The European Physical Journal Special Topics

, Volume 147, Issue 1, pp 25–43 | Cite as

The modulation instability revisited

  • H. Segur
  • D. M. Henderson
Article

Abstract.

The modulational instability (or “Benjamin-Feir instability”) has been a fundamental principle of nonlinear wave propagation in systems without dissipation ever since it was discovered in the 1960s. It is often identified as a mechanism by which energy spreads from one dominant Fourier mode to neighboring modes. In recent work, we have explored how damping affects this instability, both mathematically and experimentally. Mathematically, the modulational instability changes fundamentally in the presence of damping: for waves of small or moderate amplitude, damping (of the right kind) stabilizes the instability. Experimentally, we observe wavetrains of small or moderate amplitude that are stable within the lengths of our wavetanks, and we find that the damped theory predicts the evolution of these wavetrains much more accurately than earlier theories. For waves of larger amplitude, neither the standard (undamped) theory nor the damped theory is accurate, because frequency downshifting affects the evolution in ways that are still poorly understood.

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References

  1. M.J. Lighthill, J. Inst. Math. Appl. 1, 269 (1965) Google Scholar
  2. G.B. Whitham, J. Fluid Mech. 27, 399 (1967) Google Scholar
  3. T.B. Benjamin, J.E. Feir, J. Fluid Mech. 27, 417 (1967) Google Scholar
  4. V.E. Zakharov, Sov. Phys. JETP 24, 455 (1967) Google Scholar
  5. L.A. Ostrovsky, Sov. Phys. JETP 24, 797 (1967) Google Scholar
  6. D.J. Benney, A.C. Newell, Stud. App. Math. 46, 133 (1967) Google Scholar
  7. T.B. Benjamin, Proc. R. Soc. Lond. A 299, 59 (1967) Google Scholar
  8. V.E. Zakharov, J. Appl. Mech. Tech. Phys. 2, 190 (1968) Google Scholar
  9. B.M. Lake, H.C. Yuen, H. Rungaldier, W.E. Ferguson, J. Fluid Mech. 83, 49 (1977) Google Scholar
  10. K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986) Google Scholar
  11. J. Hammack, D. Henderson, J. Offshore Mech. Arctic Eng. 125, 48 (2003) Google Scholar
  12. J.L. Hammack, D.M. Henderson, H. Segur, J. Fluid Mech. 532, 1 (2005) Google Scholar
  13. D.M. Henderson, M.S. Patterson, H. Segur, J. Fluid Mech. 559, 413 (2006) Google Scholar
  14. O. Kimmoun, H. Branger, C. Kharif, Eur. J. Mech. B 18, 889 (1999) Google Scholar
  15. H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff, K. Socha, J. Fluid Mech. 539, 229 (2005) Google Scholar
  16. W. Craig, D.M. Henderson, M. Oscamou, H. Segur, Math. Comput. Simul. (2006) Google Scholar
  17. V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP 34, 62 (1972) Google Scholar
  18. V.E. Zakharov, A.M. Rubenchik, Sov. Phys. JETP 38, 494 (1974) Google Scholar
  19. V.E. Zakharov, V.S. Synakh, Sov. Phys. JETP 41, 465 (1976) Google Scholar
  20. C. Sulem, P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Callapse (Springer, New York, 1999) Google Scholar
  21. M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981) Google Scholar
  22. A. Hasegawa, Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, 1995) Google Scholar
  23. E.L. Ince, Ordinary Differential Equations (Dover, New York, 1956) Google Scholar
  24. V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations (Princeton University Press, 1960) Google Scholar
  25. R.B. Guenther, J.W. Lee, Partial Differential Equations of Math. Physics and Integral Equations (Dover, New York, 1988) Google Scholar
  26. K. Dysthe, Proc. Roy. Soc. A 369, 105 (1979) Google Scholar
  27. E. Lo, C.C. Mei, J. Fluid Mech. 150, 395 (1985) Google Scholar
  28. C.J. McKinstrie, R. Bingham, Phys. Fluids B 1, 230 (1989) Google Scholar
  29. J.W. Miles, Proc. Roy. Soc. A 297, 459 (1967) Google Scholar
  30. F.M. Mitschke, L.F. Mollenauer, Optics Lett. 11, 659 (1986) Google Scholar
  31. J.P. Gordon, Opt. Lett. 11, 662 (1986) Google Scholar
  32. K. Trulsen, K.B. Dysthe, J. Fluid Mech. 352, 359 (1997) Google Scholar
  33. D.R. Furhman, P.A. Madsen, J. Fluid Mech. 559, 205 (2006) Google Scholar
  34. A.K. Dhar, K.P. Das, Phys. Fluids A 3, 3021 (1991) Google Scholar
  35. M. Oronato, A.R. Osborne, M. Serio, Phys. Rev. Lett. 96, 014503 (2006) Google Scholar
  36. P.K. Shukla, I. Kourakis, M. Markland, L. Stenflo, Phys. Rev. Lett. 97, 094501 (2006) Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  • H. Segur
    • 1
  • D. M. Henderson
    • 2
  1. 1.Department of Applied MathematicsUniversity of ColoradoBoulderUSA
  2. 2.Department of Mathematics, Penn State UniversityW.G. Pritchard Fluid Mechanics LaboratoryUniversity ParkUSA

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