The supercritical BNLS standing-ring blowup conjecture

Let $\psi$ be a singular standing-ring solution of the BNLS with $d>1$ and $\sigma>4$. Let $L(t)$ be the ring-region similarity scale, $r_{\max}(t)$ its radius, and $\psi_B$ the ring profile. Supercritical BNLS standing-ring conjecture. The solution is self-similar in the ring region, with $psisimpsi_B$ for $r-r_{\max}=\mathcal{O}(L)$; $\psi_B(t,r)=\phi_B(t,x=r-r_{\max}(t))$, where $\phi_B$ is the one-dimensional BNLS asymptotic profile with exponent $\sigma$. Moreover,

$$L(t)\simkappa_{\rm B}(\sigma)\sqrt[4]{T_c-t},\qquad t\to T_c,$$

where $\kappa_{\rm B}(\sigma)>0$ is the parameter of the admissible $B$ profile. This predicts that standing-ring BNLS collapse has the same local dynamics as one-dimensional peak collapse; no resolution is supplied.