Scattering of the defocusing Calogero--Moser derivative nonlinear Schrödinger equation
Xi Chen
TL;DR
The paper resolves the long-time behavior of solutions to the defocusing CM-DNLS equation by leveraging a Gérard-type explicit formula together with a distorted Fourier transform tied to the Lax operator. It proves scattering for initial data in $L_{+}^{2,\alpha}$ ($\alpha>0$) and provides a precise description of the scattering term via spectral data, extending the analytic framework beyond rational initial data. The work also outlines how the same method yields an asymptotic bound-state/radiation decomposition for the focusing CM-DNLS and notes a natural soliton-resolution conjecture in that setting. This advances integrable PDE scattering theory by broadening the class of admissible initial data and connecting explicit spectral formulas to long-time asymptotics.
Abstract
In this paper, we study the long time behavior of solutions to the defocusing Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS). Using the Gérard-type explicit formula, we prove the scattering result of solutions to this equation with initial data in $L_{+}^{2,α}(\mathbb{R}): = \{u \in L_{+}^2(\mathbb{R}): |x|^α u \in L^2(\mathbb{R})\}$ with some $α>0$. 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 $L_{+}^{2,α}(\mathbb{R})$ with some $α>0$. This is one of the first works that apply the Gérard-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.