рус eng
«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2024. Vol 47

Soliton Solutions of the Negative Order Modified Korteweg – de Vries Equation

Author(s)
Gayrat U. Urazboev1, Iroda I. Baltaeva1, Shoira E. Atanazarova1,2

1Urgench State University, Urgench, Uzbekistan

2Khorezm branch of V. I. Romanovski Institute of Mathematics, Uzbekistan Academy of Science, Urgench, Uzbekistan

Abstract
In this paper, we study the negative order modified Korteweg-de Vries (nmKdV) equation in the class of rapidly decreasing functions. In particular, we show that the inverse scattering transform technique can be applied to obtain the time dependence of scattering data of the operator Dirac with potential being the solution of the considered problem. We demonstrate the explicit representation of one soliton solution of nmKdV based on the obtained results.
About the Authors

Gayrat U. Urazboev, Dr. Sci. (Phys.-Math.), Prof., Urgench State University, Urgench, 220100, Uzbekistan, gayrat71@mail.ru

Iroda I. Baltaeva, Cand. Sci. (Phys.-Math.), Assoc. Prof., Urgench State University, Urgench, 220100, Uzbekistan, iroda-b@mail.ru

Shoira E. Atanazarova, Master (Phys.-Math.), Junior Researcher, Khorezm Branch of V. I. Romanovski Institute of mathematics, Uzbekistan Academy of Science, Urgench, 220100, Uzbekistan, atanazarova94@gmail.com

