Singular ring solutions of critical and supercritical nonlinear Schrödinger equations

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, L² (s)) + i frac(L_t, 4 L) [α r² + (1 - α) (r - r_m (t))²]],L (t) is the ring width that vanishes at the singularity, r_m (t) = r₀ 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 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.