DOI: 10.12732/ijam.v37i1.4
INTEGRATION OF THE LOADED
NEGATIVE ORDER KORTEWEG-DE
VRIES EQUATION IN THE CLASS
OF PERIODIC FUNCTIONS
Gayrat Urazboev 1, Muzaffar
Khasanov 2,§, Otabek Ganjaev 3
1,2,3 Urgench State University
Urgench - 220100, UZBEKISTAN
Abstract. In this paper, we consider the loaded Korteweg-de Vries equation of negative order in the class of periodic functions corresponding to the eigenvalues of the corresponding spectral problem. It is shown that the considered equation 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 considered equation is determined.
The obtained results make it possible to apply the inverse problem method for solving the loaded Korteweg-de Vries equation of negative order in the class of periodic functions corresponding to the eigenvalues of the corresponding spectral problem.
How
to cite this paper?
DOI: 10.12732/ijam.v37i1.4
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2024
Volume: 37
Issue: 1
References
[1] C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Method for solving the Korteweg-de Vries equation, Physical Review Letters, 19 No 19 (1967), 1095-1097;
doi: 10.1103/PhysRevLett.19.1095.
[2] S.P. Novikov, The periodic problem for the Korteweg-de Vries equation, Functional Analysis and Its Applications, 8 (1974), 236-246; doi: 10.1007/BF01075697.
[3] B.A. Dubrovin, S.P. Novikov, Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation, Journal of Experimental and Theoretical Physics, 67, No 6 (1974), 2131-2143.
[4] V.A. Marchenko, The periodic Korteweg-de Vries problem, Matematicheskii Sbornik. Novaya Seriya, 95, No 137 (1974), 331-356.
[5] B.A. Dubrovin, Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials, Functional Analysis and Its Applications, 9 No 3 (1975), 215-223;
doi: 10.1007/BF01075598.
[6] A.R. Its, V.B. Matveev, Schrodinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation, Theoretical and Mathematical Physics, 23, No 1 (1975), 343-355; doi: 10.1007/BF01038218.
[7] P.D. Lax, Periodic solutions of the KdV equations, In: Nonlinear Wave Motion, Providence, AMS, 85-96 (1974).
[8] P.D. Lax, Periodic solutions of the KdV equation, Communications on Pure and Applied Mathematics, 28, No 1, (1975), 141-188; doi: 10.1002/cpa.3160280105.
[9] V.K. Melnikov, Method for integrating the Korteweg-de Vries equation with a self-consistent source, Joint Institute for Nuclear Research (1988).
[10] A.B. Khasanov, A.B. Yakhshimuratov, The Korteweg-de Vries equation with a self-consistent source in the class of periodic functions, Theoretical and Mathematical Physics, 164, No 2 (2010), 1008-1015; doi: 10.1007/s11232-010-0081-8.
[11] A.B. Yakhshimuratov, M.M. Matyokubov, Integration of loaded Korteweg-de Vries equation in a class of periodic functions, Russian Math., 60, No 2 (2016), 72-76;
doi: 10.3103/S1066369X16020110.
[12] A.B. Hasanov, M.M. Hasanov,
Integration of the nonlinear Shrodinger equation with an additional term in the
class of periodic functions, Theoretical and Mathematical Physics, 199,
No 1, (2019), 525-532; doi: 10.1134/S0040577919040044.
[13] M.M. Khasanov, Integration of the loaded modified Korteweg-de Vries equation in the class of periodic functions, Uzbek. Mathematical Journal, 4 (2016), 139.
[14] G.U. Urazboev, I.I. Baltaeva, I.D. Rakhimov, A generalized (G_/G)-expansion method for the loaded Korteweg-de Vries equation, Sibirskii Zhurnal Industrial’noi Matematiki, 24 (2021), 139; doi: 10.33048/SIBJIM.2021.24.410.
[15] I.I. Baltaeva, I.D. Rakhimov, M.M. Khasanov, Exact traveling wave solutions of the loaded modified Korteweg-de Vries equation, The Bulletin of Irkutsk State University. Series Mathematics, 41, (2022), 85-95; doi: 10.26516/1997-7670.2022.41.85.
