Well-posedness for the NLS hierarchy

Abstract

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 $j$th equation in the hierarchy is locally well-posed for initial data in ${\hat{H}}_r^s(\mathbb{R})$ for $s\ge \frac{j-1}{r^{\prime }}$ and $1<r\le 2$ and also in $M_{2,p}^s(\mathbb{R})$ for $s=\frac{j-1}{2}$ and $2\le p<\infty$. 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 $H^s(\mathbb{R})$ for $s\ge \frac{j-1}{2}$. 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.

To appear in J. Evol. Equ

References in corpus (8)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data
• Conserved energies for the cubic NLS in 1-d
• On the one-dimensional cubic nonlinear Schrodinger equation below L^2
• On the well-posedness problem for the derivative nonlinear Schrödinger equation
• Optimal local well-posedness for the periodic derivative nonlinear Schrodinger equation
• A priori estimates for the derivative nonlinear Schrödinger equation
• NLS in the modulation space $M_{2,q}(\mathbb{R})$
• Well-posedness for the fourth-order Schrödinger equation with third order derivative nonlinearities