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Integration of the nonlinear Korteweg–de Vries equation with an additional term

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Abstract

We use the method of the inverse spectral problem to integrate the nonlinear Korteweg–de Vries equation with an additional term in the class of periodic functions.

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References

  1. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-de Vries equation,” Phys. Rev. Lett., 19, 1095–1097 (1967).

    Article  ADS  Google Scholar 

  2. L. D. Faddeev, “Properties of the S-matrix of the one-dimensional Schrödinger equation,” Amer. Math. Soc. Transl. Ser. 2, 65, 139–166 (1967).

    MATH  Google Scholar 

  3. V. A. Marchenko, Sturm-Liouville Operators and Their Applications [in Russian], Naukova Dumka, Kiev (1977); English transl.: Sturm-Liouville Operators and Applications (Operator Theory: Advances and Applications, Vol. 22), Birkhäuser, Basel (1986).

    MATH  Google Scholar 

  4. B. M. Levitan, Inverse Sturm-Liouville Problems [in Russian], Nauka, Moscow (1984); English transl., VNU Science Press BV, Utrecht (1987).

    MATH  Google Scholar 

  5. P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Comm. Pure Appl. Math., 21, 467–490 (1968).

    Article  MathSciNet  Google Scholar 

  6. V. F. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of wave in nonlinear media,” JETP, 34, 62-69.

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

    Article  ADS  Google Scholar 

  8. 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 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  11. L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Methods in the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl., Springer, Berlin (2007).

    MATH  Google Scholar 

  12. A. B. Khasanov and G. U. Urazboev, “On the sine-Gordon equation with a self-consistent source corresponding to multiple eigenvalues,” Differ. Equ., 43, 561–570 (2007).

    Article  MathSciNet  Google Scholar 

  13. A. B. Khasanov and G. U. Urazboev, “Integration of the sine-Gordon equation with a self-consistent source of the integral type in the case of multiple eigenvalues,” Russian Math. Iz. VUZ), 53, 45–55 (2009).

    Article  MathSciNet  Google Scholar 

  14. A. B. Khasanov and A. A. Reyimberganov, “About the finite density solution of the higher nonlinear Schrödinger equation with self-consistent source [in Russian],” Ufimsk. Mat. Zh., 1, No. 4, 133–143 (2009).

    Google Scholar 

  15. A. B. Khasanov and U. A. Khoitmetov, “On integration of Korteweg-de Vries equation in a class of rapidly decreasing complex-valued functions,” Russian Math. (Iz. VUZ), 62, 68–78 (2018).

    Article  MathSciNet  Google Scholar 

  16. A. B. Khasanov and K. A. Mamedov, “On the modified Korteweg-de Vries equation with a self-consistent source of the integral type corresponding to multiple eigenvalues [in Russian],” Uzb. Matem. Zhurn., No. 4, 81–93 (2007).

    Google Scholar 

  17. A. R. Its and V. B. Matveev, “Schrödinger operators with finitegap spectrum and N-soliton solutions of the Korteweg-de Vries equation,” Theor. Math. Phys., 23, 343–355 (1975).

    Article  Google Scholar 

  18. B. A. Dubrovin and S. P. Novikov, “Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation,” JETP, 40, 1058–1063 (1975).

    ADS  MathSciNet  Google Scholar 

  19. Yu. A. Mitropolskii, N. N. Bogolyubov Jr., A. K. Prikarpatskii, and V. G. Samoilenko, Integrable Dynamical Systems: Spectral and Differential-Geometric Aspects [in Russian], Naukova Dumka, Kiev (1987).

    Google Scholar 

  20. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevsky, Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1980).

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. M. U. Schmidt, Integrable Systems and Riemann Surfaces of Infinite Genus (Mem. AMS, Vol. 122, No. 581), Amer. Math. Soc., Providence, R. I. (1996); arXiv:solv-int/9412006v1 (1994).

