## Abstract

We present new singular solutions of the nonlinear Schrödinger equation (NLS) i ψ_{t} (t, r) + ψ_{r r} + frac(d - 1, r) ψ_{r} + | ψ |^{2 σ} ψ = 0, 1 < d, frac(2, d) ≤ σ ≤ 2 . These solutions collapse with a quasi self-similar ring profile ψ_{Q}, i.e. ψ ∼ ψ_{Q}, where ψ_{Q} = frac(1, L^{1 / σ} (t)) Q (frac(r - r_{m} (t), L)) exp [i ∫_{0}^{t} frac(d s, L^{2} (s)) + i frac(L_{t}, 4 L) [α r^{2} + (1 - α) (r - r_{m} (t))^{2}]],L (t) is the ring width that vanishes at the singularity, r_{m} (t) = r_{0} L^{α} (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 L^{2} 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 language | English |
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Pages (from-to) | 55-86 |

Number of pages | 32 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 231 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jul 2007 |

## Keywords

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