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

Abstract

We consider the defocusing Calogero--Moser derivative nonlinear Schr{ö}dinger equation\begin{align*}i \partial_{t} u+\partial_{x}^2 u-2ΠD\left(|u|^{2}\right)u=0, \quad (t,x ) \in \mathbb{R} \times \mathbb{R}\end{align*}posed on $E:=\left\{u\in L^{\infty }(\mathbb{R}):u^{\prime }\in L^2(\mathbb{R}),u^{\prime \prime }\in L^2(\mathbb{R}),|u{|}^2-1\in L^2(\mathbb{R})\right\}$. We prove the global well-posedness of this equation in $E$. Moreover, we give an explicit formula for the chiral solution to this equation.