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

Abstract

We construct an explicit family of smooth finite-time blow-up solutions for the focusing Calogero--Sutherland derivative NLS given by $$ i \partial_t u = -\partial_x^2 u - 2 D Π(|u|^2) u \quad \mbox{with} \quad (t,x) \in \mathbb{R} \times \mathbb{T} , $$ where $D=-i \partial_x$ and $Π$ 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 $\| u_0 \|_{L^2}^2 < 1$ 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 $1 < \|u_0 \|_{L^2}^2 <2$. 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 $e^{i m x}$ 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 $$ \| u(t) \|_{H^s} \sim \frac{1}{(T-t)^{2s}} \quad \mbox{as} \quad \mbox{$t \nearrow T$} \quad \mbox{for all $s > 0$} $$ for some finite time $0 < T < \infty$. 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.