Scattering of the defocusing Calogero--Moser derivative nonlinear Schr\"odinger equation

TL;DR

This paper proves the scattering behavior of solutions to the defocusing Calogero--Moser derivative NLS using explicit formulas and Fourier analysis, extending understanding of long-term dynamics for broad initial data classes.

Contribution

It introduces a novel application of Gérard-type explicit formulas to analyze the long-time behavior of an integrable equation for general initial data.

Findings

Proves scattering for solutions with weighted initial data.

Characterizes the scattering term via distorted Fourier transform.

Establishes asymptotic bound-state/radiation decomposition.

Abstract

In this paper, we study the long time behavior of solutions to the defocusing Calogero--Moser derivative nonlinear Schr\"odinger equation (CM-DNLS). Using the G\'erard-type explicit formula, we prove the scattering result of solutions to this equation with initial data in $L_{+}^{2,\alpha}(\mathbb{R}): = \{u \in L_{+}^2(\mathbb{R}): |x|^{\alpha} u \in L^2(\mathbb{R})\}$ with some $\alpha>0$. We also characterize the scattering term using the distorted Fourier transform associated with the Lax operator. Following our approach developed in this paper, we can also conclude the asymptotic bound-state/radiation decomposition for global solutions to the focusing (CM-DNLS) with initial data in $L_{+}^{2,\alpha}(\mathbb{R})$ with some $\alpha>0$. This is one of the first works that apply the G\'erard-type explicit formula to study the long-time behavior of an integrable equation for a broad class of initial data, beyond the previously studied rational cases.