Finite-time blow-up solutions for the Calogero--Sutherland derivative NLS

We construct an explicit family of smooth finite-time blow-up solutions for the focusing Calogero–Sutherland derivative NLS given by $$i{\mathrm{\partial }}_{t}u=-{\mathrm{\partial }}_x^{2}u-2D\mathrm{\Pi }(\mathrm{\mid }u{\mathrm{\mid }}^2)uwith(t,x)\in \mathbb{R}\times \mathbb{T},$$ where $D=-i{\mathrm{\partial }}_x$ and $\mathrm{\Pi }$ denotes the Cauchy–Szegő projector. This is a mass-critical NLS-type equation with a Lax pair structure. The Cauchy problem is global well-posed in the class of Hardy-Sobolev spaces $H_{+}^s(\mathbb{T})=L_{+}^2(\mathbb{T})\cap H^s(\mathbb{T})$ for small $L^2$-mass $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2}^2<1$ as recently proven in [R.~Badreddine, Pure Appl. Anal. 6 (2024)]. By a non-perturbative method, we construct smooth blow-up initial data with $L^2$-mass in the entire range $1<\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2}^2<2$. The strategy is based on a stability analysis for the explicit formula for (CS) combined with a suitable choice of finite-gap potentials as initial data that bifurcate from the discrete set of trivial plane waves $e^{imx}$ with $m\in {\mathbb{Z}}_{\ge 0}$. More precisely, we find a parametrized family of smooth initial data $u_0$ in $L_{+}^2(\mathbb{T})$ such that the corresponding solution $u(t)$ of (CS) blows up with \| u(t) \|_{H^s} \sim \frac{1}{(T-t)^{2s}} \quad as \quad $t \nearrow T$ \quad for all $s > 0$ for some finite time $0<T<\mathrm{\infty }$. Moreover, we give a full description of the blow-up dynamics and we identify the unique weak limit of $u(t)$ in $L_{+}^2(\mathbb{T})$ as $t\nearrow T$. Finally, we show instability of these blow-up solutions and complement our results by showing global existence for a class of finite-gap potentials as initial data with arbitrarily large $L^2$-mass.