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

We consider the Calogero–Sutherland derivative nonlinear Schrödinger equation i∂ₜu+∂ₓ²u ± 2/i∂ₓΠ(|u|²)u=0 , x , where Π is the Szegő projector Π(∑n∈ Zû(n)eⁱⁿˣ)=∑n≥ 0û(n)eⁱⁿˣ . First, we characterize the traveling wave u₀(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 L² -critical equation, is one of the few nonlinear PDEs enjoying nontrivial traveling waves with arbitrarily small and large L² -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.