Geodesic
Integration of the loaded mKdV-sine-Gordon equation with a source
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 859-879.

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

The work is devoted to finding the solution of the Cauchy problem for the loaded modified Korteweg-de Vries–sine-Gordon (mKdV-sG) equation with a source in the class of rapidly decreasing functions. The problem is solved by the inverse scattering method. Several examples are given illustrating the application of the obtained results.
Keywords: loaded mKdV–sine-Gordon equation, inverse scattering problem, Gelfand-Levitan-Marchenko integral equation, evolution of the scattering data.
Mots-clés : Jost solutions 
@article{SEMR_2023_20_2_a44,
     author = {U. A. Hoitmetov},
     title = {Integration of the loaded {mKdV-sine-Gordon} equation with a source},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {859--879},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a44/}
}
TY  - JOUR
AU  - U. A. Hoitmetov
TI  - Integration of the loaded mKdV-sine-Gordon equation with a source
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2023
SP  - 859
EP  - 879
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a44/
LA  - en
ID  - SEMR_2023_20_2_a44
ER  - 
%0 Journal Article
%A U. A. Hoitmetov
%T Integration of the loaded mKdV-sine-Gordon equation with a source
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2023
%P 859-879
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a44/
%G en
%F SEMR_2023_20_2_a44
U. A. Hoitmetov. Integration of the loaded mKdV-sine-Gordon equation with a source. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 859-879. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a44/

[1] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, “Method for solving the sine-Gordon equation”, Phys. Rev. Lett., 30 (1973), 1262–1264 | DOI | MR

[2] M.J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia, 1981 | DOI | MR | Zbl

[3] T. Aktosun, F. Demontis, C. van der Mee, “Exact solutions to the sine-Gordon equation”, J. Math. Phys., 51:12 (2010), 123521 | DOI | MR | Zbl

[4] L. Alzaleq, V. Manoranjan, “Analytical solutions for the generalized sine-Gordon equation with variable coefficients”, Phys. Scr., 96:5 (2021), 055218 | DOI

[5] D.-Y. Chen, D.-J. Zhang, S.-F. Deng, “The novel multi-soliton solutions of the mKdV-sine Gordon equations”, J. Phys. Soc. Jpn., 71:2 (2002), 658–659 https://api.semanticscholar.org/CorpusID:119778697 | DOI | MR | Zbl

[6] J.L. Cieśliński, T. Nikiciuk, K. Waśkiewicz, “The sine-Gordon equation on time scales”, J. Math. Anal. Appl., 423:2 (2015), 1219–1230 | DOI | MR | Zbl

[7] A.T. Deresse, Y.O. Mussa, A.K. Gizaw, “Analytical solution of two-dimensional sine-Gordon equation”, Adv. Math. Phys., 2021 (2021), 6610021 | DOI | MR | Zbl

[8] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C Morris, Solitons and nonlinear wave equations, Academic Press, London etc., 1982 | DOI | MR | Zbl

[9] L.P. Eisenhart, A treatise on the differential geometry of curves and surfaces, Dover Publ., New York, 1960 https://books.google.co.uz/books?id=FWbCAgAAQBAJ&printsec=frontcover&hl=ru#v=onepage&q&f=false | Zbl

[10] I.S. Frolov, “Inverse scattering problem for a Dirac system on the whole axis”, Sov. Math., Dokl., 13 (1972), 1468–1472 | MR | Zbl

[11] V.A. Gani, A.M. Marjaneh, A. Askari, E. Belendryasova, D. Saadatmand,, “Scattering of the double sine-Gordon kinks”, Eur. Phys. J. C., 78 (2018), 345 | DOI

[12] C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, “Method for solving the Korteweg-de Vries equation”, Phys. Rev. Lett., 19 (1967), 1095–1097 | DOI | MR | Zbl

[13] D.R. Gulevich, M.B. Gaifullin, F.V. Kusmartsev, “Controlled dynamics of sine-Gordon breather in long Josephson junctions”, Eur. Phys. Jour. B, 85 (2012), 24 | DOI

[14] A.B. Hasanov, U.A. Hoitmetov, “On integration of the loaded Korteweg-de Vries equation in the class of rapidly decreasing functions”, Proc. Inst. Math. Mech. Natl. Acad. Azerb., 47:2 (2021), 250–261 | DOI | MR | Zbl

[15] U.A. Hoitmetov, “On the Cauchy problem for the mKdV-sine-Gordon equation with an additional term”, Acta Appl. Math., 184:1 (2023), 7 | DOI | MR | Zbl

[16] G. Hovhannisyan, “3 soliton solution to sine-Gordon equation on a space scale”, J. Math. Phys., 60:10 (2019), 103502 | DOI | MR | Zbl

