Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 047, 12 pages      nlin.PS/0604076      http://dx.doi.org/10.3842/SIGMA.2006.047

Internal Modes of Solitons and Near-Integrable Highly-Dispersive Nonlinear Systems

Oksana V. Charkina and Mikhail M. Bogdan
B. Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine, 47 Lenin Ave., Kharkiv, 61103 Ukraine

Received November 30, 2005, in final form April 11, 2006; Published online April 28, 2006

Abstract
The transition from integrable to non-integrable highly-dispersive nonlinear models is investigated. The sine-Gordon and φ4-equations with the additional fourth-order spatial and spatio-temporal derivatives, describing the higher dispersion, and with the terms originated from nonlinear interactions are studied. The exact static and moving topological kinks and soliton-complex solutions are obtained for a special choice of the equation parameters in the dispersive systems. The problem of spectra of linear excitations of the static kinks is solved completely for the case of the regularized equations with the spatio-temporal derivatives. The frequencies of the internal modes of the kink oscillations are found explicitly for the regularized sine-Gordon and φ4-equations. The appearance of the first internal soliton mode is believed to be a criterion of the transition between integrable and non-integrable equations and it is considered as the sufficient condition for the non-trivial (inelastic) interactions of solitons in the systems.

Key words: solitons; integrable and non-integrable equations; internal modes; dispersion.

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References

  1. Ablowitz M.J., Segur H., Solitons and the inverse scattering transform, Philadelphia, SIAM, 1981.
  2. Dashen R., Hasslacher B., Neveu A., Particle spectrum in model field theories from semiclassical functional integral techniques, Phys. Rev. D, 1975, V.11, N 12, 3424-3450.
  3. Bogdan M.M., Kosevich A.M., Voronov V.P., Generation of the internal oscillation of soliton in a one-dimensional non-integrable system, in Proceedings of IVth International Workshop "Solitons and Applications" (August 24-26, 1989, Dubna, USSR), Editors V.G. Makhankov, V.K. Fedyanin and O.K. Pashaev, Singapore, World Scientific, 1990, Part IV, 397-401.
  4. Landau L.D., Lifshitz E.M., Quantum mechanics, New York, Pergamon, 1977.
  5. Campbell D.K., Peyrard M., Sodano P., Kink-antikink interactions in the double sine-Gordon equation, Phys. D, 1986, V.19, N 2, 165-205.
  6. Campbell D.K., Schonfeld J.F., Wingate C.A., Resonance structure in kink-antikink interactions in f4 theory, Phys. D, 1983, V.9, N 1-2, 1-32.
  7. Bogdanov L.V., Zakharov V.E., The Boussinesq equation revisited, Phys. D, 2002, V.165, N 3-4, 137-162.
  8. Kivshar Yu.S., Pelinovsky D.E., Cretegny T., Peyrard M., Internal modes of solitary waves, Phys. Rev. Lett., 1998, V.80, N 23, 5032-5035.
  9. Berryman J.G., Stability of solitary waves in shallow water, Phys. Fluids, 1976, V.19, N 6, 771-777.
  10. Fal'kovich G.E., Spector M.D., Turitsyn S.K., Destruction of stationary solutions and collapse in the nonlinear string equation, Phys. Lett. A, 1983, V.99, N 6-7, 271-274.
  11. Kuznetsov E.A., Spector M.D., Fal'kovich G.E., On the stability of nonlinear waves in integrable models, Phys. D, 1984, V.10, N 3, 379-386.
  12. Yajima N., On a growing mode of the Boussinesq equation, Progr. Theoret. Phys., 1983, V.69, N 2, 678-680.
  13. Flytzanis N., Pnevmatikos S., Remoissenet M., Soliton resonances in atomic nonlinear systems, Phys. D, 1987, V.26, N 1-3, 311-320.
  14. Tajiri M., Murakami Y., On breather solutions to the Boussinesq equation, J. Phys. Soc. Japan, 1989, V.58, N 10, 3585-3590.
  15. Bogdan M.M., Kosevich A.M., Maugin G.A., Soliton complex dynamics in strongly dispersive medium, Wave Motion, 2001, V.34, N 1, 1-26, patt-sol/9902009.
  16. Charkina O.V., Bogdan M.M., Internal modes of oscillations of topological solitons in highly dispersive media, Uzhgorod Univ. Sci. Herald. Series Physics, 2005, N 17, 30-37 (in Russian).
  17. Gorshkov K.A., Ostrovsky L.A., Interaction of solitons in nonintegrable systems: direct perturbation method and applications, Phys. D, 1981, V.3, N 3, 428-438.
  18. Kawahara T., Toh S., Pulse interactions in an unstable dissipative-dispersive nonlinear system, Phys. Fluids, 1988, V.3, N 8, 2103-2111.
  19. Champneys A.R., Malomed B.A., Yang J., Kaup D.J., Embedded solitons: solitary waves in resonance with the linear spectrum, Phys. D, 2001, V.152-153, 340-354, nlin.PS/0005056.
  20. Kolossovski K., Champneys A.R., Buryak A.V., Sammut R.A., Multi-pulse embedded solitons as bound states of quasi-solitons, Phys. D, 2002, V.171, N 3, 153-177.
  21. Barashenkov I.V., Zemlyanaya E.V., Stable complexes of parametrically driven damped nonlinear Schrödinger solitons, Phys. Rev. Lett., 1999, V.83, N 13, 2568-2571.
  22. Kosevich A.M., Kovalev A.S., The supersonic motion of a crowdion. The one-dimensional model with nonlinear interaction between the nearest neighbours, Sol. State Comm., 1973, V.12, N 8, 763-765.
  23. Konno K., Kameyama W., Sanuki H., Effect of weak dislocation potential on nonlinear wave propagation in anharmonic crystal, J. Phys. Soc. Japan, 1974, V.37, N 1, 171-176.
  24. Chen D.Y., Zhang D.J., Deng S.F., The novel multi-soliton solutions of the MKdV-sine Gordon equations, J. Phys. Soc. Japan, 2002, V.71, N 2, 658-659.
  25. Hirota R., Direct method of finding exact solutions of non-linear evolution equations, in Bäcklund Transformations, Editors A. Dold and B. Eckmann, Lecture Notes in Mathematics, Vol. 515, New York, Springer-Verlag, 1980, 40-68.
  26. Leo M., Leo R.A., Soliani G., Soliton-like solutions for a dispersive nonlinear wave equation, Progr. Theoret. Phys., 1978, V.60, N 1, 100-111.
  27. Braun O.M., Zhang Fei, Kivshar Yu.S., Vazquez L., Kinks in the Klein-Gordon model with anharmonic interatomic interactions: a variational approach, Phys. Lett. A, 1991, V.157, N 4-5, 241-245.
  28. Gendelman O.V., Manevich L.I., Exact soliton-like solutions in generalised dynamical models of quasi-one-dimensional crystal, Zh. Eksper. Teoret. Fiz., 1997, V.112, N 4 (10), 1510-1515.

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