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Integration of the differential–difference sine-Gordon equation with a self-consistent source

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Abstract

We use the inverse scattering theory to integrate the differential–difference sine-Gordon equation with a self-consistent source.

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References

  1. R. Hirota, “Nonlinear partial difference equations III; Discrete sine-Gordon equation,” J. Phys. Soc. Japan, 43, 2079–2086 (1977).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. S. J. Orfanidis, “Discrete sine-Gordon equations,” Phys. Rev. D, 18, 3822–3827 (1978).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  4. L. Pilloni and D. Levi, “The inverse scattering transform for solving the discrete sine-Gordon equation,” Phys. Lett. A, 92, 5–8 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  5. F. Gesztesy and 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, “Constant mean curvature surfaces and integrable equations,” Russian Math. Surveys, 46, 1–45 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. I. Bobenko, “All constant mean curvature tori in \(R^3\), \(S^3\), \(H^3\) in terms of theta-functions,” Math. Ann., 290, 209–245 (1991).

    Article  MathSciNet  Google Scholar 

  8. L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York (1960).

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  10. D. A. Korotkin, “On some integrable cases in surface theory,” J. Math. Sci. (New York), 94, 1177–1217 (1999).

    Article  MathSciNet  Google Scholar 

  11. M. Melko and I. Sterling, “Application of soliton theory to the construction of pseudospherical surfaces in \(\mathbf R^3\),” Ann. Glob. Anal. Geom., 11, 65–107 (1993).

  12. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London (1982).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  17. D.-J. Zhang and D.-Y. Chen, “The \(N\)-soliton solutions of the sine-Gordon equation with self-consistent sources,” Phys. A, 321, 467–481 (2003).

    Article  MathSciNet  Google Scholar 

  18. A. B. Khasanov and G. U. Urazboev, “On the sine-Gordon equation with a self-consistent source,” Siberian Adv. Math., 19, 13–23 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  19. G. U. Urazboev and A. B. Khasanov, “Integrating the Korteweg–de Vries equation with a self-consistent source and ‘steplike’ initial data,” Theoret. and Math. Phys., 129, 1341–1356 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  20. I. M. Krichever, “Algebraic curves and non-linear difference equations,” Russian Math. Surveys, 33, 255–256 (1978).

    Article  ADS  MATH  Google Scholar 

  21. P. G. Grinevich and I. A. Taimanov, “Spectral conservation laws for periodic nonlinear equations of the Melnikov type,” in: Geometry, Topology, and Mathematical Physics. S. P. Novikov’s Seminar: 2006–2007 (Amer. Math. Soc. Transl. Ser. 2, Vol. 224, V. M. Buchstaber and I. M. Krichever, eds.), AMS, Providence, RI, 125–138 (2008).

    MathSciNet  MATH  Google Scholar 

  22. A. B. Khasanov and A. B. Yakhshimuratov, “The Korteweg–de Vries equation with a self-consistent source in the class of periodic functions,” Theoret. and Math. Phys., 164, 1008–1015 (2010).

    Article  ADS  MATH  Google Scholar 

  23. B. Babajanov, M. Fečkan, and G. Urazbaev, “On the periodic Toda lattice with self-consistent source,” Commun. Nonlinear Sci. Numer. Simul., 22, 1223–1234 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. B. A. Babajanov and A. B. Khasanov, “Periodic Toda chain with an integral source,” Theoret. and Math. Phys., 184, 1114–1128 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. A. B. Yakhshimuratov and B. A. Babajanov, “Integration of equations of Kaup system kind with self-consistent source in class of periodic functions,” Ufa Math. J., 12, 103–113 (2020).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Correspondence to B. A. Babajanov.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 210, pp. 375-386 https://doi.org/10.4213/tmf10152.

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Babajanov, B.A., Babadjanova, A.K. & Azamatov, A.S. Integration of the differential–difference sine-Gordon equation with a self-consistent source. Theor Math Phys 210, 327–336 (2022). https://doi.org/10.1134/S0040577922030035

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  • DOI: https://doi.org/10.1134/S0040577922030035

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