[16] G.U. Urazboev, M.M. Khasanov, I.D. Rakhimov, Generalized (G’/G)-expansion method and its applications to the loaded Burgers equation, Azerbaijan Journal of Mathematics. Series Mathematics, 13 (2023), 248-257; doi: 10.59849/2218-6816.2023.2.248.
[17] J.M. Verosky, Negative powers of Olver recursion operators, Journal of Mathematical Physics, 32, No 7 (1991), 1733-1736; doi: 10.1063/1.529234.
[18] S. Lou, Symmetries of the KdV equation and four hierarchies of the integrodifferential
KdV equations, Journal of Mathematical Physics, 35, No 5 (1994), 2390-2396;
doi: 10.1063/1.530509.
[19] A. Degasperis, M. Procesi, Asymptotic integrability. Symmetry and Perturbation
Theory, World Scientific, Singapore (1999).
[20] G. Zhang, Z. Qiao, Cuspons and smooth solitons of the Degasperis-Procesi equation under inhomogeneous boundary condition, Mathematical Physics, Analysis and Geometry, 10, No 3 (2007), 205-225; doi: 10.1007/s11040-007-9027-2.
[21] Z. Qiao, E. Fan, Negative-order Korteweg–de Vries equations, Physical Review E, 86, No 1 (2012), 016601; doi: 10.1103/PhysRevE.86.016601.
[22] J. Chen, Quasi-periodic solutions to a negative-order integrable system of 2-component KdV equation, International Journal of Geometric Methods in Modern Physics, 15, No 3 (2018), 1850040; doi: 10.1142/S0219887818500408.
[23] Z. Qiao, J. Li, Negative-order KdV equation with both solitons and kink wave solutions, Europhysics Letters, 94, No 5 (2011), 50003; doi: 10.1209/0295-5075/94/50003.
[24] S. Zhao, Y. Sun, A discrete negative order potential Korteweg-de Vries equation, Zeitschrift fur Naturforschung A, 71, No 12 (2016), 1151-1158; doi: 10.1515/zna-2016-0324.
[25] M. Rodriguez, J. Li, Qiao Z, Negative order KdV equation with no solitary traveling waves, Mathematics, 10, No 1 (2022), 48; doi: 10.3390/math10010048.
[26] A.M. Wazwaz, Negative-order KdV equations in (3+1) dimensions by using the KdV recursion operator, Waves in Random and Complex Media, 27, No 4 (2017), 768-778;
doi: 10.1080/17455030.2017.1317115.
[27] M. Kuznetsova, Necessary and sufficient conditions for the spectra of the Sturm–Liouville operators with frozen argument, Applied Mathematics Letters, 131 (2022), 108035;
doi: 10.1016/j.aml.2022.108035.
[28] E.C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, 2, Moscow (1961).
[29] W. Magnus, S. Winkler, Hill’s Equation, Interscience Publishers, New York (1966).
[30] I.V. Stankevich, A certain inverse spectral analysis problem for Hill’s equation, Doklady Akademii Nauk SSSR, 192, No 1 (1970), 34-37.
[31] V.A. Marchenko, I.V. Ostrovskii, A characterization of the spectrum of Hill’s operator, Matematicheskii Sbornik. Novaya Seriya, 97, No 139 (1975), 540-606;
doi: 10.1070/SM1975v026n04ABEH002493.
[32] E. Trubowitz, The inverse problem for periodic potentials, Communications on Pure and Applied Mathematics, 30, No 3 (1977), 321-337; doi: 10.1002/cpa.3160300305.
[33] B.M. Levitan, I.S. Sargsyan, The Sturm-Liouville and Dirac Operators, Nauka, Moscow (1988).
[34] G.U. Urazboev, M.M. Hasanov, 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, 32, No 2 (2022), 228-239;
doi: 10.35634/vm220205.
[35] M.M. Khasanov, I.D. Rakhimov, Integration of the KdV equation of negative order with a free term in the class of periodic functions, Chebyshevskii Sb., 24, No 2 (2023), 266–275; https://doi.org/10.22405/2226-8383-2023-24-2-266-275.
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