 On Dead Water: The Norwegian North Polar Expedition 1893–1896, vol. 5, ch. 15. Christiania.
Gradshteyn, I. S. & Ryzhik, I. M. 1973 Tables of Integrals Series and Products. Academic.
Joseph, R. I. 1977 Solitary waves in a finite depth fluid. J. Phys. A: Math. Gen. 10, L225.Google Scholar
Joseph, R. I. & Egri, R. 1978 Multi-soliton solutions in a finite depth fluid. J. Phys. A: Math. Gen. 11, L97.Google Scholar
Koop, C. G. & Butler, G. 1981 An investigation of internal solitary waves in a two-fluid system. J. Fluid Mech. 112, 225.Google Scholar
Korppel, A. & Banerjee, P. P. 1981 Exact decomposition of Cnoidal waves into associated solitons. Phys. Lett. 82A. 113.Google Scholar
Korteweg, D. I. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. Mag. (5) 39, 422.Google Scholar
Kubota, T., Ko, D. R. S. & Dobbs, L. D. 1978 Propagation of weakly nonlinear internal waves in a stratified fluid of finite depth. AIAA J. Hydronautics 12, 157.Google Scholar
Lipovskiy, V. D. 1986 On the nonlinear theory of internal waves in a fluid of finite depth. Bull. USSR Acad. Sci. Atmos. Oceanic Phys. 21, 665.Google Scholar
Liu, A. K., Holbrook, J. R. & Apel, J. R. 1985 Nonlinear internal wave evolution in the Sulu Sea. J. Phys. Oceanogr. 15, 1613.Google Scholar
Maslowe, S. A. & Redfkopp, L. G. 1980 Long nonlinear waves in stratified shear flows. J. Fluid Mech. 101, 321.Google Scholar
Matsuno, Y. 1980 N-soliton solution of the higher order wave equation for a fluid of finite depth. J. Phys. Soc. Japan 48, 663.Google Scholar
Miles, J. W. 1980 Solitary waves. Ann. Rev. Fluid Mech. 12, 11.Google Scholar
Miloh, T. & Tulin, M. P. 1988 A theory of dead water phenomena. Proc. Seventeenth Symp. Naval Hydrodynamics. The Hague, Netherlands
Miloh, T. & Tulin, M. P. 1989 Periodic solutions of the DABO equation as sum of repeated solitons. J. Phys. A: Math. Gen. 22, 921.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Method of Theoretical Physics. Macmillan.
Nakamura, A. & Matsuno, Y. 1980 Exact one and two-periodic wave solutions of fluids of finite depth. J. Phys. Soc. Japan 48, 653.Google Scholar
Ono, H. 1975 Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39, 1082.Google Scholar
Osborne, A. R. & Burch, T. L. 1980 Internal solitons in the Andaman Sea. Science 208, 451.Google Scholar
Parker, D. F. 1980 The decay of sawtooth solutions to the Burgers equation.. Proc. R. Soc. Lond. A 369, 409.Google Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.
Romanova, N. N. 1984 Long nonlinear waves in layers of drastic wind velocity changes. Bull. USSR Acad. Sci. Atmos. Oceanic Phys. 20, 6.Google Scholar
Santini, P. M., Ablowitz, M. J. & Fokas, A. S. 1984 On the limit from the intermediate long wave equation to the Benjamin-Ono equation. J. Math. Phys. 25, 892.Google Scholar
Segur, H. & Hammack, J. L. 1982 Soliton models of long internal waves. J. Fluid Mech. 118, 285.Google Scholar
Toda, M. 1975 Studies of a nonlinear lattice. Phys. Rep. 18, 1.Google Scholar
Ursell, F. 1953 The long wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc. 49, 685.Google Scholar
Volyak, K. I. & Krasnoslobodtsev, A. V. 1986 On the symmetry and nonstationary solutions of the Benjamin-Ono equation. Isv. Atmos. & Ocean Phys. 22, 153.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.
Whitham, G. B. 1984 Comments on periodic waves and solitons. IMA J. Appl. Maths 32, 353.Google Scholar
Zabusky, N. J. 1967 Nonlinear Partial Differential Equations (ed. W. Ames). Academic.