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

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 $L_{+}^2(\mathbb{R}):=\left\{f\in L^2(\mathbb{R}):\mathrm{supp}(\hat{f})\subset [0,+\infty )\right\}$. 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.

The distorted Fourier transform can be established without any weighted condition, which allows for a complete analysis involving purely L_{+}^2 initial data