Geodesic
Integration of negative-order modified Korteweg–de Vries equation in a class of periodic functions
Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 2, pp. 317-328 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the negative-order modified Korteweg–de Vries equation and show that it can be integrated by the inverse spectral transform method. We determine the evolution of the spectral data for the Dirac operator with periodic potential associated with a solution of the negative-order modified Korteweg–de Vries equation. The obtained results allow applying the inverse spectral transform method for solving the negative-order modified Korteweg–de Vries equation in the class of periodic functions. Important corollaries are obtained concerning the analyticity and the period of a solution in spatial variable. We show that a function constructed using the Dubrovin–Trubowitz system and the first trace formula satisfies the negative-order modified Korteweg–de Vries equation. We prove the solvability of the Cauchy problem for the infinite Dubrovin–Trubowitz system of differential equations in the class of three-times continuously differentiable periodic functions.
Keywords: negative-order modified Korteweg–de Vries equation, Dirac operator, inverse spectral problem, Dubrovin–Trubowitz system of equations
Mots-clés : trace formula. 
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G. U. Urazboev; A. B. Yakhshimuratov; M. M. Khasanov. Integration of negative-order modified Korteweg–de Vries equation in a class of periodic functions. Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 2, pp. 317-328. http://geodesic.mathdoc.fr/item/TMF_2023_217_2_a4/

[1] M. Wadati, “The exact solution of the modifed Korteweg–de Vries equation”, J. Phys. Soc. Japan, 32:6 (1972), 1681–1681 | DOI

[2] A. R. Its, Tochnoe integrirovanie v rimanovykh $\theta$-funktsiyakh nelineinogo uravneniya Shredingera i modifitsirovannogo uravneniya Kortevega–de Friza, Dis. ... kand. fiz.-matem. nauk, Leningr. gos. un-t im. A. A. Zhdanova, L., 1977

[3] A. O. Smirnov, “Ellipticheskie resheniya nelineinogo uravneniya Shredingera i modifitsirovannogo uravneniya Kortevega–de Friza”, Matem. sb., 185:8 (1994), 103–114 | DOI | MR | Zbl

[4] A. B. Yakhshimuratov, M. M. Khasanov, “Integrirovanie modifitsirovannogo uravneniya Kortevega–de Friza s samosoglasovannym istochnikom v klasse periodicheskikh funktsii”, Differents. uravneniya, 50:4 (2014), 536–543 | DOI | MR

[5] A. B. Khasanov, G. U. Urazboev, “Metod resheniya uravneniya mKdF s samosoglasovannym istochnikom”, Uzb. matem. zhurn., 1 (2003), 69–75

[6] K. A. Mamedov, “Integrirovanie uravneniya mKdF s samosoglasovannym istochnikom v klasse funktsii konechnoi plotnosti, v sluchae dvizhuschikhsya sobstvennykh znachenii”, Izv. vuzov. Matem., 2020, no. 10, 73–85 | DOI | DOI

[7] P. G. Grinevich, I. A. Taimanov, “Spectral conservation laws for periodic nonlinear equations of the Melnikov type”, Geometry, Topology, and Mathematical Physics: S. P. Novikov's Seminar: 2006–2007, American Mathematical Society Translations. Ser. 2, 224, eds. V. M. Buchstaber, I. M. Krichever, AMS, Providence, RI, 2008, 125–138 | MR

[8] A. B. Khasanov, A. B. Yakhshimuratov, “Ob uravnenii Kortevega–de Friza s samosoglasovannym istochnikom v klasse periodicheskikh funktsii”, TMF, 164:2 (2010), 214–221 | DOI | DOI

[9] A. B. Yakhshimuratov, “Integrirovanie nelineinoi sistemy Shredingera vysshego poryadka s samosoglasovannym istochnikom v klasse periodicheskikh funktsii”, TMF, 202:2 (2020), 157–169 | DOI | DOI | MR

[10] A. Yakhshimuratov, “The nonlinear Schrödinger equation with a self-consistent source in the class of periodic functions”, Math. Phys. Anal. Geom., 14:2 (2011), 153–169 | DOI | MR

[11] I. I. Baltaeva, I. D. Rakhimov, M. M. Khasanov, “Exact traveling wave solutions of the loaded modified Korteweg–de Vries equation”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 41 (2022), 85–95 | DOI | MR

[12] 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

[13] 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

[14] J. F. Gomes, G. Starvaggi França, G. R. de Melo, A. H. Zimerman, “Negative even grade mKdV hierarchy and its soliton solutions”, J. Phys. A: Math. Theor., 42:44 (2009), 445204, 11 pp. | DOI | MR

[15] A. Kundu, R. Sahadevan, L. Nalinidevi, “Nonholonomic deformation of KdV and mKdV equations and their symmetries, hierarchies and integrability”, J. Phys. A: Math. Theor., 42:11 (2009), 115213, 13 pp. | DOI | MR

[16] J. F. Gomes, G. R. de Melo, A. H. Zimerman, “A class of mixed integrable models”, J. Phys. A: Math. Theor., 42:27 (2009), 275208, 11 pp. | DOI | MR

[17] Z. Qiao, W. Strampp, “Negative order MKdV hierarchy and a new integrable Neumann-like system”, Phys. A, 313:3–4 (2002), 365–380 | DOI | MR

[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] B. M. Levitan, I. S. Sargsyan, Operatory Shturma–Liuvillya i Diraka, Nauka, M., 1988 | DOI | MR | MR | Zbl

[20] T. V. Misyura, “Kharakteristika spektrov periodicheskoi i antiperiodicheskoi kraevykh zadach, porozhdaemykh operatsiei Diraka I”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, Vyp. 30. Respublikanskii mezhvedomstvennyi nauchnyi sbornik, ed. V. A. Marchenko, Izd-vo Khark. un-ta im. A. M. Gorkogo, Kharkov, 1978, 90–101 | MR

[21] A. B. Khasanov, A. M. Ibragimov, “Ob obratnoi zadache dlya operatora Diraka s periodicheskim potentsialom”, Uzb. matem. zhurn., 3–4 (2001), 48–55

[22] A. B. Khasanov, A. B. Yakhshimuratov, “Analog obratnoi teoremy G. Borga dlya operatora Diraka”, Uzb. matem. zhurn., 3 (2000), 40–46

[23] P. B. Dzhakov, B. S. Mityagin, “Zony neustoichivosti odnomernykh periodicheskikh operatorov Shredingera i Diraka”, UMN, 61:4(370) (2006), 77–182 | DOI | DOI | MR | Zbl

[24] S. Currie, T. T. Roth, B. A. Watson, “Borg's periodicity theorems for first-order self-adjoint systems with complex potentials”, Proc. Edinb. Math. Soc., 60:3 (2017), 615–633 | DOI | MR

[25] A. B. Khasanov, Kh. N. Normurodov, U. O. Khudaerov, “Integrirovanie modifitsirovannogo uravneniya Kortevega–de Friza–sinus-Gordona v klasse periodicheskikh beskonechnozonnykh funktsii”, TMF, 214:2 (2023), 198–210 | DOI | DOI | MR