Classification of single-bubble blow-up solutions for Calogero--Moser derivative nonlinear Schrödinger equation
Uihyeon Jeong, Kihyun Kim, Taegyu Kim, Soonsik Kwon
TL;DR
The paper furnishes a sharp, universal classification of finite-time, single-bubble blow-up for the CM-DNLS equation by combining a gauge-transformed, self-dual formulation with a modulation analysis that tracks a hierarchy of higher-order conserved quantities. A key advancement is the identification of quantized blow-up rates $\lambda(t)\sim (T-t)^{2k}$ for $1\le k\le L$, distinguished from an exotic regime with $\lambda(t)\lesssim (T-t)^{2L+\tfrac{3}{2}}$, enabled by a closed finite-dimensional system incorporating radiation parameters $c_j$ and quadratic interactions $\mathfrak{c}_{i,j}$. The approach exploits the Lax hierarchy and associated conservation laws while avoiding refined integrability techniques, allowing the framework to apply beyond chiral solutions. A radially symmetric intuition motivates the mechanism, which is then extended to the full non-symmetric setting via intricate decompositions, coercivity arguments, and a Taylor-type normalization that yields the Omega-ODE system guiding rate selection. Collectively, the results provide the first rigorous quantized blow-up classification in a dispersive model and lay groundwork for multi-bubble dynamics through a robust modulation-radiation framework.
Abstract
We study the Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS), a mass-critical and completely integrable dispersive model. Recent works established finite-time blow-up constructions and soliton resolution, describing the asymptotic behaviors of blow-up solutions. In this paper, we go beyond soliton resolution and provide a sharp classification of finite-time blow-up dynamics in the \textit{single-bubble} regime. Assuming that a solution blows up at time $0<T<\infty$ with a single-soliton profile, we determine all possible blow-up rates. For initial data in $H^{2L+1}(\mathbb{R})$ with $L\ge1$, we prove a dichotomy: either the solution lies in a \emph{quantized regime}, where the scaling parameter satisfies \[ λ(t)\sim (T-t)^{2k},\qquad 1\le k\le L, \] with convergent phase and translation parameters, or it lies in an \emph{exotic regime}, where the blow-up rate satisfies $λ(t)\lesssim (T-t)^{2L+\frac 32}$. To our knowledge, this is the first classification result for quantized blow-up dynamics in the class of dispersive models. We provide a framework for identifying the quantized blow-up rates in classification problems. The proof relies on a modulation analysis combined with the hierarchy of conservation laws provided by the complete integrability of (CM-DNLS). However, it does not use \emph{more refined integrability-based techniques}, such as the inverse scattering method, the method of commuting flows, or the explicit formula. As a result, our analysis applies beyond the chiral solutions.