Classification of single-bubble blow-up solutions for Calogero--Moser derivative nonlinear Schr\"odinger equation

TL;DR

This paper classifies finite-time blow-up behaviors of solutions to the Calogero--Moser derivative nonlinear Schr"odinger equation, identifying quantized and exotic blow-up rates in the single-bubble regime.

Contribution

It provides the first classification of quantized blow-up rates for dispersive models using a modulation analysis and integrability hierarchy, without relying on inverse scattering techniques.

Findings

Identifies all possible blow-up rates in the single-bubble regime.

Establishes a dichotomy between quantized and exotic blow-up regimes.

Provides a framework for classifying blow-up dynamics in integrable dispersive equations.

Abstract

We study the Calogero--Moser derivative nonlinear Schr\"odinger 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 \[ \lambda(t)\sim (T-t)^{2k},\qquad 1\le k\le L, \] with convergent phase and translation parameters, or it lies in an \emph{exotic regime}, where the blow-up rate satisfies $\lambda(t)\lesssim (T-t)^{2L+\frac 32}$. 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.