In this talk, I will present direct and inverse scattering for the focusing nonlinear Schrödinger equation with initial data approaching elliptic traveling waves as \(x\to\pm\infty\). Under suitable assumptions, we characterize the scattering data and establish a uniquely solvable Riemann–Hilbert formulation. For a suitable class of reflectionless data, this formulation also arises as a continuum limit of multisoliton solutions, yielding a full soliton gas that remains nonvanishing at both spatial infinities.
This is joint work with Tamara Grava, Robert Jenkins and Xiaofan Zhang.