Traveling waves & finite gap potentials for the Calogero-Sutherland Derivative nonlinear Schrödinger equation

Abstract: We consider the Calogero-Sutherland derivative nonlinear Schr\"odinger equation \begin{equation}\tag{CS} i\partial_tu+\partial_x^2u\,\pm\,\frac{2}{i}\,\partial_x\Pi(|u|^2)u=0\,,\qquad x\in\mathbb{T}\,, \end{equation} where $\Pi$ is the Szeg\H{o} projector $$\Pi\Big(\sum_{n\in \mathbb{Z}}\widehat{u}(n)\mathrm{e}^{inx}\Big)=\sum_{n\geq 0 }\widehat{u}(n)\mathrm{e}^{inx}\,.$$ First, we characterize the traveling wave $u_0(x-ct)$ solutions to the defocusing equation (CS$^-$), and prove for the focusing equation (CS$^+$), that all the traveling waves must be either the constant functions or plane waves or rational functions. A noteworthy observation is that the (CS)-equation is one of the fewest nonlinear PDE enjoying nontrivial traveling waves with arbitrary small and large $L^2$-norms. Second, we study the finite gap potentials, and show that they are also rational functions, containing the traveling waves, and they can be grouped into sets that remain invariant under the system's evolution.