Ring-type singular solutions of the biharmonic nonlinear Schrödinger equation

We present new singular solutions of the biharmonic nonlinear Schrödinger equation (NLS) iψt(t,x)− ψ + |ψ |2σψ = 0, x ∈ R , 4/d σ 4. These solutions collapse with the quasi-self-similar ring profile ψQB , where |ψQB(t, r)| ∼ 1 L2/σ (t) QB ( r − rmax(t) L(t) ) , r = |x|, L(t) is the ring width that vanishes at singularity, rmax(t) ∼ r0L(t) is the ring radius, and α = (4 − σ)/(σ (d − 1)). The blowup rate of these solutions is 1/(3 + α) for 4/d σ < 4, and slightly faster than 1/4 for σ = 4. These solutions are analogous to the ring-type solutions of the NLS. Mathematics Subject Classification: 35Q55, 35G25 (Some figures in this article are in colour only in the electronic version)