A single pole and a duble pole in the inverse scattering transform method

1Vakhnenko, VO
1S. I. Subbotin Institute of Geophysics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2017, 7:10-17
https://doi.org/10.15407/dopovidi2017.07.010
Section: Mathematics
Language: Ukrainian
Abstract: 

For the discrete part of spectral data in the inverse scattering transform method, the double poles and a single pole are taken into account. The scope of application for the suggested spectral data is demonstrated through the analysis of the Vakhnenko—Parkes equation that allows new solutions to be obtained. This approach can be applied to other integrable nonlinear equations.

Keywords: double poles, inverse problem, spectral data
References: 
  1. Kraenkel, R. A., Leblond, H. & Manna, M. A. (2014). An integrable evolution equation for surface waves in deep water. J. Phys. A: Math. Theor., 47, No. 2, 025208 (17pp). doi:https://doi.org/10.1088/1751-8113/47/2/025208
  2. Sazonov, S. V. & Ustinov, N. V. (2017). Nonlinear propagation of vector extremely short pulses in a medium symmetric and asymmetric molecules. J. Exp. Theor. Phys., 124, No. 2, P. 213-230. doi:10.1134/S1063776117010150
  3. Vakhnenko, V. O. (1999). High-frequency soliton-like waves in a relaxing medium. J. Math. Phys., 40, pp. 2011-2020. doi:https://doi.org/10.1063/1.532847
  4. Kuetche, V. K. (2015). Barothropic relaxing media under pressure perturbations: Nonlinear dynamics. Dynamics Atmosph. Oceans., 72, pp. 21-37. doi:https://dx.doi.org/10.1016/j.dynatmoce.2015.10.001
  5. Kuetche, V. K. (2016). Inhomogeneous exchange within ferrites: Magnetic solitons and their interactions. J. Magnetism Magnetic Materials, 398, pp. 70-81. doi:https://dx.doi.org/10.1016/j.jmmm.2015.08.120
  6. Vakhnenko, V. O. & Parkes, E. J. (2002). The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method. Chaos, Solitons and Fractals, 13(9), pp. 1819-1826. doi:https://dx.doi.org/10.1016/S0960-0779(01)00200-4
  7. Ye, Y., Song, J., Shen, S. & Di, Y. (2012). New coherent structures of the Vakhnenko-Parkes equation. Results in Physics, 2, pp. 170-174. doi:https://dx.doi.org/10.1016/j.rinp.2012.09.011
  8. Vakhnenko, V. O. & Parkes, E. J. (2016). Approach in theory of nonlinear evolution equations: the Vakhnenko-Parkes equation. Advances in Mathematical Physics, 2016, Article ID 2916582, 39 p. doi:https://dx.doi.org/10.1155/2016/2916582
  9. Vakhnenko, V. A. (1992). Solitons in a nonlinear model medium. J. Phys. A: Math. Gen., 25, pp. 4181-4187. doi:https://doi.org/10.1088/0305-4470/25/15/025
  10. Roshid, H., Kabir, M. R., Bhowmik, R. C. & Datta, B. K. (2014). Investigation of solitary wave solutions for Vakhnenko-Parkes equation via exp-function and exp(−ϕ(ξ)) -expansion method. SpringerPlus. 3, 692 (10 pp). doi:https://doi.org/10.1186/2193-1801-3-692
  11. Vakhnenko, V. O. & Parkes, E. J. (2016). The inverse problem for some special spectral data. Chaos, Solitons and Fractals, 82, pp. 116-124. doi:https://dx.doi.org/10.1016/j.chaos.2015.11.012
  12. Caudrey, P. J. (1982). The inverse problem for a general N × N spectral equation. Physica D, D6, pp. 51-66. doi:https://doi.org/10.1016/0167-2789(82)90004-5
  13. Satsuma, J. & Kaup, D. J. (1977). A Bdcklund transformation for a higher order Korteweg-de Vries equation. J. Phys. Society Japan, 43, pp. 692-697. doi:https://dx.doi.org/10.1143/JPSJ.43.692
  14. Hirota, R. (1980). Direct methods in soliton theory. Solitons (Eds. R. K. Bullough, P. J. Caudrey), New York, Berlin: Springer, pp. 157-176.
  15. Vakhnenko, V. O. & Parkes, E. J. (2012). The singular solutions of a nonlinear evolution equation taking continuous part of the spectral data into account in inverse scattering method. Chaos, Solitons and Fractals, 45, pp. 846-852. doi:https://dx.doi.org/10.1016/j.chaos.2012.02.019