Singular ring solutions of critical and supercritical nonlinear Schrödinger equations

Abstract

We present new singular solutions of the nonlinear Schrödinger equation (NLS) i ψ t ( t , r ) + ψ r r + d − 1 r ψ r + | ψ | 2 σ ψ = 0 , 1 < d , 2 d ≤ σ ≤ 2 . These solutions collapse with a quasi self-similar ring profile ψ Q , i.e. ψ ∼ ψ Q , where ψ Q = 1 L 1 / σ ( t ) Q ( r − r m ( t ) L ) exp [ i ∫ 0 t d s L 2 ( s ) + i 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 = 2 − σ σ ( d − 1 ) . The blowup rate of these solutions is 1 1 + α for 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 .