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Home » Issues » Archive of Issues » 2023 - 1 issue » pp. 156-170

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Uzbekistan Urgench
Year
2023
Volume
33
Issue
1
Pages
156-170
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Section Mathematics
Title Integration of the Korteweg-de Vries equation with loaded terms and a self-consistent source in the class of rapidly decreasing functions
Author(-s) Hoitmetov U.A.a, Khasanov T.G.a
Affiliations Urgench State Universitya
Abstract In this paper, we solve the Cauchy problem for the Korteweg-de Vries equation with loaded terms and a self-consistent source in the class of rapidly decreasing functions. To solve this problem, the method of the inverse scattering problem is used. The evolution of the scattering data of the self-adjoint Sturm-Liouville operator, whose coefficient is a solution of the Korteweg-de Vries equation with loaded terms and a self-consistent source, is obtained. Examples are given to illustrate the application of the obtained results.
Keywords loaded Korteweg-de Vries equation, Jost solutions, inverse scattering problem, Gelfand-Levitan-Marchenko integral equation, evolution of the scattering data
UDC 517.956
MSC 34L25, 35P25, 47A40, 37K15
DOI 10.35634/vm230111
Received 6 October 2022
Language English
Citation Hoitmetov U.A., Khasanov T.G. Integration of the Korteweg-de Vries equation with loaded terms and a self-consistent source in the class of rapidly decreasing functions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 1, pp. 156-170.
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, no. 19, pp. 1095-1097. https://doi.org/10.1103/PhysRevLett.19.1095
  2. Faddeev L.D. Properties of the $S$-matrix of the one-dimensional Schrödinger equation, Nine papers on partial differential equations and functional analysis, Providence, RI: Amer. Math. Soc., 1967, pp. 139-166. https://zbmath.org/?q=an:0181.56704
  3. Marchenko V.A. Operatory Shturma-Liuvillya i ikh prilozheniya (Sturm-Liouville operators and their applications), Kiev: Naukova Dumka, 1977.
  4. Levitan B.M. Obratnye zadachi Shturma-Liuvillya (Inverse Sturm-Liouville problems), Moscow: Nauka, 1984.
  5. Lax P.D. Integrals of nonlinear equations of evolution and solitary waves, Communications on Pure and Applied Mathematics, 1968, vol. 21, issue 5, pp. 467-490. https://doi.org/10.1002/cpa.3160210503
  6. Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevskii L.P. Teoriya solitonov. Metod obratnoi zadachi (Theory of solitons. Inverse problem method), Moscow: Nauka, 1980. https://zbmath.org/?q=an:0598.35003
  7. Ablowitz M.J., Segur H. Solitons and the inverse scattering transform, SIAM Philadelphia, 1981. https://doi.org/10.1137/1.9781611970883
  8. Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.C. Solitons and nonlinear wave equations, London: Academic Press, 1982.
  9. Mel'nikov V.K. Integration method of the Korteweg-de Vries equation with a self-consistent source, Physics Letters A, 1988, vol. 133, issue 9, pp. 493-496. https://doi.org/10.1016/0375-9601(88)90522-1
  10. Mel'nikov V.K. Integration of the Korteweg-de Vries equation with a source, Inverse Problems, 1990, vol. 6, no. 2, pp. 233-246. https://doi.org/10.1088/0266-5611/6/2/007
  11. 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
  12. Claude C., Latifi A., Leon J. Nonlinear resonant scattering and plasma instability: an integrable model, Journal of Mathematical Physics, 1991, vol. 32, issue 12, pp. 3321-3330. https://doi.org/10.1063/1.529443
  13. Khasanov A.B., Matyakubov M.M. Integration of the nonlinear Korteweg-de Vries equation with an additional term, Theoretical and Mathematical Physics, 2020, vol. 203, issue 2, pp. 596-607. https://doi.org/10.1134/S0040577920050037
  14. Khasanov A.B., Khasanov T.G. The Cauchy problem for the Korteweg-de Vries equation in the class of periodic infinite-gap functions, Zapiski Nauchnykh Seminarov POMI, 2021, vol. 506, pp. 258-278. https://www.mathnet.ru/eng/znsl7154
  15. 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
