Quantized blow-up dynamics for Calogero--Moser derivative nonlinear Schr\"odinger equation

TL;DR

This paper constructs smooth finite-time blow-up solutions with quantized blow-up rates for the integrable Calogero--Moser derivative nonlinear Schrödinger equation, utilizing a novel modulation analysis approach based on the equation's integrable structure.

Contribution

The paper introduces a new modulation analysis method using nonlinear adapted derivatives and conservation laws to construct blow-up solutions in an integrable NLS variant.

Findings

Constructed solutions exhibit discrete blow-up rates.

Utilized nonlinear adapted derivatives aligned with the Lax pair structure.

Simplified analysis by replacing energy bootstrap with hierarchy of conservation laws.

Abstract

We consider the Calogero--Moser derivative nonlinear Schr\"odinger equation (CM-DNLS), an $L^2$-critical nonlinear Schr\"odinger type equation enjoying a number of numerous structures, such as nonlocal nonlinearity, self-duality, pseudo-conformal symmetry, and complete integrability. In this paper, we construct smooth finite-time blow-up solutions to (CM-DNLS) that exhibit a sequence of discrete blow-up rates, so-called \emph{quantized blow-up rates}. Our strategy is a forward construction of the blow-up dynamics based on modulation analysis. Our main novelty is to utilize the \emph{nonlinear adapted derivative} suited to the \textit{Lax pair structure} and to rely on the \emph{hierarchy of conservation laws} inherent in this structure to control higher-order energies. This approach replaces a repulsivity-based energy method in the bootstrap argument, which significantly simplifies the analysis compared to earlier works. Our result highlights that the integrable structure remains a powerful tool, even in the presence of blow-up solutions. In (CM-DNLS), one of the distinctive features is \emph{chirality}. However, our constructed solutions are not chiral, since we assume the radial (even) symmetry in the gauge transformed equation. This radial assumption simplifies the modulation analysis.