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Theoretical and Mathematical Physics

, Volume 201, Issue 1, pp 1514–1520 | Cite as

Nonlinear Evolutionary Schrödinger Equation in a Two-Dimensional Domain

  • Sh. M. NasibovEmail author
Article
  • 9 Downloads

Abstract

We consider a mixed problem for a nonlinear evolutionary Schrödinger equation in a two-dimensional domain and study the smoothness of solutions and their destruction.

Keywords

nonlinear evolutionary Schrödinger equation global solvability destruction 

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Notes

Conflicts of interest. The author declares no conflicts of interest.

References

  1. 1.
    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.Google Scholar
  2. 2.
    V. N. Lugovoi and A. M. Prokhorov, “Theory of the propagation of high-power laser radiation in a nonlinear medium,” Sov. Phys. Usp., 16, 658–679 (1974).ADSCrossRefGoogle Scholar
  3. 3.
    K. Rypdal and J. J. Rasmussen, “Blow-up in nonlinear Schroedinger equations: I. A general review,” Phys. Scr., 33, 481–504 (1986); “Blow-up in nonlinear Schroedinger equations: II. Similarity structure of the blow-up singularity,” Phys. Scr., 33, 498–504 (1986).ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Sh. M. Nasibov, “On stability, collapse, decay, and self-trapping of solutions of a nonlinear Schr¨odinger equation,” Sov. Math. Dokl., 32, 784–788 (1985).zbMATHGoogle Scholar
  5. 5.
    Sh. M. Nasibov, “Optimal constants in some Sobolev inequalities and their applications to the nonlinear Schr¨odinger equation,” Soviet Math. Dokl., 40, 110–115 (1990).MathSciNetGoogle Scholar
  6. 6.
    A.B. Ahabat, “On the Cauchy problem for the Ginzburg–Landau equation [in Russian],” in: Dynamics of a Continuous Medium (No. 1), Inst. Hydrodynamics, Siberian Branch, Russ. Acad. Sci., Novosibirsk (1999), pp. 180–194.Google Scholar
  7. 7.
    I. Segal, “Nonlinear semi-groups,” Ann. Math., 78, 339–364 (1963).MathSciNetCrossRefGoogle Scholar
  8. 8.
    H. Brezis and T. Gallouet, “Nonlinear Schrödinger evolution equations,” Nonlin. Anal., 4, 677–681 (1980).CrossRefGoogle Scholar
  9. 9.
    Sh. M. Nasibov, “On an inequality of Trudinger type and its application to a nonlinear Schrödinger equation,” Math. Notes, 80, 740–743 (2006).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sh. M. Nasibov, “A nonlinear equation of Schrödinger type,” Differ. Equ., 16, 660–670 (1980).Google Scholar
  11. 11.
    L. Nirenberg, “On elliptic partial differential equations,” Ann. Sci. Norm. Sup. Pisa, 13, 115–162 (1959).MathSciNetzbMATHGoogle Scholar
  12. 12.
    R. T. Glassey, “On the blowing up of solutions to the Cauchy problem for the nonlinear Schrödinger equation,” J. Math. Phys., 18, 1794–1797 (1977).ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    S. I. Pokhozhaev, “Eigenfunctions of the equation Δu + λf(u) = 0,” Sov. Math. Dokl., 6, 1408–1411 (196zbMATHGoogle Scholar
  14. 14.
    L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, R. I. (1998).zbMATHGoogle Scholar
  15. 15.
    S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Radiophysics and Quantum Electronics, 14, 1062–1070 (1971).ADSCrossRefGoogle Scholar
  16. 16.
    O. I. Kudryashov, “Singularities of the solutions of nonlinear equations of Ginzburg–Landau type,” Sib. Math. J., 16, 665–667 (1975).MathSciNetCrossRefGoogle Scholar
  17. 17.
    D. H. Sattinger, “On global solution of nonlinear hyperbolic equations,” Arch. Rational Mech. Anal., 30, 148–172 (1968).ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    J.-L. Lions, Quelques méthodes de résolution des probl`emes aux limites non linéaires, Gauthier-Villars, Paris (1969).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Institute of Applied MathematicsBaku State UniversityBakuAzerbaijan

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