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On the Collapse of Solutions of the Cauchy Problem for the Cubic Schrödinger Evolution Equation

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Abstract

It is proved that, for some initial data, the solutions of the Cauchy problem for the cubic Schrödinger evolution equation blow up in finite time whose exact value is estimated from above. In addition, lower bounds for the blow-up rate of the solution in certain norms are obtained.

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References

  1. V. E. Zakharov and A. B. Shabat, “Exact theory of the two–dimensional self–focusing and one–dimensional modulation of waves in nonlinear media,” Zh. Éxper. Teoret. Fiz. 61 (1), 118–134 (1971).

    MathSciNet  Google Scholar 

  2. V. I. Lugovoi and A. M. Prokhorov, “The theory of propagation of powerful laser radiation in nonlinear medium,” Uspekhi Fiz. Nauk 111 (10), 203–247 (1973).

    Article  Google Scholar 

  3. H. Brezis and T. Gallouet, “Nonlinear Schrödinger evolution equations,” Nonlinear Anal. 4 (4), 677–681 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  4. Sh. M. Nasibov, “A nonlinear equation of Schrödinger type,” Differ. Uravn. 16 (4), 660–670 (1980)

    Google Scholar 

  5. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear media and nonlinear media (the moment method),” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14 (9), 1353–1363 (1971).

    Google Scholar 

  6. O. I. Kudryashov, “On the singularities of the solutions of nonlinear equations of Ginzburg–Landau type,” Sibirsk. Mat. Zh. 16 (4), 866–868 (1975).

    MathSciNet  MATH  Google Scholar 

  7. R. T. Classey, “On the blowing–up of solutions to the Cauchy problem for nonlinear Schrödinger equations,” J. Math. Phys. 18 (9), 1794–1797 (1977).

    Article  MathSciNet  Google Scholar 

  8. Sh. M. Nasibov, “A sharp constant in a Sobolev–Nirenberg inequality and its application to the Schrödinger equation,” Izv. Ross. Akad. Nauk Ser. Mat. 73 (3), 127–150 (2009) [Izv. Math. 73 (3), 555–577 (2009)].

    Article  MathSciNet  MATH  Google Scholar 

  9. Sh. M. Nasibov, “On optimal constants in some Sobolev inequalities and their application to the nonlinear Schrödinger equation,” Dokl. Akad. Nauk SSSR 307 (3), 538–542 (1989).

    MathSciNet  Google Scholar 

  10. Sh. M. Nasibov, “Some remarks on the fundamental Gross–Sobolev, Hirschman, Heisenberg–Pauli–Weyl inequalities and the entropy inequality,” Dokl. Akad. Nauk 415 (5), 589–591 (2007).

    MathSciNet  MATH  Google Scholar 

  11. Sh. M. Nasibov, “On the stability, blow–up, decay, and self–channelling of the solutions of a nonlinear Schrödinger equation,” Dokl. Akad. Nauk SSSR 285 (4), 807–811 (1985).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics, Baku State University, AZ 1148, Azerbaijan

    Sh. M. Nasibov

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  1. Sh. M. Nasibov
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Correspondence to Sh. M. Nasibov.

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Russian Text © Sh. M. Nasibov, 2019, published in Matematicheskie Zametki, 2019, Vol. 105, No. 1, pp. 76–83.

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Nasibov, S.M. On the Collapse of Solutions of the Cauchy Problem for the Cubic Schrödinger Evolution Equation. Math Notes 105, 64–70 (2019). https://doi.org/10.1134/S0001434619010073

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  • DOI: https://doi.org/10.1134/S0001434619010073

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