The defocusing Calogero--Moser derivative nonlinear Schr{\"o}dinger equation with a nonvanishing condition at infinity

Abstract

We consider the defocusing Calogero--Moser derivative nonlinear Schr{\"o}dinger equation\begin{align*}i \partial_{t} u+\partial_{x}^2 u-2\Pi D\left(|u|^{2}\right)u=0, \quad (t,x ) \in \mathbb{R} \times \mathbb{R}\end{align*}posed on $E := \left\{u \in L^{\infty}(\mathbb{R}): u' \in L^{2}(\mathbb{R}), u'' \in L^{2}(\mathbb{R}), |u|^{2}-1 \in L^{2}(\mathbb{R})\right\}$. We prove the global well-posedness of this equation in $E$. Moreover, we give an explicit formula for the chiral solution to this equation.