Singular ring solutions of critical and supercritical nonlinear Schrödinger equations

Gadi Fibich*, Nir Gavish, Xiao Ping Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We present new singular solutions of the nonlinear Schrödinger equation (NLS) i ψt (t, r) + ψr r + frac(d - 1, r) ψr + | ψ |2 σ ψ = 0, 1 < d, frac(2, d) ≤ σ ≤ 2 . These solutions collapse with a quasi self-similar ring profile ψQ, i.e. ψ ∼ ψQ, where ψQ = frac(1, L1 / σ (t)) Q (frac(r - rm (t), L)) exp [i ∫0t frac(d s, L2 (s)) + i frac(Lt, 4 L) [α r2 + (1 - α) (r - rm (t))2]],L (t) is the ring width that vanishes at the singularity, rm (t) = r0 Lα (t) is the ring radius and α = frac(2 - σ, σ (d - 1)). The blowup rate of these solutions is frac(1, 1 + α) for frac(2, d) ≤ σ < 2 and 1 < d (0 < α ≤ 1), and a square root with a loglog correction (the loglog law) when σ = 2 and 1 < d (α = 0). Therefore, the NLS has solutions that collapse with any blowup rate p for 1 / 2 ≤ p < 1. This study extends the results of [G. Fibich, N. Gavish, X. Wang, New singular solutions of the nonlinear Schrödinger equation, Physica D 211 (2005) 193-220] for σ = 1 and d = 2, and of [P. Raphael, Existence and stability of a solution blowing up on a sphere for a L2 super critical non linear Schrödinger equation, Duke Math. J. 134 (2) (2006) 199-258] for σ = 2 and d = 2, to all 2 / d ≤ σ ≤ 2 and 1 < d.

Original languageEnglish
Pages (from-to)55-86
Number of pages32
JournalPhysica D: Nonlinear Phenomena
Volume231
Issue number1
DOIs
StatePublished - 1 Jul 2007

Keywords

  • Blowup rate
  • Collapse
  • Nonlinear Schrödinger equation
  • Ring profile
  • Self-similar solution
  • Singularity
  • Supercritical collapse

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