Scaling and long-time behavior conjecture for semiclassical DS II

Consider rapidly decreasing smooth initial data in $L_{2}(\mathbb{R}^{2})$ with a single maximum, and let $t_c$ be the critical time of the corresponding semiclassical DS II solution. Let $\epsilon$ be the dispersion parameter. Semiclassical DS II scaling conjecture. The difference between semiclassical and dispersive DS II solutions at $t_c$ scales as $\epsilon^{2/7}$ in the defocusing case and as $\epsilon^{2/5}$ in the focusing case. For $t\gg t_c$, defocusing solutions exhibit rapid modulated oscillations near critical points, while focusing solutions with distinct cusp times and locations exhibit a cusped oscillatory zone; with symmetry under interchange of $x$ and $y$, focusing solutions instead blow up in finite time $t^*$ for sufficiently small $\epsilon$, with no oscillatory zone for $|\Psi|^2$, and $t^*-t_c\propto\epsilon$, with $t^*>t_c$. The source presents these as numerical conclusions summarized in a conjecture environment; their general validity remains open.