Scattering of the defocusing Calogero--Moser derivative nonlinear Schrödinger equation

Abstract: In this paper, we study the defocusing Calogero--Moser derivative nonlinear Schr\"odinger equation. Using the G\'erard-type explicit formula, we prove the scattering result of solutions to this equation with initial data in $H_{+}^{1,\alpha}(\mathbb{R}) := {u\in H_{+}^1(\mathbb{R}) : |x|^{\alpha}u\in L^2(\mathbb{R})}$ for $\alpha>1/4$. We also characterize the scattering using the distorted Fourier transform associated with the Lax operator. 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.