Supercritical BNLS peak-type singularity conjecture

Let $\psi$ be a peak-type singular solution of the supercritical BNLS, let $L(t)$ be its collapsing length scale, let $T_{\rm c}$ be the collapse time, and set $\rho=r/L(t)$. Define

$$\tau(t)=\int_{s=0}^{t}\frac{1}{L^{4}(s)}\,ds.$$

Let $B(\rho)$ be the self-similar profile, let $R(\rho)$ be the ground-state profile from the stationary equation, let $H[B]$ be the Hamiltonian, and let $\kappa>0$. Supercritical BNLS peak-type singularity conjecture. The collapsing core approaches

$$\psi(t,r)\sim\psi_B(t,r),\qquad 0\leq r\leq r_c,$$

where

$$\psi_B(t,r)=\frac{1}{L^{2/\sigma}(t)}B(\rho)e^{i\tau(t)}.$$

The profile satisfies

$$-B(\rho)+i\frac{\kappa^4}{4}\left(\frac{2}{\sigma}B+\rho B'\right)-\Delta_\rho B+|B|^{2\sigma}B=0,$$

with

$$B(0)=1,\qquad B'(0)=B”'(0)=0,\qquad H[B]=0.$$

It differs from the ground state, $B(\rho)\neq R(\rho)$, and is the unique admissible solution with unique $\kappa=\kappa(\sigma,d)$ for which $|B(\rho)|$ is monotonically decreasing and

$$B(\rho)\sim\rho^{-2/\sigma-i4/\kappa^4}\qquad(\rho\to\infty).$$

The blowup rate is exactly quartic-root:

$$L(t)\sim\kappa(T_{\rm c}-t)^{1/4},$$

and

$$\kappa=\lim_{t\to T_{\rm c}}\frac{L(t)}{(T_{\rm c}-t)^{1/4}}=\kappa(\sigma,d).$$

Thus $\kappa$ is universal and independent of the initial condition. This is a comprehensive formal conjecture about supercritical peak-type collapse, combining the predicted profile, its asymptotics, uniqueness, and the universal quartic-root rate. The source supplies no resolution.