[17] A.B. Khasanov, “The inverse problem of scattering theory for a system of two nonselfadjoint first-order equations”, Sov. Math., Dokl., 30 (1984), 145–148 | MR | Zbl

[18] A.B. Khasanov, U.A. Hoitmetov, “On integration of the loaded mKdV equation in the class of rapidly decreasing functions”, Izv. Irkutsk. Gos. Univ., Ser. Mat., 38 (2021), 19–35 | DOI | MR | Zbl

[19] 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”, Russ. Math., 65:7 (2021), 43–57 | DOI | MR | Zbl

[20] A.B. Khasanov, G.U. Urazboev, “On the sine-Gordon equation with a self-consistent source”, Sib. Adv. Math., 19:1 (2009), 13–23 | DOI | MR | Zbl

[21] A.B. Khasanov, G.U. Urazboev, “Integration of the sine-Gordon equation with a self-consistent source of the integral type in the case of multiple eigenvalues”, Rus. Math., 53:3 (2009), 45–55 | DOI | MR | Zbl

[22] A.H. Khater, O.H. El-Kalaawy, D.K. Callebaut, “Bäcklund transformations and exact solutions for Alfvén solitons in a relativistic electron-positron plasma”, Phys. Scr., 58:6 (1998), 545–548 | DOI | MR

[23] K. Konno, W. Kameyama, H. Sanuki, “Effect of weak dislocation potential on nonlinear wave propagation in anharmonic crystal”, J. Phys. Soc. Jpn., 37 (1974), 171–176 | DOI

[24] A.I. Kozhanov, “Nonlinear loaded equations and inverse problems”, Comput. Math. Math. Phys., 44:4 (2004), 657–678 | MR | Zbl

[25] H. Leblond, D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation”, Phys. Rep., 523:2 (2013), 61–126 | DOI | MR

[26] H. Leblond, D. Mihalache, “Few-optical-cycle solitons: modified Korteweg–de Vries sine-Gordon equation versus other non-slowly-varying-envelope-approximation models”, Phys. Rev. A, 79 (2009), 063835 | DOI

[27] H. Leblond, S.V. Sazonov, I.V. Melnikov, D. Mihalache, F. Sanchez, “Few-cycle nonlinear optics of multicomponent media”, Phys. Rev. A, 74 (2006), 063815 | DOI

[28] S. Matsutani, H. Tsuru, “Reflectionless quantum wire”, J. Phys. Soc. Jpn., 60:11 (1991), 3640–3644 | DOI | MR

[29] A.M. Nakhushev, Equations of mathematical biology, Vysshaya Shkola, M., 1995 https://www.studmed.ru/nahushev-am-uravneniya-matematicheskoy-biologii_5f9b3ede6d5.html | Zbl

[30] S.P. Novikov, S.V. Manakov, L.P. Pitaevskii, V.E. Zakharov, Theory of solitons. The inverse scattering method, Plenum Press, New York, 1984 https://scholar.google.com/scholar_lookup?&title=Theory | MR | Zbl

[31] W. Schief, “An infinite hierarchy of symmetries associated with hyperbolic surfaces”, Nonlinearity, 8:1 (1995), 1–9 | DOI | MR | Zbl

[32] G.U. Urazboev, U.A. Xoitmetov, A.K. Babadjanova, “Integration of the matrix modified Korteweg-de Vries equation with an integral-type source”, Theor. Math. Phys., 203:3 (2020), 734–746 | DOI | MR | Zbl

[33] M. Wadati, “The exact solution of the modified Korteweg–de Vries equation”, J. Phys. Soc. Jpn., 32 (1972), 1681 | DOI | MR

[34] A.M. Wazwaz, “N-soliton solutions for the integrable modified KdV-sine-Gordon equation”, Phys. Scr., 89:6 (2014), 065805 | DOI | MR

[35] A.M. Wazwaz, “Travelling wave solutions for the MKdV-sine-Gordon and the MKdV-sinh-Gordon equations by using a variable separated ODE method”, Appl. Math. Comput., 181:2 (2006), 1713–1719 | DOI | MR | Zbl

[36] L.V. Yakushevich, Nonlinear physics of DNA, 2nd ed., Wiley-VCH, Weinheim, 2004 ; 1998 | DOI | Zbl

[37] V.E. Zakharov, L.A. Takhtadzhyan, L.D. Faddeev, “Complete description of solutions of the sine-Gordon equation”, Sov. Phys. Dokl., 19 (1974), 824–826 | MR | Zbl

[38] S.-L. Zhao, H.-H. Hao, D.-J. Zhang, “The multi-soliton solutions for the non-isospectral mKdV-sine Gordon equation”, Mod. Phys. Lett. B, 23:4 (2009), 607–621 | DOI | Zbl