Collapse in the nonlinear Schrödinger equation of critical dimension {σ=1, D=2}

  • Yu. N. Ovchinnikov
  • I. M. Sigal
Methods of Theoretical Physics


Collapsing solutions to the nonlinear Schrödinger equation of critical dimension {σ=1, D=2} are analyzed in the adiabatic approximation. A three-parameter set of solutions is obtained for the scale factor λ(t). It is shown that the Talanov solution lies on the separatrix between the regions of collapse and convenient expansion. A comparison with numerical solutions indicates that weakly collapsing solutions provide a good initial approximation to the collapse problem.

PACS numbers

02.30.Jr 03.65.Ge 


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  1. 1.
    V. I. Talanov, Pis’ma Zh. Éksp. Teor. Fiz. 11, 303 (1970) [JETP Lett. 11, 199 (1971)].Google Scholar
  2. 2.
    V. E. Zakharov, Zh. Éksp. Teor. Fiz. 62, 1746 (1972) [Sov. Phys. JETP 35, 908 (1972)].Google Scholar
  3. 3.
    G. M. Fraiman, Zh. Éksp. Teor. Fiz. 88, 390 (1985) [Sov. Phys. JETP 61, 228 (1985)].ADSMathSciNetGoogle Scholar
  4. 4.
    M. J. Landman, G. C. Papanicolaou, C. Sulem, and P. L. Sulem, Phys. Rev. A 38, 3837 (1988).CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    A. I. Smirnov and G. M. Fraiman, Physica D (Amsterdam) 52, 2 (1991).ADSMathSciNetGoogle Scholar
  6. 6.
    D. Pelinovskii, Physica D (Amsterdam) 119, 301 (1998).ADSGoogle Scholar
  7. 7.
    G. Perelman, in Nonlinear Dynamics and Renormalization Group, Ed. by I. M. Sigal and C. Sulem (American Mathematical Society, Providence, 2001), CRM Proceedings and Lecture Notes, Vol. 27, p. 147.Google Scholar
  8. 8.
    Yu. N. Ovchinnikov and I. M. Sigal, Zh. Éksp. Teor. Fiz. 116, 67 (1999) [JETP 89, 35 (1999)].MathSciNetGoogle Scholar
  9. 9.
    Catherine Sulem and Pierre-Louis Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer-Verlag, New York, 1999).Google Scholar

Copyright information

© MAIK "Nauka/Interperiodica" 2002

Authors and Affiliations

  • Yu. N. Ovchinnikov
    • 1
    • 2
  • I. M. Sigal
    • 3
  1. 1.Max-Planck Institute for Physics of Complex SystemsDresdenGermany
  2. 2.Landau Institute for Theoretical PhysicsRussian Academy of SciencesMoscowRussia
  3. 3.Department of MathematicsUniversity of TorontoTorontoCanada

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