Finite-time blow-up for the Calogero--Sutherland derivative NLS on $\mathbb{T}$

Abstract

We study finite-time blow-up for the focusing Calogero--Sutherland derivative NLS on the torus with smooth initial data in the Hardy space $L_{+}^2(\mathbb{T})$. For finite-gap potentials as initial data, the explicit solution formula reduces the dynamics to a finite-dimensional analytic family of contractions. This yields a complete description of every finite-time blow-up solution in this class: near blow-up time $T>0$, the solution decomposes into finitely many concentrating universal blow-up profiles and a smooth remainder, each bubble carries one unit of $L^2$-mass, the concentration points are distinct, and the blow-up rates are quantized according to $$ \|u(t) \|_{H^s(\mathbb{T})} \sim_{s,u_0} \frac{1}{(T-t)^{2sν}} \quad \mbox{as} \quad t \nearrow T \quad \mbox{for any $s > 0$} $$ with some integer $\nu \ge 1$. Within an explicit subclass of finite-gap potentials, we identify an exact resonance condition that yields the existence of finite-time blow-up solutions. By a non-perturbative method, we construct single- and multi-bubble blow-up solutions with rate $\nu =1$ for every prescribed $L^2$-mass in $(1,\infty )\setminus \{2\}$. In the complementary non-resonant regime, we prove global existence with uniform Sobolev bounds. The construction also shows instability of the blow-up solutions and non-chiral finite-time blow-up examples. To the best of our knowledge, these results provide the first explicit, non-perturbative construction of finite-time blow-up for an integrable NLS-type equation on the torus, together with a complete classification of the singular dynamics within a natural finite-gap class.

This is a revised and largely extended version of the previous version. We have added the complete analysis of blow-up dynamics for initial data given by finite-gap potentials, along with the explicit construction of single- and multi-bubble blow-up solutions. Comments are welcome