Singular solutions of the biharmonic nonlinear Schrödinger equation

G. Baruch*, G. Fibich, E. Mandelbaum

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider singular solutions of the L2-critical biharmonic nonlinear Schrödinger equation. We prove that the blowup rate is bounded by a quartic-root, the solution approaches a quasi-self-similar profile, and a finite amount of L2-norm, which is no less than the critical power, concentrates into the singularity. We also prove the existence of a ground-state solution. We use asymptotic analysis to show that the blowup rate of peak-type singular solutions is slightly faster than that of a quartic-root, and the self-similar profile is given by the ground-state standing wave. These findings are verified numerically (up to focusing levels of 108) using an adaptive grid method. We also use the spectral renormalization method to compute the ground state of the standing-wave equation, and the critical power for collapse, in one, two, and three dimensions.

Original languageEnglish
Pages (from-to)3319-3341
Number of pages23
JournalSIAM Journal on Applied Mathematics
Volume70
Issue number8
DOIs
StatePublished - 2010

Keywords

  • Biharmonic
  • Blowup
  • High-order dispersion
  • NLS
  • Nonlinear Schrödinger
  • Self-similar solutions

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