Geodesic
Scattering theory for the loaded negative order Korteweg--de Vries equation
Čebyševskij sbornik, Tome 25 (2024) no. 2, pp. 169-180.

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In this paper, we consider the loaded negative order Korteweg–de Vries equation. The evolution of the spectral data of the Sturm–Liouville operator with a potential associated with the solution of the loaded negative order Korteweg–de Vries equation is determined. The obtained results make it possible to apply the inverse problem method to solve the loaded negative order Korteweg–de Vries equation in the class of rapidly decreasing functions. An example of the given problem is given with graphs of the solution.
Keywords: Sturm–Liouville operator, loaded equation, loaded negative order Korteweg–de Vries equation, soliton solution, inverse scattering problems. 
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G. U. Urazboev; I. I. Baltaeva; O. B. Ismoilov. Scattering theory for the loaded negative order Korteweg--de Vries equation. Čebyševskij sbornik, Tome 25 (2024) no. 2, pp. 169-180. http://geodesic.mathdoc.fr/item/CHEB_2024_25_2_a9/

[1] Gardner C. S., Greene J. M., Kruskal M. D., Miura R. M., “Method for Solving Korteweg-Devries Equation”, Phys. Rev. Lett., 19 (1967), 1095–1097 | DOI | MR

[2] Lax P. D., “Integrals of Nonlinear Equations of Evolution and Solitary Waves”, Communications on Pure and Applied Mathematics, 21 (1968), 467–490 | DOI | MR | Zbl

[3] Leon J., Latifi A., “Solution of an initial-boundary value problem for coupled nonlinear waves”, J. Phys. A, 23 (1990), 1385–1403 | DOI | MR | Zbl

[4] Mel'nikov V. K., “Exact solutions of the Korteweg-de Vries equation with a self-consistent source”, Phys. Lett., 128 (1988), 488–492 | DOI | MR

[5] Mel'nikov V. K., “Integration of the Korteweg-de Vries equation with a source”, Inverse Problem, 6 (1990), 233–246 | DOI | MR | Zbl

[6] Khasanov A. B., Khasanov T. G., “Integration of a Nonlinear Korteweg-de Vries Equation with a Loaded Term and a Source”, J. Appl. Ind. Math., 16 (2022), 227–239 | DOI | MR | Zbl

[7] Khasanov A. B., Urazboev G.U., “Integration of the sine-Gordon equation with a self-consistent source of the integral type in the case of multiple eigenvalues”, Russ Math., 53 (2009), 45–55 | DOI | MR | Zbl

[8] 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”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 32 (2022), 228–239 | DOI | MR | Zbl

[9] Verosky J. M., “Negative powers of Olver recursion operators”, J. Math. Phys., 32 (1991), 1733–1736 | DOI | MR | Zbl

[10] Lou S. Y., “Symmetries of the KdV equation and four hierarchies of the integrodifferential KdV equation”, J. Math. Phys., 35 (1994), 2390–2396 | DOI | MR | Zbl

[11] Qiao Z., Li J., “Negative-order KdV equation with both solitons and kink wave solutions”, Europhys. Lett., 94 (2011), 50003 | DOI | MR

[12] Zhijun Q., Engui F., “Negative-order Korteweg-de Vries equations”, Phys. Rev. E, 86 (2012), 016601 | DOI

[13] Rodriguez M., Li J., Qiao Z., “Negative Order KdV Equation with No Solitary Traveling Waves”, Mathematics, 10 (2022), 48–58 | DOI

[14] Urazboev G. U., Baltaeva I. I., Ismoilov O.B., “Integration of the negative order Korteweg-de Vries equation by the inverse scattering method”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 33 (2023), 523–533 | DOI | MR | Zbl

[15] Kneser A., “Belastete Integralgleihungen”, Rendiconti del Circolo Mathematiko di Palermo, 38 (1914), 169–197 | DOI

[16] Lichtenstein L., Vorlesungen uber einege Klassen nichtlinear Integral gleichungen und Integral differential gleihungen nebst, Anwendungen, Springer, Berlin, 1931

[17] Nakhushev A. M., “The Darboux problem for a certain degenerate second order loaded integrodifferential equation”, Differential Equations, 12 (1976), 103–108 | MR | Zbl

[18] Kozhanov A. I., “Nonlinear loaded equations and inverse problems”, Zh. Vychisl. Mat. Mat. Fiz., 44 (2004), 1095–1097 | MR

[19] Baltaeva U., Baltaeva I., Agarwal P., “Cauchy problem for a high-order loaded integro-differential equation”, Mathematical Methods in the Applied Sciences, 45 (2022), 8115–8124 | DOI | MR | Zbl

[20] Urazboev G. U., Baltaeva I. I., “Integration of Camassa-Holm equation with a self-consistent source of integral type”, Ufa Mathematical Journal, 14 (2022), 77–86 | DOI | MR

[21] Yakhshimuratov A. B., Matyokubov M. M., “Integration of a Loaded Korteweg-de Vries Equation in a Class of Periodic Functions”, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 69 (2016), 87–92 | MR

[22] Urazboev G. U., Baltaeva I. I., Rakhimov I. D., “The Generalized (G'/G)-Expansion Method for the Loaded Korteweg-de Vries Equation”, Journal of Applied and Industrial Mathematics, 15 (2021), 679–685 | DOI | MR

[23] Feckan M., Urazboev G., Baltaeva I., “Inverse scattering and loaded Modified Korteweg-de Vries equation”, Journal of Siberian Federal University. Mathematics and Physics, 15 (2022), 176–185 | MR | Zbl