Solitary-Waves in Self-Induced Transparency

  • Olivier Martin
  • Spiros V. Branis
Part of the NATO ASI Series book series (NSSB, volume 284)


Numerous systems are known to give rise to velocity or wavelength “selection”: crystal growth, fluid flow, shock waves, etc... Mathematically, the velocity or wavelength appear as non-linear eigenvalues1,2. It is often the case that the “unperturbed” problem (e.g., the zero-surface tension limit for crystal growth and fluid flow) has a continuous non-linear eigenvalue spectrum because of some underlying symmetry. The addition of a perturbation will usually break this symmetry and will lead to a discrete spectrum, that is to “selection”. Exactly integrable PDEs provide a particularly interesting ground for selection studies. They typically have a continuum of soliton solutions with different velocities or amplitudes. Adding a generic perturbation destroys their exact integrability. Solitons should disappear or become solitary-waves under such perturbations. There are many examples where ordinary perturbations destroy the family of solitons, leaving a single solitary-wave3. The effects of singular perturbations are more subtle, but have been investigated in the last few years for the cases of the standard exactly integrable PDEs (KdV, NLS, SG; see several of the articles in these proceedings). The conclusion of these works is that higher derivatives destroy all solitary-waves; generally, steady-state solutions have capillary waves going out all the way to infinity from the main peak. The purpose of this article is to report on some work4,5 on the coupled Maxwell-Bloch equations. In the slowly varying envelope approximation (SVEA), these PDEs reduce to an exactly integrable system which can be treated by the inverse scattering transform6,7. Also, to all orders in perturbation theory, there is a continuum of solitary-wave solutions.


Singular Perturbation Electric Field Amplitude Pulse Solution Exact Integrability Solitary Pulse 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G.I. Barenblatt, “Similarity, Self-Similarity, and Intermediate Asymptotics” (Consultants Bureau, New York, 1978).Google Scholar
  2. 2.
    “Dynamics of Curved Fronts”, edited by P. Pelcé, (Academic, New York, 1988).Google Scholar
  3. 3.
    V. Hakim, P. Jakobsen, and Y. Pomeau, Europhys. Lett. 11, 19 (1990).ADSCrossRefGoogle Scholar
  4. 4.
    S.V. Branis, O. Martin, and J.L. Birman, Phys. Rev. Lett. 65, 2638 (1990).ADSCrossRefGoogle Scholar
  5. 5.
    S.V. Branis, O. Martin, and J.L. Birman, Phys. Rev. A 43, 1549 (1991).ADSCrossRefGoogle Scholar
  6. 6.
    M.J Ablowitz, D.J. Kaup, and A.C. Newell, J. Math. Phys. 15 1852 (1974).ADSCrossRefGoogle Scholar
  7. 7.
    G. L. Lamb Jr.. Rev. Mod. Phys. 43. 99 (1971).MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    S.L. McCall and E. L. Hahn. Bull. Am. Phys. Soc. 10, 1189 (1965);Google Scholar
  9. 8a.
    S.L. McCall and E. L. Hahn. Phys. Rev. Lett. 18, 908 (1967)ADSCrossRefGoogle Scholar
  10. 8b.
    S.L. McCall and E. L. Hahn. Phys. Rev. 183, 457 (1969).ADSCrossRefGoogle Scholar
  11. 9.
    Z. Bialynicka-Birula, Phys. Rev. A 10, 999 (1974).ADSCrossRefGoogle Scholar
  12. 10.
    O. Akimoto, and K. Ikeda, J. Phys. A 10, 425 (1977).ADSCrossRefGoogle Scholar
  13. 11.
    R.A. Marth, D.A. Holmes, and J.H. Eberly, Phys. Rev. A 9, 2733 (1974).ADSCrossRefGoogle Scholar
  14. 12.
    L. Allen and J.H. Eberly, “Optical Resonance and Two-Level Atoms” (Dover, New York, 1987).Google Scholar
  15. 13.
    F. Bloch, Phys. Rev. 70, 460 (1946).ADSCrossRefGoogle Scholar
  16. 14.
    G. L. Lamb Jr., Phys. Rev. Lett. 31, 196 (1973);ADSCrossRefGoogle Scholar
  17. 14a.
    G. L. Lamb Jr., Phys, Rev, A 9, 422 (1974).Google Scholar
  18. 15.
    V. E. Zakharov and A. B. Shabat. Zh. Eksp. Teor. Fiz. 61, 118 (1971)Google Scholar
  19. 15a.
    V. E. Zakharov and A. B. Shabat. [Sov. Phys. JETP 34. 62 (1972)].MathSciNetADSGoogle Scholar
  20. 16.
    H.A. Hauss, Rev Mod. Phys. 51, 331 (1979).ADSCrossRefGoogle Scholar
  21. 17.
    J-M. Vanden-Broeck, Phys. Fluids 26(8), 2033 (1983).MathSciNetADSCrossRefGoogle Scholar
  22. 18.
    M. Kruskal, H. Segur, Aeronautical Research Associates of Princeton, Technical Memo, 85–25, 1985 (to appear in Studies in Applied Math.. 1991).Google Scholar
  23. 19.
    D.A. Kessler, J. Koplik, and H. Levine, Adv, Phys. 37, 255 (1988).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Olivier Martin
    • 1
  • Spiros V. Branis
    • 2
  1. 1.Department of PhysicsCity CollegeNew YorkUSA
  2. 2.Department of PhysicsEmory UniversityAtlantaUSA

Personalised recommendations