    Google Scholar 

  24. B. A. Dubrovin, “Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials,” Funct. Anal. Appl., 9, 215–223 (1975).

    Article  Google Scholar 

  25. P. G. Grinevich and I. A. Taimanov, “Spectral conservation laws for periodic nonlinear equations of the Melnikov type,” in: Geometry, Topology, and Mathematical Physics (Amer. Math. Soc. Transl. Ser. 2, Vol. 224, V. M. Buchstaber and I. M. Krichever, eds.), Amer. Math. Soc., Providence, R. I. (2008), pp. 125–138.

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  27. A. Yakshimuratov, “The nonlinear Schrödinger equation with a self-consistent source in the class of periodic functions,” Math. Phys. Anal. Geom., 14, 153–169 (2011).

    Article  MathSciNet  Google Scholar 

  28. A. O. Smirnov, “Elliptic solutions of the nonlinear Schrödinger equation and the modified Korteweg-de Vries equation,” Russian Acad. Sci. Sb. Math., 82, 461–470 (1995).

    MathSciNet  Google Scholar 

  29. P. D. Lax, “Almost periodic solutions of the KdV equation,” SIAM Rev., 18, 438–462 (1976).

    Article  MathSciNet  Google Scholar 

  30. A. V. Domrin, “Meromorphic extension of solutions of soliton equations,” Izv. Math., 74, 461–480 (2010).

    Article  MathSciNet  Google Scholar 

  31. E. C. Titchmarsh, Eigenfunction Expansions with Second-Order Differential Operators, Vol. 1, Clarendon, Oxford (1946).

  32. I. V. Stankevich, “An inverse problem of spectral analysis for Hill’s equation [in Russian],” Dokl. Akad. Nauk SSSR, 192, 34–37 (1970).

    MathSciNet  Google Scholar 

  33. N. I. Akhiezer, “Continuous analogues of orthogonal polynomials on a system of intervals [in Russian],” Dokl. Akad. Nauk SSSR, 141, 263–266 (1961).

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. H. Hochstadt, “On the determination of Hill’s equation from its spectrum,” Arch. Rational Mech. Anal., 19, 353–362 (1965).

    Article  ADS  MathSciNet  Google Scholar 

  36. H. P. McKean and P. van Moerbeke, “The spectrum of Hill’s equation,” Invent. Math., 30, 217–274 (1975).

    Article  ADS  MathSciNet  Google Scholar 

  37. H. Flaschka, “On the inverse problem for Hill’s operator,” Arch. Rational Mech. Anal., 59, 293–309 (1975).

    Article  ADS  MathSciNet  Google Scholar 

  38. H. Hochstadt, “Estimates on the stability intervals for the Hill’s equation,” Proc. AMS, 14, 930–932 (1963).

    MathSciNet  MATH  Google Scholar 

  39. B. M. Levitan and G. Sh. Guseinov, “Calculation of the leading term of the asymptotic gap length of the periodic Sturm-Liouville problem [in Russian],” Serdica Bulgarsko Matematichesko Spisanie, 3, 273–280 (1977).

    Google Scholar 

  40. H. Hochstadt, “A generalization of Borg’s inverse theorem for Hill’s equations,” J. Math. Anal. Appl., 102, 599–605 (1984).

    Article  MathSciNet  Google Scholar 

  41. G. Borg, “Eine Umkehrung der Sturm-Liouvillschen Eigenwertaufgabe, Bestimmung der Differentialgleichung durch die Eigenwerte,” Acta Math., 78, 1–96 (1946).

    Article  MathSciNet  Google Scholar 

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Correspondence to A. B. Khasanov or M. M. Matyakubov.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 203, No. 2, pp. 192–204, May, 2020.

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Khasanov, A.B., Matyakubov, M.M. Integration of the nonlinear Korteweg–de Vries equation with an additional term. Theor Math Phys 203, 596–607 (2020). https://doi.org/10.1134/S0040577920050037

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  • DOI: https://doi.org/10.1134/S0040577920050037

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