Perturbation Theory for the Compound Soliton of the Gardner’s Equation; Their Interaction and Evolution in a Media with Variable Parameters

  • Irina A. Soustova
  • Konstantin A. Gorshkov
  • Alexey V. Ermoshkin
  • Lev A. Ostrovsky
  • Yuliya I. Troitskaya
Chapter
Part of the Springer Oceanography book series (SPRINGEROCEAN)

Abstract

This paper is a brief review of the results of an approximate description of the evolution and interaction of composite solitons, obtained by the authors in 2001–2016. As one of the applications of the theory, the features of the evolution of intense internal waves in the shelf zone of the ocean are analyzed.

Notes

Acknowledgements

This study was supported by the Russian Foundation for Basic Research (projects 16-55-52022, 18-05-00292 and 17-05-41117 RGS). Numerical code development and numerical modeling were supported by the Russian Science Foundation (grant 15-17-20009).

References

  1. 1.
    Battjes, J. A., Zitman, T. J., & Holtheijsen, L. H. (1987). A re-analysis of the spectra observed in JONSWAP. Journal of Physical Oceanography, 17, 1288–1295.CrossRefGoogle Scholar
  2. 2.
    Ermoshkin, A. V., Bakhanov, V. V., & Bogatov, N. A. (2015). Development of an empirical model for Radar backscattering cross section of the ocean surface at grazing angles. Sovremennye problemi distantsionnogo zondirovaniya Zemli iz kosmosa, 12(4), 51–59.Google Scholar
  3. 3.
    Gorshkov, K. A., & Ostrovsky, L. A. (1981). Interaction of solitons in nonintegrable systems. Physica D: Nonlinear Phenomena, 3, 428–438.CrossRefGoogle Scholar
  4. 4.
    Grimshaw, R., Pelinovsky, E., & Talipova, T. (2007). Modeling internal solitary waves in the coastal ocean. Surveys In Geophysics, 28, 273–287.CrossRefGoogle Scholar
  5. 5.
    Gorshkov, К. A., & Soustova, I. A. (2001). Interaction of solitons as compound structures in the Gardner model. Radiophysics and Quantum Electronics, 44(5–6), 502–512.Google Scholar
  6. 6.
    Gorshkov, K. A., Ostrovsky, L. A., Soustova, I. A., & Irisov, V. G. (2004). Perturbation theory for kinks and application for multisoliton interactions in hydrodynamics. Physical Review E, 69, 1–10.CrossRefGoogle Scholar
  7. 7.
    Gorshkov, K. A., Ostrovsky, I. A., & Soustova, I. A. (2011). Dynamics of strongly nonlinear solitons in the two-layer fluid. Studies in Applied Mathematics, 126(1), 49–73.CrossRefGoogle Scholar
  8. 8.
    Gorshkov, К. A., Ostrovsky, L. A., Soustova, I. A., & Shevz, L. M. (2011). The interaction of intense internal waves within the framework of Choi-Camassa equation. Izvestiya RAN, Atmosphere and Ocean Physics., 47(3), 339–347.Google Scholar
  9. 9.
    Gorshkov, K. A., Soustova, I. A., Ermoshkin, A. V., & Zaitseva, N. V. (2012). Evolution of the composite soliton of the Gardner equation in media with variable parameter. Radiophysics and Quantum Electronics, 55(5), 380–392.CrossRefGoogle Scholar
  10. 10.
    Gorshkov, K. A., Soustova, I. A., & Ermoshkin, A. V. (2016). Field structure of a quasisoliton approaching the critical point. Radiophysics Quantum Electronics, 58(10), 738–744.CrossRefGoogle Scholar
  11. 11.
    Grimshaw, R., Pelinovsky, E., & Talipova, T. (1999). Solitary wave transformation in a medium with sing-variable quadratic nonlinearity and cubic nonlinearity. Physica D: Nonlinear Phenomena, 132, 40–62.CrossRefGoogle Scholar
  12. 12.
    Gorshkov, K.A., Dolina, I. S., Soustova, I. A., & Troitskaya, Y. I. (2003). Transformation of short waves in a nonuniform flow field on the ocean surface. The effect of wind growth rate modulation. Radiophysics Quantum Electronics 46(7), 464–485.Google Scholar
  13. 13.
    Hughes, B. (1978). The effect of internal waves on surface wind waves. 2. Theoretical analysis. Journal of Geophysical Research, 83(1), 455–465.CrossRefGoogle Scholar
  14. 14.
    Kropfli, R. A., Ostrovski, L. A., Stanton, T. P., Skirta, E. A., Keane, A. N., & Irisov, V. (1999). Relationships between strong internal waves in the coastal zone and their radar and radiometric signatures. Journal of Geophysical Research, 104, 3133–3148.CrossRefGoogle Scholar
  15. 15.
    Lamb, J. L. (1983). Introduction to the theory of solitons (p. 295).Google Scholar
  16. 16.
    Morozov, E. G., Paka, V. T., & Bakhanov, V. V. (2008). Strong internal tides in the Kara Gates Strait. Geophysical Research Letters, 35, L166603.CrossRefGoogle Scholar
  17. 17.
    Nakoulima, O., Zahybo, N., Pelynovsky, E., Talipova, T., Slunyaev, A., & Kurkin, A. (2004). Analytical and numerical studies of the variable-coefficient Gardner equation. Applied Mathematics and Computation, 152, 449–471.CrossRefGoogle Scholar
  18. 18.
    Serebryany, A. N., & Pao, H. P. (2008). Transition of a nonlinear internal wave through an overturning point on a shelf. Doklady Earth Science, 420(4), 714–719.CrossRefGoogle Scholar
  19. 19.
    Trevorrow, M. (1998). Observations of internal solitary waves near the Oregon coast with an inverted echo sounder. Journal of Geophysical Research, 103, 7671–7680.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Irina A. Soustova
    • 1
  • Konstantin A. Gorshkov
    • 1
  • Alexey V. Ermoshkin
    • 1
  • Lev A. Ostrovsky
    • 1
  • Yuliya I. Troitskaya
    • 1
  1. 1.Institute of Applied Physics, Russian Academy of SciencesNizhny NovgorodRussia

Personalised recommendations