Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-27T04:33:35.736Z Has data issue: false hasContentIssue false
  • English
  • Français

Transverse instability of surface solitary waves. Part 2. Numerical linear stability analysis

Published online by Cambridge University Press:  17 June 2010

TAKESHI KATAOKA
TAKESHI KATAOKA*
Affiliation:
Department of Mechanical Engineering, Graduate School of Engineering, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan
*
†Email address for correspondence: kataoka@mech.kobe-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

In a previous work, Kataoka & Tsutahara (J. Fluid Mech., vol. 512, 2004a, p. 211) proved the existence of longitudinally stable but transversely unstable surface solitary waves by asymptotic analysis for disturbances of small transverse wavenumber. In the present paper, the same transverse instability is examined numerically for the whole range of solitary-wave amplitudes and transverse wavenumbers of disturbances. Numerical results show that eigenvalues and eigenfunctions of growing disturbance modes agree well with those obtained by the asymptotic analysis if the transverse wavenumber of the disturbance is small. As the transverse wavenumber increases, however, the growth rate of the disturbance, which is an increasing function for small wavenumbers, reaches a maximum and finally falls to zero at some finite wavenumber. Thus, there is a high-wavenumber cutoff to the transverse instability. For higher amplitude, solitary waves become longitudinally unstable, and the dependence of the eigenvalues on the transverse wavenumber exhibits various complicated patterns. We found that such eigenvalues versus transverse wavenumber can be simply grouped into three basic classes.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 657 , 25 August 2010 , pp. 126 - 170
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alexander, J. C., Pego, R. L. & Sachs, R. L. 1997 On the transverse instability of solitary waves in the Kadomtsev–Petviashivili equation. Phys. Lett. A 226, 187–192.CrossRefGoogle Scholar
Allen, M. A. & Rowlands, G. 1997 On the transverse instabilities of solitary waves. Phys. Lett. A 235, 145–146.CrossRefGoogle Scholar
Amick, C. J. & Toland, J. F. 1981 On solitary water-waves of finite amplitude. Arch. Ration. Mech. Anal. 76, 9–95.CrossRefGoogle Scholar
Benjamin, T. B. 1972 The stability of solitary waves. Proc. R. Soc. Lond. A 328, 153–183.Google Scholar
Bridges, T. J. 2001 Transverse instability of solitary-wave states of the water-wave problem. J. Fluid Mech. 439, 255–278.CrossRefGoogle Scholar
Byatt-Smith, J. G. & Longuet-Higgins, M. S. 1976 On the speed and profile of steep solitary waves. Proc. R. Soc. Lond. A 350, 175–189.Google Scholar
Craig, W. & Sternberg, P. 1988 Symmetry of solitary waves. Commun. Partial Differ. Equ. 13, 603–633.CrossRefGoogle Scholar
Jeffrey, A. & Kakutani, T. 1970 Stability of the Burgers shock wave and the Korteweg–de Vries soliton. Indiana Univ. Math. J. 20, 463–468.CrossRefGoogle Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in a weakly dispersing medium. Sov. Phys. Dokl. 15, 539–541.Google Scholar
Kataoka, T. 2006 a The stability of finite-amplitude interfacial solitary waves. Fluid Dyn. Res. 38, 831–867.CrossRefGoogle Scholar
Kataoka, T. 2006 b On the superharmonic instability of surface gravity waves on fluid of finite depth. J. Fluid Mech. 547, 175–184.CrossRefGoogle Scholar
Kataoka, T. 2008 Transverse instability of interfacial solitary waves. J. Fluid Mech. 611, 255–282.CrossRefGoogle Scholar
Kataoka, T. & Tsutahara, M. 2004 a Transverse instability of surface solitary waves. J. Fluid Mech. 512, 211–221.CrossRefGoogle Scholar
Kataoka, T. & Tsutahara, M. 2004 b Instability of solitary wave solutions to long-wavelength transverse perturbations in the generalized Kadomtsev–Petviashvili equation with negative dispersion. Phys. Rev. E 70, 016604.CrossRefGoogle ScholarPubMed
Kivshar, Y. S. & Pelinovsky, D. E. 2000 Self-focusing and transverse instabilities of solitary waves. Phys. Rep. 331, 117–195.CrossRefGoogle Scholar
Kuznetsov, E. A., Spector, M. D. & Fal'Kovich, G. E. 1984 On the stability of nonlinear waves in integrable models. Physica D 10, 379–386.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Fox, M. J. H. 1996 Asymptotic theory for the almost-highest solitary wave. J. Fluid Mech. 317, 1–19.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Tanaka, M. 1997 On the crest instabilities of steep surface waves. J. Fluid Mech. 336, 51–68.CrossRefGoogle Scholar
McLean, J. W. 1982 a Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.CrossRefGoogle Scholar
McLean, J. W. 1982 b Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331–341.CrossRefGoogle Scholar
Saffman, P. G. 1985 The superharmonic instability of finite-amplitude water waves. J. Fluid Mech. 159, 169–174.CrossRefGoogle Scholar
Tanaka, M. 1986 The stability of solitary waves. Phys. Fluids 29, 650–655.CrossRefGoogle Scholar
Tanaka, M., Dold, J. W., Lewy, M. & Peregrine, D. H. 1987 Instability and breaking of a solitary wave. J. Fluid Mech. 185, 235–248.CrossRefGoogle Scholar
Turner, R. E. L. & Vanden-Broeck, J.-M. 1988 Broadening of interfacial solitary waves. Phys. Fluids 31, 2486–2490.CrossRefGoogle Scholar
Wilkinson, J. H. 1965 The Algebraic Eigenvalue Problem. Oxford University Press.Google Scholar
Zakharov, V. E. 1975 Instability and nonlinear oscillations of solitons. JETP Lett. 22, 172–173.Google Scholar