Well-posedness for the dNLS hierarchy

We prove well-posedness for higher-order equations in the so-called dNLS hierarchy (also known as part of the Kaup-Newell hierarchy) in almost critical Fourier-Lebesgue and in modulation spaces. Leaning in on estimates proven by the author in a previous instalment Adams (2024), where a similar well-posedness theory was developed for the equations of the NLS hierarchy, we show the $j$th equation in the dNLS hierarchy is locally well-posed for initial data in $\hat H^s_r(\mathbb{R})$ for $s \ge \frac{1}{2} + \frac{j-1}{r'}$ and $1 < r \le 2$ and also in $M^s_{2, p}(\mathbb{R})$ for $s \ge \frac{j}{2}$ and $2 \le p < \infty$. Supplementing our results with corresponding ill-posedness results in Fourier-Lebesgue and modulation spaces shows optimality. Our arguments are based on the Fourier restriction norm method in Bourgain spaces adapted to our data spaces and the gauge-transformation commonly associated with the dNLS equation. For the latter we establish bi-Lipschitz continuity between appropriate modulation spaces and that even for higher-order equations `bad' cubic nonlinear terms are lifted from the equation.