Skip to main content
SpringerLink
Log in

Integrating the modified Korteweg–de Vries– sine-Gordon equation in the class of periodic infinite-gap functions

  • Research Articles
  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

The inverse spectral problem method is used to integrate the nonlinear modified Korteweg–de Vries–sine-Gordon equation in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of six-times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly converging functional series constructed with the use of a solution of a system of Dubrovin equations and the first trace formula satisfies the modified Korteweg–de Vries–sine-Gordon equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. A.-M. Wazwaz, “\(N\)-soliton solutions for the integrable modified KdV-sine-Gordon equation,” Phys. Scr., 89, 065805, 5 pp. (2014).

    Article  ADS  Google Scholar 

  2. M. Wadati, “The exact solution of the modified Korteweg–de Vries equation,” J. Phys. Soc. Japan, 32, 1681–1681 (1972).

    Article  ADS  Google Scholar 

  3. A. V. Zhiber, R. D. Murtazina, I. T. Habibullin, and A. B. Shabat, “Characteristic Lie rings and integrable models in mathematical physics,” Ufa Math. J., 4, 17–85 (2012).

    MathSciNet  MATH  Google Scholar 

  4. V. E. Zakharov, L. A. Takhtadzhyan, and L. D. Faddeev, “Complete description of solutions of the ‘sine-Gordon’ equation,” Sov. Phys. Dokl., 19, 824–826 (1974).

    MATH  Google Scholar 

  5. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Sugur, “Method for solving the sine-Gordon equation,” Phys. Rev. Lett., 30, 1262–1264 (1973).

    Article  ADS  MathSciNet  Google Scholar 

  6. V. O. Kozel and V. P. Kotlyarov, Finite-zone solutions of the sine-Gordon equation (preprint FTINT Akad. Nauk Ukr. SSR, No. 9-77), Kharkov (1977).

    Google Scholar 

  7. V. O. Kozel and V. P. Kotlyarov, “Almost periodical solutions of the equation,” Dopov. Akad. Nauk Ukr. SSR, Ser. A, 10, 878–881 (1976).

    MATH  Google Scholar 

  8. B. A. Dubrovin and S. M. Natanzon, “Real two-zone solutions of the sine-Gordon equation,” Funct. Anal. Appl., 16, 21–33 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. O. Smirnov, “3-Elliptic solutions of the sine-Gordon equation,” Math. Notes, 62, 368–376 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Springer, New York (1984).

    MATH  Google Scholar 

  11. Yu. A. Mitropol’skii, N. N. Bogolyubov, Jr., A. K. Prikarpatskii, and V. G. Samoilenko, Integrable Dynamic Systems: Spectral and Differential-Geometric Aspects [in Russian], Naukova Dumka, Kiev (1987).

    Google Scholar 

  12. V. B. Matveev, “30 years of finite-gap integration theory,” Philos. Trans. Roy. Soc. London Ser. A, 366, 837–875 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  13. E. L. Ince, Ordinary Differential Equations, Dover, New York (1956).

    Google Scholar 

  14. P. B. Djakov and B. S. Mityagin, “Instability zones of periodic 1-dimensional Schrödinger and Dirac operators,” Russian Math. Surveys, 61, 663–766 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. A. B. Hasanov and M. M. Hasanov, “Integration of the nonlinear Schrödinger equation with an additional term in the class of periodic functions,” Theoret. and Math. Phys., 199, 525–532 (2019).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. A. B. Khasanov and M. M. Matjakubov, “Integration of the nonlinear Korteweg–de Vries equation with an additional term,” Theoret. and Math. Phys., 203, 596–607 (2020).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. A. B. Hasanov and T. G. Hasanov, “The Cauchy problem for the Korteweg–de Vries equation in the class of periodic infinite-gap functions,” (Zap. Nauchn. Sem. POMI, Vol. 506), POMI, St. Petersburg (2021), pp. 258–278.

