Complete integrability of the Calogero--Sutherland DNLS equation on $L^2_+(\mathbb T)$

Abstract

We establish the complete integrability of the defocusing Calogero--Sutherland derivative nonlinear Schrödinger equation \begin{equation}\label{CS-abstract}\tag{CS} i\partial_t u +\partial_x^2 u + 2i u \partial_x\,Π\!\left(|u|^2\right)=0,\qquad x\in\mathbb T, \end{equation} by constructing a nonlinear Fourier transform \begin{equation*} Φ: u\in L^2_+(\mathbb T)\mapsto\ (ζ_n(u))_{n\geq 0}\in \ell^2(\mathbb N_0),\qquad ζ_n(u)=\sqrt{γ_n(u)}\;e^{\,i\arg\langle u\mid f_n\rangle}, \end{equation*} built from the eigenvalues $(λ_n)$, the spectral gaps $γ_n:=λ_n-λ_{n-1}-1\geq 0$ and the eigenfunctions $(f_n)$ of the Lax operator $L_u$. We prove that $Φ$ is a norm-preserving homeomorphism. In these coordinates, the flow becomes the explicit rotation $ζ_n(t)=e^{-iω_n t}ζ_n(0)$ for all $n$, whose frequencies $ω_n$ depend only on the conserved spectrum $(λ_n)$. As consequences, every orbit of \eqref{CS-abstract} is precompact in $L^2_+(\mathbb T)$, and $t\mapsto u(t)$ is Bohr almost periodic. For finite-gap data, we obtain quasi-periodicity in time and a sharp criterion for genuine periodicity.