Geodesic
Integration of the general loaded Korteweg--de Vries equation with an integral type source in the class of rapidly decreasing complex-valued functions
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2021), pp. 52-66.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, the evolution of the scattering data of the Sturm-Liouville operator is derived by the method of the inverse scattering problem, the potential of which is a solution to the general loaded Korteweg-de Vries equation in the class of rapidly decreasing complex-valued functions.
Keywords: Korteweg-de Vries equation, Sturm-Liouville operator, inverse scattering theory, Gelfand-Levitan-Marchenko integral equation.
Mots-clés : Jost solution 
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A. B. Khasanov; U. A. Hoitmetov. Integration of the general loaded Korteweg--de Vries equation with an integral type source in the class of rapidly decreasing complex-valued functions. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2021), pp. 52-66. http://geodesic.mathdoc.fr/item/IVM_2021_7_a5/

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

[2] Faddeev L. D., “Svoistva S-matritsy odnomernogo uravneniya Shredingera”, Tr. in-ta im. V.A. Steklova, 73, 1964, 314–336 | Zbl

[3] Marchenko V. A., Operatory Shturma–Liuvillya i ikh prilozheniya, Nauk. dumka, Kiev, 1977 | MR

[4] Levitan B. M., Obratnye zadachi Shturma–Liuvillya, Nauka, M., 1984 | MR

[5] Lax P. D., “Integrals of Nonlinear Equations of Evolution and Solitary Waves”, Comm. Pure and Appl. Math., 21:5 (1968), 467–490 | DOI | MR | Zbl

[6] Mel'nikov V. K., “A direct method for deriving a multisoliton solution for the problem of interaction of waves on the $x, y$ plane”, Comm. Math. Phys., 112:4 (1987), 639–652 | DOI | MR

[7] Mel'nikov V. K., “Integration method of the Korteweg-de Vries equation with a self-consistent source”, Phys. Lett. A, 133:9 (1988), 493–496 | DOI | MR

[8] Mel'nikov V. K., “Integration of the Korteweg-de Vries equation with a source”, Inverse Probl., 6:2 (1990), 233–246 | DOI | MR

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

[10] Claude C., Leon J., Latifi A., “Nonlinear resonant scattering and plasma instability: an integrable model”, J. Math. Phys., 32 (1991), 3321–3330 | DOI | MR | Zbl

[11] Shchesnovich V. S., Doktorov E. V., “Modified Manakov system with self-consistent source”, Phys. Lett. A, 213:1–2 (1996), 23–31 | DOI | MR | Zbl

[12] Kneser A., “Belastete Integralgleichungen”, Rendiconti del Circolo Matematico di Palermo, 37 (1914), 169–197 | DOI | Zbl

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

[14] Nakhushev A. M., “Nagruzhennye uravneniya i ikh prilozheniya”, Differents. uravneniya, 19:1 (1983), 86–94 | MR | Zbl

[15] Nakhushev A. M., Uravneniya matematicheskoi biologii, Vyssh. shk., M., 1995

[16] Kozhanov A. I., “Nelineinye nagruzhennye uravneniya i obratnye zadachi”, Zh. vychisl. matem. i matem. fiz., 44:4 (2004), 694–716 | MR | Zbl

[17] Khasanov A. B., Khoitmetov U. A., “Ob integrirovanii uravneniya Kortevega–de Friza v klasse bystroubyvayuschikh kompleksnoznachnykh funktsii”, Izv. vuzov. Matem., 2018, no. 3, 79–90 | Zbl

[18] Zamonov M. Z., Khasanov A. B., Khoitmetov U. A., “Integrirovanie uravneniya KdF s samosoglasovannym istochnikom integralnogo tipa v klasse bystroubyvayuschikh kompleksnoznachnykh funktsii”, Izv. AN RTadzh. Otd. Fiz-matem., khim. i geolog. nauk., 2007, no. 4(129), 7–21

[19] Khoitmetov U. A., “Integrirovanie obschego uravneniya KdF s samosoglasovannym istochnikom integralnogo tipa”, Dokl. AN RTadzh., 50:4 (2007), 307–311 | MR

[20] Khasanov A. B., Matyakubov M. M., “Integrirovanie nelineinogo uravneniya Kortevega–de Friza s dopolnitelnym chlenom”, Teor. i matem. fiz., 203:2 (2020), 192–204 | MR | Zbl

[21] Yakhshimuratov A. B., Matekubov M. M., “Integrirovanie nagruzhennogo uravneniya Kortevega–de Friza v klasse periodicheskikh funktsii”, Izv. vuzov. Matem., 2016, no. 2, 87–92 | MR | Zbl

[22] Blaschak V. A., “Analog obratnoi zadachi teorii rasseyaniya dlya nesamosopryazhennogo operatora. I”, Differents. uravneniya, 4:8 (1968), 1519–1533

[23] Blaschak V. A., “Analog obratnoi zadachi teorii rasseyaniya dlya nesamosopryazhennogo operatora. II”, Differents. uravneniya, 4:10 (1968), 1915–1924