    MathSciNet  Google Scholar 

  18. A. B. Khasanov and T. Zh. Allanazarova, “On the modified Korteweg–De-Vries equation with loaded term,” Ukr. Math. J., 73, 1783–1809 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. V. Domrin, “Remarks on the local version of the inverse scattering method,” Proc. Steklov Inst. Math., 253, 37–50 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. A. V. Domrin, “Real-analytic solutions of the nonlinear Schrödinger equation,” Trans. Moscow Math. Soc., 75, 173–183 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  21. G. A. Mannonov and A. B. Khasanov, “The Cauchy problem for a nonlinear Hirota equation in the class of periodic infinite-zone functions,” Algebra i Analiz, 34, 139–172 (2022).

    Google Scholar 

  22. U. B. Muminov and A. B. Khasanov, “Integration of a defocusing nonlinear Schrödinger equation with additional terms,” Theoret. and Math. Phys., 211, 514–531 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  23. U. B. Muminov and A. B. Khasanov, “The Cauchy problem for the defocusing nonlinear Schrödinger equation with a loaded term,” Siberian Advances in Mathematics, 32, 277–298 (2022).

    Article  Google Scholar 

  24. H. P. McKean and E. Trubowitz, “Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points,” Commun. Pure Appl. Math., 29, 143–226 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  25. H. P. McKean and E. Trubowitz, “Hill’s surfaces and their theta functions,” Bull. Amer. Math. Soc., 84, 1052–1085 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  26. M. U. Schmidt, “Integrable Systems and Riemann Surfaces of Infinite Genus,” (Memoirs of the American Mathematical Society, Vol. 122, no. 581), AMS, Providence, RI (1996); arXiv: solv-int/9412006.

    Article  MathSciNet  Google Scholar 

  27. T. V. Misjura, “Characterization of the spectra of the periodic and antiperiodic boundary value problems that are generated by the Dirac operator. I [in Russian]” (Teor. Funktsii Funktsional. Anal. i Prilozhen., Vol. 30), Vishcha shkola, Kharkiv (1978), pp. 90–101.

    Google Scholar 

  28. A. B. Khasanov and A. M. Ibragimov, “On an inverse problem for the Dirac operator with periodic potential [in Russian],” Uzbek. Matem. Zh., 3–4, 48–55 (2001).

    MathSciNet  Google Scholar 

  29. E. Trubowitz, “The inverse problem for periodic potentials,” Commun. Pure Appl. Math., 30, 321–337 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  30. A. B. Khasanov and A. B. Yakhshimuratov, “An analogue of the inverse theorem of G. Borg for the Dirac operator [in Russian],” Uzb. Matem. Zh., 3–4, 40–46 (2000).

    MATH  Google Scholar 

  31. S. Currie, T. T. Roth, and B. A. Watson, “Borg’s periodicity theorems for first-order self-adjoint systems with complex potentials,” Proc. Edinb. Math. Soc., 60, 615–633 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  32. D. Bättig, B. Grébert, J.-C. Guillot, and T. Kappeler, “Folation of phase space for the cubic nonlinear Schrödinger equation,” Composito Math., 85, 163–199 (1993).

    MATH  Google Scholar 

  33. B. Grébert and J.-C. Guillot, “Gap of one-dimensional periodic AKNS systems,” Forum Math., 5, 459–504 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Korotayev, “Inverse problem and estimates for periodic Zakharov–Shabat systems,” J. Reine Angew. Math., 583, 87–115 (2005).

    Article  MathSciNet  Google Scholar 

  35. E. Korotayev and D. Mokeev, “Dubrovin equation for periodic Dirac operator on the half-line,” Appl. Anal., 101, 337–365 (2020).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Samarkand State University, Samarkand, Uzbekistan

    A. B. Khasanov & Kh. N. Normurodov

  2. Samarkand Architectural and Construction Institute, Samarkand, Uzbekistan

    U. O. Khudaerov

Authors
  1. A. B. Khasanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kh. N. Normurodov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. U. O. Khudaerov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. B. Khasanov.

Ethics declarations

The authors declare no conflicts of interest.

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 214, pp. 198–210 https://doi.org/10.4213/tmf10365.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khasanov, A.B., Normurodov, K.N. & Khudaerov, U.O. Integrating the modified Korteweg–de Vries– sine-Gordon equation in the class of periodic infinite-gap functions. Theor Math Phys 214, 170–182 (2023). https://doi.org/10.1134/S0040577923020022

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577923020022

Keywords

Search

Navigation