Geodesic
Integration of the differential–difference sine-Gordon equation with a self-consistent source
Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 3, pp. 375-386 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We use the inverse scattering theory to integrate the differential–difference sine-Gordon equation with a self-consistent source.
Keywords: differential–difference sine-Gordon equation, self-consistent source, discrete Dirac-type operator, scattering data, inverse scattering method. 
@article{TMF_2022_210_3_a2,
     author = {B. A. Babajanov and A. K. Babadjanova and A. Sh. Azamatov},
     title = {Integration of the~differential{\textendash}difference {sine-Gordon} equation with a~self-consistent source},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {375--386},
     year = {2022},
     volume = {210},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2022_210_3_a2/}
}
TY  - JOUR
AU  - B. A. Babajanov
AU  - A. K. Babadjanova
AU  - A. Sh. Azamatov
TI  - Integration of the differential–difference sine-Gordon equation with a self-consistent source
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2022
SP  - 375
EP  - 386
VL  - 210
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2022_210_3_a2/
LA  - ru
ID  - TMF_2022_210_3_a2
ER  - 
%0 Journal Article
%A B. A. Babajanov
%A A. K. Babadjanova
%A A. Sh. Azamatov
%T Integration of the differential–difference sine-Gordon equation with a self-consistent source
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2022
%P 375-386
%V 210
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2022_210_3_a2/
%G ru
%F TMF_2022_210_3_a2
B. A. Babajanov; A. K. Babadjanova; A. Sh. Azamatov. Integration of the differential–difference sine-Gordon equation with a self-consistent source. Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 3, pp. 375-386. http://geodesic.mathdoc.fr/item/TMF_2022_210_3_a2/

[1] R. Hirota, “Nonlinear partial difference equations III; Discrete sine-Gordon equation”, J. Phys. Soc. Japan, 43:6 (1977), 2079–2086 | DOI | MR | Zbl

[2] S. J. Orfanidis, “Discrete sine-Gordon equations”, Phys. Rev. D, 18:10 (1978), 3822–3827 | DOI | MR

[3] D. Levi, O. Ragnisco, M. Bruschi, “Extension of the Zakharov–Shabat generalized inverse method to solve differential-difference and difference-difference equations”, Nuovo Cimento A, 58:1 (1980), 56–66 | DOI | MR

[4] L. Pilloni, D. Levi, “The inverse scattering transform for solving the discrete sine-Gordon equation”, Phys. Lett. A, 92:1 (1982), 5–8 | DOI | MR

[5] F. Gesztesy, H. Holden, A combined sine-Gordon and modified Korteweg–de Vries hierarchy and its algebro-geometric solutions, arXiv: solv-int/9707010

[6] A. I. Bobenko, “Poverkhnosti postoyannoi srednei krivizny i integriruemye uravneniya”, UMN, 46:4(280) (1991), 3–42 | DOI | MR | Zbl

[7] A. I. Bobenko, “All constant mean curvature tori in $R^3$, $S^3$, $H^3$ in terms of theta-functions”, Math. Ann., 290:2 (1991), 209–245 | DOI | MR

[8] L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York, 1960 | MR

[9] N. M. Ercolani, H. Knörrer, E. Trubowitz, “Hyperelliptic curves that generate constant mean curvature tori in $\mathbb R^3$”, Integrable Systems. The Verdier Memorial Conference Actes du Colloque International de Luminy, Progress in Mathematics, 115, eds. O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, Birkhäuser, Boston, MA, 1993, 81–114 | DOI | MR

[10] D. A. Korotkin, “On some integrable cases in surface theory”, Zap. nauchn. sem. POMI, 234 (1996), 65–124 | DOI | MR | Zbl

[11] M. Melko, I. Sterling, “Application of soliton theory to the construction of pseudospherical surfaces in $\mathbf R^3$”, Ann. Glob. Anal. Geom., 11:1 (1993), 65–107 | DOI | MR

[12] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, 1982 | MR

[13] A. B. Borisov, V. V. Kiseliev, “Topological defects in incommensurate magnetic and crystal structures and quasi-periodic solutions of the elliptic sine-Gordon equation”, Phys. D, 31:1 (1988), 49–64 | DOI | MR

[14] A. B. Borisov, V. V. Kiseliev, “Inverse problem for an elliptic sine-Gordon equation with an asymptotic behaviour of the cnoidal-wave type”, Inverse Problems, 5:6 (1999), 959–982 | DOI | MR

[15] A. C. Ting, H. H. Chen, Y. C. Lee, “Exact solutions of a nonlinear boundary value problem: The vortices of the two-dimensional sinh-Poisson equation”, Phys. D, 26:1–3 (1987), 37–66 | DOI | MR

[16] R. Hirota, “Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett., 27:18 (1971), 1192–1994 | DOI

[17] D.-J. Zhang, D.-Y. Chen, “The $N$-soliton solutions of the sine-Gordon equation with self-consistent sources”, Phys. A, 321:3–4 (2003), 467–481 | DOI | MR

[18] A. B. Khasanov, G. U. Urazboev, “Ob uravnenii sin-Gordon s samosoglasovannym istochnikom”, Matem. tr., 11:1 (2008), 153–166 | DOI | MR

[19] G. U. Urazboev, A. B. Khasanov, “Integrirovanie uravneniya Kortevega–de Friza s samosoglasovannym istochnikom pri nachalnykh dannykh tipa ‘stupenki’ ”, TMF, 129:1 (2001), 38–54 | DOI | DOI | MR | Zbl

[20] I. M. Krichever, “Algebraicheskie krivye i nelineinye raznostnye uravneniya”, UMN, 33:4(202) (1978), 215–216 | DOI | MR | Zbl

[21] 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, Amer. Math. Soc. Transl. Ser. 2, 224, eds. V. M. Buchstaber, I. M. Krichever, AMS, Providence, RI, 2008, 125–138 | MR | Zbl

[22] 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

[23] B. Babajanov, M. Fečkan, G. Urazbaev, “On the periodic Toda lattice with self-consistent source”, Commun. Nonlinear Sci. Numer. Simul., 22:1–3 (2015), 1223–1234 | DOI | MR

[24] B. A. Babazhanov, A. B. Khasanov, “O periodicheskoi tsepochke Tody s integralnym istochnikom”, TMF, 184:2 (2015), 253–268 | DOI | DOI | MR

[25] X. Liu, Y. Zeng, “On the Toda lattice equation with self-consistent sources”, J. Phys. A: Math. Gen., 38:41 (2005), 8951–8965, arXiv: nlin/0510014 | DOI | MR

[26] B. A. Babajanov, M. Fečkan, G. U. Urazbaev, “On the periodic Toda lattice hierarchy with an integral source”, Commun. Nonlinear Sci. Numer. Simul., 52 (2017), 110–123 | DOI | MR

[27] A. B. Yakhshimuratov, B. A. Babajanov, “Integration of equations of Kaup system kind with self-consistent source in class of periodic functions”, Ufimsk. matem. zhurn., 12:1 (2020), 104–114 | DOI | MR

[28] Y. Hanif, U. Saleem, “Exact solutions of semi-discrete sine-Gordon equation”, Eur. Phys. J. Plus, 134:5 (2019), 200, 9 pp. ; V. B. Matveev, M. A. Salle, Darboux Transformation and Solitons, Springer, Berlin, 1991 | DOI | MR

[29] X. Kou, D.-J. Zhang, Y. Shi, S.-L. Zhao, “Generating solutions to discrete sine-Gordon equation from modified Bäcklund transformation”, Commun. Theor. Phys., 55:4 (2011), 545–550 | DOI | MR