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Integration of the Matrix Modified Korteweg-de Vries Equation with an Integral-Type Source

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Abstract

We derive time evolution equations for the scattering data for the matrix Zakharov-Shabat system with a potential that is the solution of the matrix modified Korteweg-de Vries equation with an integral-type source.

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References

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP, 34, 62–69 (1972).

    ADS  MathSciNet  Google Scholar 

  4. M. Wadati, “The modified Korteweg-de Vries equation,” J. Phys. Soc. Japan, 34, 1289–1296 (1973).

    Article  ADS  MathSciNet  Google Scholar 

  5. M. Wadati, “Generalized matrix form of the inverse scattering method,” in: Solitons (Topics Curr. Phys., Vol. 17, R. K. Bullough and P. J. Caudrey, eds.), Springer, Berlin (1980), pp. 287–299.

    Chapter  Google Scholar 

  6. V. K. Mel’nikov, “A direct method for deriving a multi-soliton solution for the problem of interaction of waves on the x, y plane,” Commun. Math. Phys., 112, 639–652 (1987); “Integration of the nonlinear Schroedinger equation with a self-consistent source,” Commun. Math. Phys., 137, 359–381 (1991).

    Article  ADS  Google Scholar 

  7. V. K. Mel’nikov, “Integration method of the Korteweg-de Vries equation with a self-consistent source,” Phys. Lett. A, 133, 493–496 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  8. J. Leon and A. Latifi, “Solution of an initial-boundary value problem for coupled nonlinear waves,” J. Phys. A: Math. Gen., 23, 1385–1403 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  9. F. A. Khalilov and E. Ya. Khruslov, “Matrix generalisation of the modified Korteweg-de Vries equation,” Inverse Problems, 6, 193–204 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  10. N. Bondarenko, G. Freiling, and G. Urazboev, “Integration of the matrix KdV equation with self-consistent source,” Chaos, Solitons and Fractals, 49, 21–27 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  11. U. A. Xoitmetov, “On finding multi-soliton solutions of the matrix Korteweg-de Vries equation with a self-consistent source [in Russian],” Dokl. AN RUz, No. 6, 3–5 (2016).

    Google Scholar 

  12. G. Urazboev and A. K. Babadjanova, “On the integration of the matrix modified Korteweg-de Vries equation with a self-consistent source,” Tamkang J. Math., 50, 281–291 (2019).

    Article  MathSciNet  Google Scholar 

  13. F. Demontis and C. Van der Mee, “Marchenko equations and norming constants of the matrix Zakharov-Shabat system,” Oper. Matrices, 2, 79–113 (2008).

    Article  MathSciNet  Google Scholar 

  14. A. B. Khasanov and G. U. Urazboev, “Integration of the mKdV equation with a self-consistent source [in Russian],” in: Works 2nd Intl. Conf. “Functional Spaces. Differential Operators. Problems of Mathematical Education” Dedicated to the 80th Birthday of L. D. Kudryavtsev, Fizmatlit, Moscow (2003), pp. 340–349.

    Google Scholar 

  15. Y.-B. Zeng, Y.-J. Shao, and W.-X. Ma, “Integral-type Darboux transformations for the mKdV hierarchy with self-consistent sources,” Commun. Theor. Phys., 38, 641–644 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  16. F. Demontis and C. van der Mee, “Characterization of scattering data for the AKNS system,” Acta Appl. Math., 131, 29–47 (2014).

    MathSciNet  MATH  Google Scholar 

  17. F. Demontis, Matrix Zakharov-Shabat System and Inverse Scattering Transform: Scattering Theory of Zakharov-Shabat Systems, Lap Lambert Academic Publ., Saarbrucken, Germany (2012).

    Google Scholar 

  18. T. Aktosun, F. Demontis, and C. van der Mee, “Exact solutions to the sine-Gordon equation,” J. Math. Phys., 51, 123521 (2010).

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgments

The authors are grateful to a referee for the valuable recommendations and corrections.

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Authors and Affiliations

  1. Al-Khorezmi Urgench State University, Urgench, Uzbekistan

    G. U. Urazboev, U. A. Xoitmetov & A. K. Babadjanova

Authors
  1. G. U. Urazboev
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  2. U. A. Xoitmetov
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  3. A. K. Babadjanova
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Correspondence to G. U. Urazboev, U. A. Xoitmetov or A. K. Babadjanova.

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The authors declare no conflicts of interest.

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 203, No. 3, pp. 351–364, June, 2020.

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Urazboev, G.U., Xoitmetov, U.A. & Babadjanova, A.K. Integration of the Matrix Modified Korteweg-de Vries Equation with an Integral-Type Source. Theor Math Phys 203, 734–746 (2020). https://doi.org/10.1134/S0040577920060033

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  • DOI: https://doi.org/10.1134/S0040577920060033

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