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

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 $E:=\{u\in L^{\mathrm{\infty }}(\mathbb{R}):u^{\mathrm{\prime }}\in L^2(\mathbb{R}),u^{\mathrm{\prime }\mathrm{\prime }}\in L^2(\mathbb{R}),\mathrm{\mid }u{\mathrm{\mid }}^2-1\in L^2(\mathbb{R})\}$. We prove the global well-posedness of this equation in $E$. Moreover, we give an explicit formula for the chiral solution to this equation.