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On the evolution of a solitary wave for very weak nonlinearity

Published online by Cambridge University Press:  12 April 2006

John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, La Jolla

Abstract

The initial-value problem for a one-dimensional gravity wave of amplitude a and characteristic length l in water of depth d is examined for 0 < a/d [Lt ] d2/l2 [Lt ] 1. A preliminary reduction leads to a Korteweg-de Vries (KdV) equation in which the nonlinear term is O(ε) relative to the linear terms, where ε = 3al2/4d3 [Lt ] 1 is a measure of nonlinearity/dispersion. The linear approximation (ε ↓ 0) is found to be valid if and only if $\epsilon\tau^{\frac{1}{3}}\ll 1 $ where $\tau = \frac{1}{2}(d/l)^2(gd)^{\frac{1}{2}}({\rm time})/l $ is the slow time in the KdV equation. The asymptotic solution of the KdV equation is obtained with the aid of inverse-scattering theory and is found to comprise not only a decaying wave train that is qualitatively similar to that predicted by the linear approximation, but also a soliton of amplitude 3V2/4d3 = Oa) if V > 0, where V is the cross-sectional area of the initial displacement, or of amplitude = O3a) if V = 0 (there is no soliton if V < 0). This soliton is fully evolved, and dominates the solution, only for $\epsilon \tau^{\frac{1}{3}} \gg 1$ if V > 0 or $\epsilon^2\tau^{\frac{1}{3}}\gg 1 $ if V = 0, but nonlinearity already has significant effects for $\epsilon\tau^{\frac{1}{3}} = O(1)$.

Type
Research Article
Copyright
© 1978 Cambridge University Press

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References

Ablowitz, M. & Newell, A. 1973 The decay of the continuous spectrum for solutions of the Korteweg–deVries equation. J. Math. Phys. 14, 12771284.Google Scholar
Ablowitz, M. & Segur, H. 1977 Asymptotic solutions of the Korteweg–deVries equation. Stud. Appl. Math. 57, 1344.Google Scholar
Abramowitz, M. & Stegun, I. 1964 Handbook of Mathematical Functions. Washington: Nat. Bur. Stand.
Berezin, Y. A. & Karpman, V. I. 1964 Theory of nonstationary finite-amplitude waves in a low-density plasma. Sov. Phys. J. Exp. Theor. Phys. 19, 12651271.Google Scholar
Berezin, Y. A. & Karpman, V. I. 1967 Nonlinear evolution of disturbances in plasmas and other dispersive media. Sov. Phys. J. Exp. Theor. Phys. 24, 10491056.Google Scholar
Dealfaro, V. & Regge, T. 1965 Potential Scattering. North Holland.
Goldberger, M. & Watson, K. 1964 Collision Theory. Wiley.
Gwyther, R. F. 1900 The general motion of long waves, with an examination of the direct reflexion of a solitary wave. Phil. Mag. 50(5), 349–352.Google Scholar
Karpman, V. I. 1967 The structure of two-dimensional flow around bodies in dispersive media. Sov. Phys. J. Exp. Theor. Phys. 25, 11021111.Google Scholar
Miles, J. W. 1977 Obliquely interacting solitary waves. J. Fluid Mech. 79, 157169.Google Scholar
Miura, R. M. 1976 The Korteweg-deVries equation: a survey of results. SIAM Rev. 18, 412459.Google Scholar
Scott Russell, J. 1844 Report on waves. Rep. 14th Meeting Brit. Ass. Adv. Sci., York, pp. 311390.Google Scholar
Segur, H. 1973 The Korteweg–de Vries equation and water waves. Part 1. Solutions of the equation. J. Fluid Mech. 59, 721736.Google Scholar
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc. 49, 685694.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Zakharov, V. E. & Manakov, S. V. 1976 Asymptotic behaviour of non-linear wave systems integrated by the inverse scattering method. Sov. Phys. J. Exp. Theor. Phys. 44, 106112.Google Scholar