Abstract:
It is shown that “steplike” solutions of the Korteweg–de Vries equation with a self-consistent source can be found by the inverse scattering method for the Sturm–Liouville operator on the entire real line.
Citation:
G. U. Urazboev, A. B. Khasanov, “Integrating the Korteweg–de Vries Equation with a Self-Consistent Source and “Steplike” Initial Data”, TMF, 129:1 (2001), 38–54; Theoret. and Math. Phys., 129:1 (2001), 1341–1356
\Bibitem{UraKha01}
\by G.~U.~Urazboev, A.~B.~Khasanov
\paper Integrating the Korteweg--de Vries Equation with a Self-Consistent Source and ``Steplike'' Initial Data
\jour TMF
\yr 2001
\vol 129
\issue 1
\pages 38--54
\mathnet{http://mi.mathnet.ru/tmf518}
\crossref{https://doi.org/10.4213/tmf518}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1904747}
\zmath{https://zbmath.org/?q=an:1044.35077}
\elib{https://elibrary.ru/item.asp?id=5033697}
\transl
\jour Theoret. and Math. Phys.
\yr 2001
\vol 129
\issue 1
\pages 1341--1356
\crossref{https://doi.org/10.1023/A:1012463310382}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000172400800005}
Linking options:
https://www.mathnet.ru/eng/tmf518
https://doi.org/10.4213/tmf518
https://www.mathnet.ru/eng/tmf/v129/i1/p38
This publication is cited in the following 32 articles:
Gayrat U. Urazboev, Iroda I. Baltaeva, Shoira E. Atanazarova, “Soliton solutions of the negative order modified Korteweg – de Vries equation”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 47 (2024), 63–77
Bazar Babajanov, Michal Fečkan, Aygul Babadjanova, “On the Differential-Difference Sine-Gordon Equation with an Integral Type Source”, Mathematica Slovaca, 73:6 (2023), 1499
Yuqin Yao, Hui Zhou, Fei Li, “The integrability, equivalence and solutions of two kinds of integrable deformed fourth-order matrix NLS equations”, Nonlinear Dyn, 111:9 (2023), 8673
F. Li, Yuqin Yao, “Multisoliton and rational solutions for the extended fifth-order KdV equation in fluids with self-consistent sources”, Theoret. and Math. Phys., 210:2 (2022), 184–197
B. A. Babajanov, A. K. Babadjanova, A. Sh. Azamatov, “Integration of the differential–difference sine-Gordon equation with a self-consistent source”, Theoret. and Math. Phys., 210:3 (2022), 327–336
G. U. Urazboev, M. M. Khasanov, “Integrirovanie uravneniya Kortevega-de Friza otritsatelnogo poryadka s samosoglasovannym istochnikom v klasse periodicheskikh funktsii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 32:2 (2022), 228–239
A. A. Reyimberganov, I. D. Rakhimov, “The soliton solutions for the nonlinear Schrödinger equation with self-consistent source”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 36 (2021), 84–94
Hasanov A.B. Hoitmetov U.A., “On Integration of the Loaded Korteweg-de Vries Equation in the Class of Rapidly Decreasing Functions”, Proc. Inst. Math. Mech., 47:2 (2021), 250–261
K. A. Mamedov, “Integration of mKdV equation with a self-consistent source in the class of finite density functions in the case of moving eigenvalues”, Russian Math. (Iz. VUZ), 64:10 (2020), 66–78
A. B. Yakhshimuratov, M. M. Matyokubov, “Integration of loaded Korteweg–de Vries equation in a class of periodic functions”, Russian Math. (Iz. VUZ), 60:2 (2016), 72–76
M. M. Matyoqubov, A. B. Yakhshimuratov, “Integration of higher Korteweg-de Vries equation with a self-consistent source in class of periodic functions”, Ufa Math. J., 5:1 (2013), 102–111
Urazboev G., “Integrating the Toda Lattice with Self-Consistent Source via Inverse Scattering Method”, Math. Phys. Anal. Geom., 15:4 (2012), 401–412
A. B. Yakhshimuratov, “Integrirovanie uravneniya Kortevega-de Friza so spetsialnym svobodnym chlenom v klasse periodicheskikh funktsii”, Ufimsk. matem. zhurn., 3:4 (2011), 144–150
A. B. Khasanov, A. B. Yakhshimuratov, “The Korteweg–de Vries equation with a self-consistent source in the class of periodic functions”, Theoret. and Math. Phys., 164:2 (2010), 1008–1015
Sun, YP, “On the non-isospectral Kadomtsev-Petviashvili equation with self-consistent sources”, Nonlinear Analysis-Theory Methods & Applications, 70:4 (2009), 1610
G. U. Urazboev, “Toda lattice with a special self-consistent source”, Theoret. and Math. Phys., 154:2 (2008), 260–269
Wang, HY, “On the Pfaffianized-KP equation with self-consistent sources”, Journal of Mathematical Analysis and Applications, 338:1 (2008), 82
“On the two-dimensional Leznov lattice equation with self-consistent sources”, Journal of Physics A-Mathematical and Theoretical, 40:42 (2007), 12691
Wang, HY, “Pfaffian solution of a semi-discrete BKP-type equation and its source generation version”, Journal of Physics A-Mathematical and Theoretical, 40:44 (2007), 13385
Wang, HY, “Integrability of the semi-discrete Toda equation with self-consistent sources”, Journal of Mathematical Analysis and Applications, 330:2 (2007), 1128