The stable-manifold conjecture for blowup in the quadratic nonlinear Schrödinger equation

Let $\mathcal{W}^s(0)\subseteq C(\mathbb{T}^1,\mathbb{R})$ be the strong stable manifold of the zero equilibrium for the nonlinear heat equation

$$u_t=u_{xx}+u^2.$$

Consider real initial data $u_0\in C(\mathbb{T}^1,\mathbb{R})$ with summable Fourier coefficients for the quadratic nonlinear Schrödinger equation

$$i u_t=u_{xx}+u^2.$$

Stable-manifold conjecture. Such initial data are globally well-posed under the nonlinear Schrödinger dynamics if and only if $u_0\notin\mathcal{W}^s(0)$.

Equivalently, the conjecture identifies the codimension-one set of real initial data producing blowup with the strong stable manifold of the zero equilibrium for the nonlinear heat equation. The paper offers numerical and heuristic evidence, but no proof is given.