For citation
Urazboev G. U., Baltaeva I. I., Atanazarova Sh. E. Soliton Solutions of the Negative Order Modified Korteweg – de Vries Equation. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 47, pp. 63–77. https://doi.org/10.26516/1997-7670.2024.47.63
Keywords
negative order modified Korteweg – de Vries equation, soliton, inverse scattering transform, scattering data, potential, reflection coefficient
UDC
517.957
MSC
35P25, 35P30, 35Q51, 35Q53, 37K15
DOI
https://doi.org/10.26516/1997-7670.2024.47.63
References
  1. Ablowitz M.J., Been J.B., Carr L.D. Fractional integrable and related discrete nonlinear Schrodinger equations. Physics Letter A, 2022, vol. 452, pp. 128459. https://doi.org/10.1016/j.physleta.2022.128459
  2. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H. The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems. Studies in Applied Mathematics,1974, vol. LII, no.4, pp. 249–315.
  3. Baltaeva I.I., Rakhimov I.D., Khasanov M.M. Exact traveling wave solutions of the loaded modified Korteweg-de Vries equation. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 41, pp. 85–95. https://doi.org/10.26516/1997-7670.2022.41.85
  4. Chen J. Quasi-periodic solutions to the negative-order KdV hierarchy. Int. J. Geom. Methods Mod. Phys, 2018, vol. 15, no. 3, 1850040.https://doi.org/10.1142/S0219887818500408
  5. Demontis F. Exact solutions of the modified Korteweg–de Vries equation. Theor. Math. Phys., 2011, vol. 168, pp. 886–897. https://doi.org/10.1007/s11232-011-0072-4
  6. Urazboev G., Babadjanova A.K. On the integration of the matrix modified Korteweg-de Vries equation with a self-consistent source. Tamkang Journal of Mathematics, 2019, vol. 50, pp. 281–291.https://doi.org/10.5556/j.tkjm.50.2019.3355
  7. Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M. Method for Solving the Korteweg de Vries Equation. Physical Review Letters, 1967, vol. 19, pp. 1095–1097.http://dx.doi.org/10.1103/PhysRevLett.19.1095
  8. Jingqun Wang, Lixin Tian, Yingnan Zhang. Breather solutions of a negative order modified Korteweg-de Vries equation and its nonlinear stability. Physics Letters A, 2019, vol. 383, pp. 1689–1697. https://doi.org/10.1016/j.physleta.2019.02.042
  9. Khasanov A.B., Allanazarova T.Zh. On the modified Korteweg-de Vries equation with loaded term. Ukrainian Mathematical Journal, 2022, vol. 73, pp. 1783–1809.https://doi.org/10.1007/s11253-022-02030-4
  10. Khasanov A.B., Yakhshimuratov A.B. The Korteweg-de Vries Equation with a Self-Consistent Source in the Class of Periodic Functions. Theor. Math. Phys., 2010, vol. 164, pp. 1008–1015.https://doi.org/10.4213/tmf6535
  11. Lax P.D. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl.Math., 1968, vol. 21, pp. 467–490.https://doi.org/10.1002/cpa.3160210503
  12. Olver P.J. Evolution equations possessing infinitely many symmetries. J.Math.Phys., 1977, vol. 18, pp. 1212–1215. https://doi.org/10.1063/1.523393
  13. Qiao Z., Li J. Negative-order KdV equation with both solitons and kink wave solutions. Europhysics Letters, 2011, vol. 94, no. 5, 50003.https://doi.org/10.1209/0295-5075/94/50003
  14. Qiao Z.J., Fan E.G. Negative-order Kortewe-de Vries equations. Phys. Rev., 2012, vol. 86, 016601. https://doi.org/10.1103/PHYSREVE.86.016601
  15. Reyimberganov A.A., Rakhimov I.D. The soliton solution for the nonlinear Schrodinger equation with self-consistent source. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 36, pp. 84–94.https://doi.org/10.54708/23040122_2022_14_1_77
  16. Shengyang Yuan, Jian Xu. On a Riemann-Hilbert problem for the negative – order KdV equation. Applied Mathematics Letters, 2022, vol. 132, 108106.https://doi.org/10.1016/j.aml.2022.108106
  17. Urazboev G.U., Khasanov M.M., Baltaeva I.I. Integration of the Negative Order Korteweg-de Vries Equation with a Special Source. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 44, pp. 31–43. (in Russian)https://doi.org/10.26516/1997-7670.2023.44.31
  18. Urazboev G.U., Khasanov M.M. Integration of the negative order Korteweg- De Vries equation with a self-consistent source in the class of periodic functions. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 2022, vol. 32, pp. 228–239. https://doi.org/10.35634/vm210209
  19. Urazboev G.U, Khasanov A.B. “Integrating the Korteweg–de Vries Equation with a Self-Consistent Source and “Steplike” Initial Data”. Theoret. and Math. Phys., 2001, vol. 129, pp. 38–54. https://doi.org/10.4213/tmf518
  20. Urazboev G.U., Babadjanova A.K., Zhuaspayev T.A. Integration of the periodic Harry Dym equation with a source. Wave Motion., 2022, vol. 113, 102970.https://doi.org/10.1016/j.wavemoti.2022.102970
  21. Urazboev G.U., Baltaeva I.I. Integration of the Camassa-Holm equation with a self-consistent source. Ufa Mathematical journal, 2022, vol. 14, pp. 77–86.https://doi.org/10.54708/23040122_2022_14_1_77
  22. Verosky J.M. Negative powers of Olver recursion operators. J. Math. Phys., 1991, vol. 32, pp. 1733–1736. https://ui.adsabs.harvard.edu/link_gateway/1991JMP32.1733V/doi:10.1063/1.529234
  23. Wadati M. The modified Korteweg–de Vries equation. J. Phys. Soc. Jpn., 1973, vol. 34, pp. 1289–1296.https://doi.org/10.1143/JPSJ.34.1289
  24. Wazwaz A.M., Xu G.Q. Negative order modified KdV equations: multiple soliton and multiple singular soliton solutions. Mathematical methods in the Applied Sciences, 2016, vol. 39.http://dx.doi.org/10.1002/mma.3507
  25. Zakharov V.E., Shabat A.B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Zh. Eksp. Teor. Fiz., 1971. vol. 61, pp. 118–134.
  26. Zhang Da-Jun, Zhao Song-Lin, Sun Ying-Ying, Zhou Jhou. Solutions to the modified Korteweg–de Vries equation. Reviews in Mathematical Physics, 2014, vol. 26, 1430006. http://dx.doi.org/10.1142/S0129055X14300064

Full text (english)
Загружается...