A Priori Estimates for the Derivative Nonlinear Schrödinger Equation

Abstract

We prove low regularity a priori estimates for the derivative nonlinear Schr\"{o}dinger equation in Besov spaces with positive regularity index conditional upon small $L^2$-norm. This covers the full subcritical range. We use the power series expansion of the perturbation determinant introduced by Killip--Vi\c{s}an--Zhang for completely integrable PDE. This makes it possible to derive low regularity conservation laws from the perturbation determinant.