Traveling waves and finite gap potentials for the Calogero–Sutherland derivative nonlinear Schrödinger equation

Abstract

We consider the Calogero–Sutherland derivative nonlinear Schrödinger equation $i{\partial }_{t}u+{\partial }_x^{2}u\pm \frac{2}{i}{\partial }_x\Pi (|u{|}^2)u=0$, $x\in \mathbb{T}$, where $\Pi$ is the Szegő projector $\Pi ({\sum }_{n\in \mathbb{Z}}\hat{u}(n){\mathrm{e}}^{inx})={\sum }_{n\ge 0}\hat{u}(n){\mathrm{e}}^{inx}$. 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 constant functions, or plane waves, or rational functions. A noteworthy observation is that the (CS) equation, which is an $L^2$-critical equation, is one of the few nonlinear PDEs enjoying nontrivial traveling waves with arbitrarily 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 evolution of the system.