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 (J. Evol. Equ., 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 Ĥˢᵣ(R) H ^ r s ( R ) for s ≥ 1/2 + j-1/r' s ≥ 1 2 + j - 1 r ′ and 1 < r ≤ 2 1 < r ≤ 2 and also in Mˢ2, p(R) M 2 , p s ( R ) for s ≥ j/2 s ≥ j 2 and 2 ≤ p < ∞ 2 ≤ p < ∞ . 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, which is connected to the complete integrability of the Kaup-Newell system. For the gauge-transformation we establish bi-Lipschitz continuity between modulation spaces appropriate for use as data spaces. Furthermore we establish that also for higher-order equations in the dNLS hierarchy cubic nonlinear terms with an unfavourable distribution of derivatives are lifted from the equation.