phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2022 - 2 issue » pp. 228-239

Archive of Issues

Uzbekistan Urgench
Year
2022
Volume
32
Issue
2
Pages
228-239
<<
>>
Section Mathematics
Title Integration of the negative order Korteweg–de Vries equation with a self-consistent source in the class of periodic functions
Author(-s) Urazboev G.U.a, Hasanov M.M.a
Affiliations Urgench State Universitya
Abstract In this paper, we consider the negative order Korteweg–de Vries equation with a self-consistent integral source. It is shown that the negative-order Korteweg–de Vries equation with a self-consistent integral source can be integrated by the method of the inverse spectral problem. The evolution of the spectral data of the Sturm–Liouville operator with a periodic potential associated with the solution of the negative order Korteweg–de Vries equation with a self-consistent integral source is determined. The obtained results make it possible to apply the inverse problem method to solve the negative order Korteweg–de Vries equation with a self-consistent source in the class of periodic functions.
Keywords negative-order KdV, self-consistent source, inverse spectral problem
UDC 517.946
MSC 35P25, 35P30, 35Q51, 35Q53, 37K15
DOI 10.35634/vm220205
Received 28 March 2022
Language Russian
Citation Urazboev G.U., Hasanov 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, issue 2, pp. 228-239.
References
  1. 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, issue 19, pp. 1095-1097. http://doi.org/10.1103/PhysRevLett.19.1095
  2. Novikov S.P. The periodic problem for the Korteweg-de Vries equation, Functional Analysis and Its Applications, 1974, vol. 8, issue 3, pp. 236-246. https://doi.org/10.1007/BF01075697
  3. Dubrovin B.A., Novikov S.P. Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation, Soviet Physics - JETP, 1975, vol. 40, no. 6, pp. 1058-1063.
  4. Marchenko V.A. The periodic Korteweg-de Vries problem, Mathematics of the USSR-Sbornik, 1974, vol. 24, issue 3, pp. 319-344. http://doi.org/10.1070/SM1974v024n03ABEH002189
  5. Dubrovin B.A. Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials, Functional Analysis and Its Applications, 1975, vol. 9, issue 3, pp. 215-223. https://doi.org/10.1007/BF01075598
  6. Its A.R., Matveev V.B. Schrödinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg-de Vries equation, Theoretical and Mathematical Physics, 1975, vol. 23, issue 1, pp. 343-355. https://doi.org/10.1007/BF01038218
  7. Lax P. Periodic solutions of the KdV equations, Nonlinear wave motion, Providence: AMS, 1974, pp. 85-96. https://www.ams.org/mathscinet-getitem?mr=0344645
  8. Lax P.D. Periodic solutions of the KdV equation, Communications on Pure and Applied Mathematics, 1975, vol. 28, issue 1, p. 141-188. https://doi.org/10.1002/cpa.3160280105
  9. Mel'nikov V.K. Metod integrirovaniya uravneniya Kortevega-de Vrisa s samosoglasovannym istochnikom (A method for integrating the Korteweg-de Vries equation with a self-consistent source), Preprint no. 2-88-11/798. Dubna: Joint Institute for Nuclear Research, 1988. https://s3.cern.ch/inspire-prod-files-d/d3b9e5b61499cfe7675c6be753aeca7c
  10. Leon J., Latifi A. Solution of an initial-boundary value problem for coupled nonlinear waves, Journal of Physics A: Mathematical and General, 1990, vol. 23, no. 8, pp. 1385-1403. https://doi.org/10.1088/0305-4470/23/8/013
  11. Urasboev G.U., Khasanov A.B. Integrating the Korteweg-de Vries equation with a self-consistent source and “steplike” initial data, Theoretical and Mathematical Physics, 2001, vol. 129, no. 1, pp. 1341-1356. https://doi.org/10.1023/A:1012463310382
  12. Khasanov A.B., Urazboev G.U. Integration of the general KdV equation with the right hand side in the class of rapidly decreasing functions, Uzbek Mathematical Journal, 2003, no. 2, pp. 53-59.
