Geodesic
Soliton solutions of the negative-order nonlinear Schrödinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 2, pp. 263-273 Cet article a éte moissonné depuis la source Math-Net.Ru

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We discuss the integration of the Cauchy problem for the negative-order nonlinear Schrödinger equation in the class of rapidly decreasing functions via the inverse scattering problem method. In particular, we obtain the time dependence of scattering data of the Zakharov–Shabat system with the potential that is a solution of the considered problem. We give an explicit representation of the one-soliton solution of the negative-order nonlinear Schrödinger equation based on the obtained results.
Keywords: negative-order nonlinear Schrödinger equation, recursion operator, inverse scattering transform, scattering data, Zakharov–Shabat system
Mots-clés : soliton. 
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G. U. Urazboev; I. I. Baltaeva; A. K. Babadjanova. Soliton solutions of the negative-order nonlinear Schrödinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 2, pp. 263-273. http://geodesic.mathdoc.fr/item/TMF_2024_219_2_a4/

[1] V. E. Zakharov, A. B. Shabat, “Tochnaya teoriya dvumernoi samofokusirovki i odnomernoi avtomodulyatsii voln v nelineinykh sredakh”, ZhETF, 61:1 (1971), 118–134 | MR

[2] A. Hasegawa, M. Matsumoto, Optical Solitons in Fibers, Springer Series in Photonics, 7, Springer, Berlin, 2002 | DOI

[3] A. Hasegawa, F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers. I. Anomalous dispersion”, Appl. Phys. Lett., 23:3 (1973), 142–144 | DOI

[4] A. Hasegawa, F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers. II. Normal dispersion”, Appl. Phys. Lett., 23:4 (1973), 171–172 | DOI

[5] M. Wadati, “The modified Korteweg–de Vries equation”, J. Phys. Soc. Japan, 34:5 (1973), 1289–1296 | DOI | MR

[6] P. J. Olver, “Evolution equations possessing infinitely many symmetries”, J. Math. Phys., 18:6 (1977), 1212–1215 | DOI | MR

[7] J. M. Verosky, “Negative powers of Olver recursion operators”, J. Math. Phys., 32:7 (1991), 1733–1736 | DOI | MR

[8] Z. J. Qiao, E. G. Fan, “Negative-order Korteweg–de Vries equations”, Phys. Rev. E, 86:1 (2012), 016601, 20 pp. | DOI

[9] J. B. Chen, “Quasi-periodic solutions to a negative-order integrable system of 2-component KdV equation”, Internat. J. Geom. Methods Modern Phys., 15:3 (2018), 1850040, 34 pp. | DOI | MR

[10] Z. J. Qiao, J. B. Li, “Negative-order KdV equation with both solitons and kink wave solutions”, Europhys. Lett., 94:5 (2011), 50003, 5 pp. | DOI

[11] S.-L. Zhao, Y.-Y. Sun, “A discrete negative order potential Korteweg–de Vries equation”, Z. Naturforsch. A, 71:12 (2016), 1151–1158 | DOI

[12] M. Rodriguez, J. Li, Z. Qiao, “Negative order KdV equation with no solitary traveling waves”, Mathematics, 10:1 (2022), 48, 20 pp. | DOI

[13] A.-M. Wazwaz, “Negative-order KdV equations in $(3+1)$ dimensions by using the KdV recursion operator”, Waves Random Complex Media, 27:4 (2017), 768–778 | DOI | MR

[14] G. U. Urazboev, I. I. Baltaeva, O. B. Ismoilov, “Integrirovanie uravneniya Kortevega–de Friza otritsatelnogo poryadka metodom obratnoi zadachi rasseyaniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 33:3 (2023), 523–533 | MR

[15] 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 | DOI | MR

[16] G. U. Urazboev, M. M. Khasanov, I. I. Baltaeva, “Integrirovanie uravneniya Kortevega–de Friza otritsatelnogo poryadka s istochnikom spetsialnogo vida”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 44 (2023), 31–43 | DOI | MR

[17] M. M. Khasanov, I. D. Rakhimov, “Integrirovanie uravneniya KdF otritsatelnogo poryadka so svobodnym chlenom v klasse periodicheskikh funktsii”, Chebyshevskii sb., 24:2 (2023), 266–275 | DOI

[18] J. Wang, L. Tian, Y. Zhang, “Breather solutions of a negative order modified Korteweg–de Vries equation and its nonlinear stability”, Phys. Lett. A, 383:15 (2019), 1689–1697 | DOI | MR

[19] A.-M. Wazwaz, “A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions”, Waves Random Complex Media, 28:3 (2018), 533–543 | DOI | MR

[20] G. U. Urazboev, A. B. Yakhshimuratov, M. M. Khasanov, “Integrirovanie modifitsirovannogo uravneniya Kortevega–de Friza otritsatelnogo poryadka v klasse periodicheskikh funktsii”, TMF, 217:2 (2023), 317–328 | DOI | DOI | MR

[21] M. I. Ismailov, C. Sabaz, Inverse scattering method for nonlinear Klein–Gordon equation coupled with a scalar field, arXiv: 2212.02092

[22] J. Ji, S.-L. Zhao, D.-J. Zang, “Soliton solutions for a negative order AKNS equation”, Commun. Theor. Phys., 50:5 (2008), 1033–1035 | DOI | MR

[23] L. M. Ling, B.-F. Feng, Z. N. Zhu, “General soliton solutions to a coupled Fokas–Lenells equation”, Nonlinear Anal. Real World Appl., 40 (2018), 185–214 | DOI | MR

[24] V. S. Gerdjikov, R. I. Ivanov, “Multicomponent Fokas–Lenells equations on Hermitian symmetric spaces”, Nonlinearity, 34:2 (2021) | MR

[25] M. J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Cambridge Texts in Applied Mathematics, 47, Cambridge Univ. Press, New York, 2011 | DOI | MR