  16. 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 (in Russian). https://doi.org/10.35634/vm220205
  17. Matyoqubov M.M., Yakhshimuratov A.B. Integration of higher Korteweg-de Vries equation with a self-consistent source in class of periodic functions, Ufa Mathematical Journal, 2013, vol. 5, issue 1, pp. 102-111. https://doi.org/10.13108/2013-5-1-102
  18. Li L., Xie Y., Zhu Sh. New exact solutions for a generalized KdV equation, Nonlinear Dynamics, 2018, vol. 92, issue 2, pp. 215-219. https://doi.org/10.1007/s11071-018-4050-3
  19. Yakhshimuratov A.B. Integration of a higher-order nonlinear Schrödinger system with a self-consistent source in the class of periodic functions, Theoretical and Mathematical Physics, vol. 202, no. 2, pp. 137-149. https://doi.org/10.1134/S0040577920020014
  20. Muminov U.B., Khasanov A.B. Integration of a defocusing nonlinear Schrödinger equation with additional terms, Theoretical and Mathematical Physics, 2022, vol. 211, no. 1, pp. 514-531. https://doi.org/10.1134/S0040577922040067
  21. Babajanov B.A., Babadjanova A.K., Azamatov A.Sh. Integration of the differential-difference sine-Gordon equation with a self-consistent source, Theoretical and Mathematical Physics, 2022, vol. 210, no. 3, pp. 327-336. https://doi.org/10.1134/S0040577922030035
  22. Babajanov B.A., Khasanov A.B. Periodic Toda chain with an integral source, Theoretical and Mathematical Physics, 2015, vol. 184, no. 2, pp. 1114-1128. https://doi.org/10.1007/s11232-015-0321-z
  23. Karunakar P., Chakraverty S. Effect of Coriolis constant on geophysical Korteweg-de Vries equation, Journal of Ocean Engineering and Science, 2019, vol. 4, issue 2, pp. 113-121. https://doi.org/10.1016/j.joes.2019.02.002
  24. Rizvi S.T.R., Seadawy A.R., Ashraf F., Younis M., Iqbal H., Baleanu D. Lump and interaction solutions of a geophysical Korteweg-de Vries equation, Results in Physics, 2020, vol. 19, 103661. https://doi.org/10.1016/j.rinp.2020.103661
  25. Hasanov A.B., Hoitmetov U.A. On integration of the loaded Korteweg-de Vries equation in the class of rapidly decreasing functions, Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 2021, vol. 47, no. 2, pp. 250-261. https://doi.org/10.30546/2409-4994.47.2.250
  26. Hoitmetov U.A. Integrating the loaded KdV equation with a self-consistent source of integral type in the class of rapidly decreasing complex-valued functions, Matematicheskie Trudy, 2021, vol. 24, no. 2, pp. 181-198 (in Russian). https://doi.org/10.33048/mattrudy.2021.24.211
  27. Nakhushev A.M. Uravneniya matematicheskoi biologii (Equations of mathematical biology), Moscow: Vysshaya Shkola, 1995.
  28. Kozhanov A.I. Nonlinear loaded equations and inverse problems, Computational Mathematics and Mathematical Physics, 2004, vol. 44, no. 4, pp. 657-678. https://zbmath.org/?q=an:1114.35148
  29. Sagdullayeva M.M. Nonlocal problem with the integral condition for a loaded heate equation, Vestnik KRAUNC. Fiziko-matematičeskie Nauki, 2021, vol. 34, no. 1, pp. 47-56 (in Russian). https://doi.org/10.26117/2079-6641-2021-34-1-47-56
  30. Abdullayev V.M. Numerical solution of a boundary value problem for a loaded parabolic equation with nonlocal boundary conditions, Vestnik KRAUNC. Fiziko-matematičeskie Nauki, 2020, vol. 32, no. 3, pp. 15-28 (in Russian). https://doi.org/10.26117/2079-6641-2020-32-3-15-28
  31. Sabitov K.B. Initial-boundary problem for parabolic-hyperbolic equation with loaded summands, Russian Mathematics, 2015, vol. 59, issue 6, p. 23-33. https://doi.org/10.3103/S1066369X15060055
  32. Lugovtsov A.A. Propagation of nonlinear waves in an inhomogeneous gas-liquid medium. Derivation of wave equations in the Korteweg-de Vries approximation, Journal of Applied Mechanics and Technical Physics, 2009, vol. 50, issue 2, pp. 327-335. https://doi.org/10.1007/s10808-009-0044-8
  33. Lugovtsov A.A. Propagation of nonlinear waves in a gas-liquid medium. Exact and approximate analytical solutions of wave equations, Journal of Applied Mechanics and Technical Physics, 2010, vol. 51, issue 1, pp. 44-50. https://doi.org/10.1007/s10808-010-0007-0
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