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Effect of a small dispersion on self-focusing in a spatially one-dimensional case

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Abstract

The effect of a small dispersion on the self-focusing of solutions of equations of nonlinear geometric optics in a spatially one-dimensional case has been studied. This effect in the leading order is described by a universal special solution of the nonlinear Schrödinger equation, k]which is isomonodromic. The analytical and asymptotic properties of this universal solution have been considered.

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References

  1. A. V. Gurevich and A. B. Shvartsburg, Sov. Phys. JETP 31, 1084 (1970).

    ADS  Google Scholar 

  2. A. V. Gurevich and A. B. Shvartsburg, Nonlinear Theory of Radio Wave Propagation in the Ionosphere (Nauka, Moscow, 1973) [in Russian].

    Google Scholar 

  3. A. B. Shvartsburg, Geometrical Optics in Nonlinear Wave Theory (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  4. C. K. Zhdanov and B. A. Trubnikov, Unstable Quasi-Gaseous Media (Nauka, Moscow, 1991) [in Russian].

    Google Scholar 

  5. A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989; Am. Math. Soc., RI, Providence, 1992).

    MATH  Google Scholar 

  6. V. I. Talanov, JETP Lett. 2, 138 (1965).

    ADS  Google Scholar 

  7. E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, Physica D 87, 201 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  8. M. V. Fedoryuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  9. A. V. Kitaev, Zap. LOMI 187, 53 (1991).

    Google Scholar 

  10. B. I. Suleimanov, Zap. LOMI 187, 110 (1991).

    Google Scholar 

  11. B. I. Suleimanov, Math. Notes 52, 1146 (1992).

    Article  MathSciNet  Google Scholar 

  12. B. I. Suleimanov and I. T. Khabibullin, Theor. Math. Phys. 97, 1250 (1993).

    Article  Google Scholar 

  13. B. I. Suleimanov, JETP Lett. 58, 849 (1993).

    ADS  Google Scholar 

  14. B. I. Suleimanov, J. Exp. Theor. Phys. 78, 583 (1994).

    ADS  Google Scholar 

  15. V. R. Kudashev, arXiv: putt-soll 9404002 (1994).

    Google Scholar 

  16. A. V. Kitaev, J. Math. Phys. 35, 2934 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  17. V. R. Kudashev and B. I. Suleimanov, JETP Lett. 62, 382 (1995).

    ADS  Google Scholar 

  18. V. R. Kudashev and B. I. Suleimanov, Theor. Math. Phys. 118, 325 (1999).

    Article  Google Scholar 

  19. V. R. Kudashev and B. I. Suleimanov, J. Appl. Math. Mech. (Engl. Transl.) 65, 441 (2001).

    Article  Google Scholar 

  20. P. N. Garifullin and B. I. Suleimanov, J. Exp. Theor. Phys. 110, 133 (2010).

    Article  ADS  Google Scholar 

  21. B. Dubrovin, Commun. Math. Phys. 267, 117 (2006).

    Article  ADS  Google Scholar 

  22. V. A. Baikov, R. K. Gazizov, and N. Kh. Ibragimov, J. Appl. Math. Mech. (Engl. Transl.) 30, 204 (1989).

    ADS  Google Scholar 

  23. A. R. Its, Math. USSR Izv. 26, 497 (1986).

    Google Scholar 

  24. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1971).

    ADS  Google Scholar 

  25. B. Dubrovin, T. Grava, and C. Klein, J. Nonlin. Sci. 19, 57 (2009).

    Article  ADS  Google Scholar 

  26. M. Bertola and A. Tovbis, Coom. Pure Appl. Math. 66, 678 (2013).

    Article  Google Scholar 

  27. A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, and A. S. Fokas, Painleve Transcendents: Riemann Problem Method (Regulyar. Khaotich. Dinamika, Moscow, Izhevsk, 2005) [in Russian].

    MATH  Google Scholar 

  28. R. Garnier, Ann. Sci. Ecole Norm. 29 (4), 1 (1912).

    Google Scholar 

  29. A. V. Kitaev and A. H. Vartanian, Inverse Problems 20, 1165 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  30. M. Jimbo and T. Miwa, Physica D 2, 407 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  31. V. E. Zakharov and E. A. Kuznetsov, Sov. Phys. JETP 64, 773 (1986).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics with Computer Center, Ufa Scientific Center, Russian Academy of Sciences, Ufa, 450008, Russia

    B. I. Suleimanov

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  1. B. I. Suleimanov
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Correspondence to B. I. Suleimanov.

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Original Russian Text © B.I. Suleimanov, 2017, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2017, Vol. 106, No. 6, pp. 375–380.

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Suleimanov, B.I. Effect of a small dispersion on self-focusing in a spatially one-dimensional case. Jetp Lett. 106, 400–405 (2017). https://doi.org/10.1134/S0021364017180126

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  • DOI: https://doi.org/10.1134/S0021364017180126

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