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

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 $H_{+}^{1,\alpha }(\mathbb{R}):=\{u\in H_{+}^1(\mathbb{R}):\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }u\in L^2(\mathbb{R})\}$ with some $\alpha >0$. We also characterize the scattering term using the distorted Fourier transform associated with the Lax operator. 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.