Well-posedness for the NLS hierarchy

We prove well-posedness for higher-order equations in the so-called NLS hierarchy (also known as part of the AKNS hierarchy) in almost critical Fourier-Lebesgue spaces and in modulation spaces. We show the jth equation in the hierarchy is locally well-posed for initial data in Hˢᵣ(R) for s ≥ j-1/r' and 1 < r ≤ 2 and also in Mˢ2, p(R) for s = j-1/2 and 2 ≤ p < ∞. Supplementing our results with corresponding ill-posedness results in Fourier-Lebesgue spaces shows optimality. Using the conserved quantities derived in Koch-Tataru (2018) we argue that the hierarchy equations are globally well-posed for data in Hˢ(R) for s ≥ j-1/2. Our arguments are based on the Fourier restriction norm method in Bourgain spaces adapted to our data spaces and bi- & trilinear refinements of Strichartz estimates.