Traveling waves & finite gap potentials for the Calogero-Sutherland Derivative nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the Calogero-Sutherland derivative nonlinear Schrödinger equation, characterizing its traveling wave solutions and finite gap potentials, revealing their rational function structure and invariance properties under evolution.

Contribution

It provides a complete characterization of traveling waves and finite gap potentials for the equation, highlighting their rational form and invariance, which is novel for this class of PDEs.

Findings

Traveling waves are either constant, plane waves, or rational functions.

Finite gap potentials are rational functions including traveling waves.

Invariant sets of solutions under the system's evolution are identified.

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.