Nonlinear Dynamics of Particle-Like States of Multidimensional Fields

  • I. S. Aranson
  • K. A. Gorshkov
  • A. S. Lomov
  • M. I. Rabinovich
Part of the NATO ASI Series book series (NSSB, volume 225)


In recent five-ten years there have appeared quite a number of papers concerned with various multidimensional spatial patterns in nonlinear media: periodic and quasiperiodic lattices, localized (particle-like) solutions, spiral structures, vortices, and so on (Makhan’kov, 1983; Haken, 1983; Linde, 1984; Kuramoto, 1984). As a rule, unified methods are applied for the analysis of the structures of different origins and the results obtained can be used in different fields of physics and not only physics. This means that we witness the advent of a new field of nonlinear dynamics -the theory of structures in nonlinear media. Many problems of this theory were posed and solved before, in the theory of nonlinear waves, in the first place. However, they were, ordinarily, concerned (except, perhaps, hydrodynamics) with one- dimensional structures: solitons, shock waves, dissipative structures (combustion fronts, etc.). The inverstigation of multidimensional structures that was necessitated by astrophysics, the physics of atmosphere and ocean, the nonlinear field theory microelectronics, gave new formulations of the problems.


Localize Structure Nonlocalized State Gradient Form Nonlinear Schrodinger Equation Stable Equilibrium State 
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  1. Aranson I. S., Gorshkov K. A. and Rabinovich M. I., 1989, Phys. Lett. A, 139:65.ADSCrossRefGoogle Scholar
  2. Aranson I. S. and Rabinovich, M. I., 1939, J. Physics A, 22.Google Scholar
  3. Chernikov A. A., Sagdeev R.Z., Usikov D. A. and Zaslavsky G. M., 1987, Phys. Lett. A, 125:101.MathSciNetADSCrossRefGoogle Scholar
  4. Dieker S. B., Pindak R. and Meyer R. B., Phys. Rev. Lett.,1986, 56:1819.ADSCrossRefGoogle Scholar
  5. Gorshkov K. A., Lomov A. S. and Rabinovich M. I., 1989, Phys. Lett. A, 137:50.CrossRefGoogle Scholar
  6. Gorshkov K. A. and Ostrovsky L. A., 1981, Physica D, 3:428.ADSzbMATHCrossRefGoogle Scholar
  7. Hagan, P. S., 1982, SIAM J Appl. Math., 42:726.MathSciNetCrossRefGoogle Scholar
  8. Haken, H., 1983, “Synergetics. Hierarchy of Instabilities in Self-Organized Systems and Devices”. Springer, N. Y.Google Scholar
  9. Hobart, R. H., 1963, Proc. Phys. Soc., 32:201.MathSciNetADSCrossRefGoogle Scholar
  10. Joets A. and Ribotta R., 1988, Phys. Rev. Lett., 60:2164.ADSCrossRefGoogle Scholar
  11. Kolodner P., Bensimon P. and Surko C. H., 1988, Phys. Rev. Lett., 60:1723.ADSCrossRefGoogle Scholar
  12. Kuramoto Y., 1984, “Chemical Oscillations. Waves and Turbulence”, Springer, BerlinzbMATHCrossRefGoogle Scholar
  13. Linde H., 1984, Topological Similarities in Dissipative Structures, in: “Self-Organization Autowaves and Structures Far from Equilibrium,” V.I. Krinski, ed, Springer, Berlin.Google Scholar
  14. Lonngren K. and Scott E., 1978, “Solitons in Action,” Academic Press, N.Y.zbMATHGoogle Scholar
  15. Malomed B. A., 1986, DAN SSSR, 291:327.MathSciNetGoogle Scholar
  16. Makhan’kov, V. G., 1983, Fiz. Elementarn. Chastits i Jadra, 14:123, (in Russian).MathSciNetGoogle Scholar
  17. Moses E. and Steinberg V., 1986, Phys. Rev. A, 34:693.ADSCrossRefGoogle Scholar
  18. Olsen, J. L., 1985, J. Low Temp. Phys., 61:167.CrossRefGoogle Scholar
  19. Rehberg I., Rasenat S., Fineberg J., Juarer M. and Steinberg V., 1988, Phys. Rev. Lett., 61:2443.ADSCrossRefGoogle Scholar
  20. Swift J. and Hohenberg P. C., 1977, Phys. Rev. A, 15:319.ADSCrossRefGoogle Scholar
  21. Walgraff D., 1989, Flow Field Effects on Dynamical Instabilities, in: “Instabilities and Nonequilibrium Structures, II,” E. Torapegui and D. Villaroel ed., Kluner Academic Publisher.Google Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • I. S. Aranson
    • 1
  • K. A. Gorshkov
    • 1
  • A. S. Lomov
    • 1
  • M. I. Rabinovich
    • 1
  1. 1.Institute of Applied PhysicsUSSR Academy of SciencesGorkyUSSR

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