  13. Khasanov A.B., Yakhshimuratov A.B. The Korteweg-de Vries equation with a self-consistent source in the class of periodic functions, Theoretical and Mathematical Physics, 2010, vol. 164, issue 2, pp. 1008-1015. https://doi.org/10.1007/s11232-010-0081-8
  14. Bondarenko N., Freiling G., Urazboev G. Integration of the matrix KdV equation with self-consistent sources, Chaos, Solitons and Fractals, 2013, vol. 49, pp. 21-27. https://doi.org/10.1016/j.chaos.2013.02.010
  15. Yakhshimuratov A.B., Khasanov M.M. Integration of the modified Korteweg-de Vries equation with a self-consistent source in the class of periodic functions, Differential Equations, 2014, vol. 50, issue 4, pp. 533-540. https://doi.org/10.1134/S0012266114040119
  16. Khasanov A.B., Hoitmetov U.A. Integration of the general loaded Korteweg-de Vries equation with an integral type source in the class of rapidly decreasing complex-valued functions, Russian Mathematics, 2021, vol. 65, issue 7, pp. 43-57. https://doi.org/10.3103/S1066369X21070069
  17. Hasanov A.B., Hasanov M.M. Integration of the nonlinear Schrödinger equation with an additional term in the class of periodic functions, Theoretical and Mathematical Physics, 2019, vol. 199, issue 1, pp. 525-532. https://doi.org/10.1134/S0040577919040044
  18. Yakhshimuratov A.B., Matyokubov M.M. Integration of a loaded Korteweg-de Vries equation in a class of periodic functions, Russian Mathematics, 2016, vol. 60, issue 2, pp. 72-76. https://doi.org/10.3103/S1066369X16020110
  19. Babadjanova A., Kriecherbauer T., Urazboev G. The periodic solutions of the discrete modified KdV equation with a self-consistent source, Applied Mathematics and Computation, 2020, vol. 376, 125136. http://doi.org/10.1016/j.amc.2020.125136
  20. Urazboev G.U., Babadjanova A.K., Saparbaeva D.R. Integration of the Harry Dym equation with an integral type source, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 2, pp. 285-295. https://doi.org/10.35634/vm210209
  21. Verosky J.M. Negative powers of Olver recursion operators, Journal of Mathematical Physics, 1991, vol. 32, issue 7, pp. 1733-1736. https://doi.org/10.1063/1.529234
  22. Lou S. Symmetries of the KdV equation and four hierarchies of the integrodifferential KdV equations, Journal of Mathematical Physics, 1994, vol. 35, issue 5, pp. 2390-2396. http://doi.org/10.1063/1.530509
  23. Degasperis A., Procesi M. Asymptotic integrability, Symmetry and perturbation theory, Singapore: World Scientific, 1999, pp. 23-37.
  24. Zhang G., Qiao Z. Cuspons and smooth solitons of the Degasperis-Procesi equation under inhomogeneous boundary condition, Mathematical Physics, Analysis and Geometry, 2007, vol. 10, issue 3, pp. 205-225. https://doi.org/10.1007/s11040-007-9027-2
  25. Qiao Z., Fan E. Negative-order Korteweg-de Vries equations, Physical Review E, 2012, vol. 86, issue 1, 016601. https://doi.org/10.1103/PhysRevE.86.016601
  26. Chen J. Quasi-periodic solutions to a negative-order integrable system of 2-component KdV equation, International Journal of Geometric Methods in Modern Physics, 2018, vol. 15, issue 3, 1850040. https://doi.org/10.1142/S0219887818500408
  27. 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
  28. Zhao S., Sun Y. A discrete negative order potential Korteweg-de Vries equation, Zeitschrift für Naturforschung A, 2016, vol. 71, issue 12, pp. 1151-1158. https://doi.org/10.1515/zna-2016-0324
  29. Rodriguez M., Li J., Qiao Z. Negative order KdV equation with no solitary traveling waves, Mathematics, 2022, vol. 10, no. 1, article 48. https://doi.org/10.3390/math10010048
  30. Wazwaz A.-M. Negative-order KdV equations in (3+1) dimensions by using the KdV recursion operator, Waves in Random and Complex Media, 2017, vol. 27, issue 4, pp. 768-778. https://doi.org/10.1080/17455030.2017.1317115
  31. Kuznetsova M. Necessary and sufficient conditions for the spectra of the Sturm-Liouville operators with frozen argument, Applied Mathematics Letters, 2022, vol. 131, 108035. https://doi.org/10.1016/j.aml.2022.108035
  32. Titchmarsh E.C. Eigenfunction expansions associated with second-order differential equations. Part 2, London: Oxford University Press, 1958.
  33. Magnus W., Winkler W. Hill's equation, New York: Interscience Publishers, 1966.
  34. Stankevich I.V. A certain inverse spectral analysis problem for Hill's equation, Doklady Akademii Nauk SSSR, 1970, vol. 192, no. 1, pp. 34-37. http://mi.mathnet.ru/eng/dan35384
  35. Marchenko V.A., Ostrovskii I.V. A characterization of the spectrum of Hill's operator, Mathematics of the USSR-Sbornik, 1975, vol. 26, no. 4, pp. 493-554. http://doi.org/10.1070/SM1975v026n04ABEH002493
  36. Trubowitz E. The inverse problem for periodic potentials, Communications on Pure and Applied Mathematics, 1977, vol. 30, issue 3, pp. 321-337. https://doi.org/10.1002/cpa.3160300305
  37. Levitan B.M., Sargsyan I.S. Operatory Shturma-Liuvillya i Diraka (Sturm-Liouville and Dirac operators), Moscow: Nauka, 1988.
Full text
/doc/issues-2022/22-02-05.pdf
<< Previous article
Next article >>