The $L^2$-critical stochastic NLS blow-up conjecture

Let $u_0 \in H^1(\mathbb R)$ and let $u(t)$, $t>0$, be an evolution of the stochastic nonlinear Schrödinger equation with $\sigma=2$ and the specified space-time white-noise coefficient. Let $Q$ denote the ground-state profile. The $L^2$-critical blow-up conjecture. In the multiplicative (Stratonovich) noise case, sufficiently localized initial data with $\\|u_0\\|_{L^2}>\\|Q\\|_{L^2}$ blows up in finite positive random time with positive probability. In the additive noise case, sufficiently localized initial data blows up in finite random time almost surely. If a solution blows up at a random positive time $T(\omega)>0$, then its blow-up has the self-similar profile $Q$ and, as $t\to T(\omega)$,

$$\\|\nabla u(t,\cdot)\\|_{L^2_x}\sim \frac{1}{L(t)},\qquad L(t)\sim \left(\frac{2\pi(T-t)}{\ln|\ln(T-t)|}\right)^{1/2}.$$

Moreover, the core satisfies

$$u_c(t,x)\sim \frac{1}{L(t)^{1/2}}Q\left(\frac{x-x(t)}{L(t)}\right)e^{i\gamma(t)},$$

with $\gamma(t)\to\gamma_0$ and $x(t)\to x_c$. Conditionally on finite-time blow-up, $x_c$ is Gaussian; no conditioning is necessary in the additive case. This conjecture describes the stochastic analogue of the deterministic log-log blow-up regime; the supplied source gives numerical confirmation, while a general rigorous result is not established here.