Microscopic conservation laws for the derivative Nonlinear Schrödinger equation

Abstract

Compared with macroscopic conservation law for the solution of the derivative nonlinear Schrödinger equation (DNLS) with small mass in Klaus and Schippa (A priori estimates for the derivative nonlinear Schrödinger equation. Accepted by Funkcial. Ekvac), we show the corresponding microscopic conservation laws for the Schwartz solutions of DNLS with small mass. The new ingredient is to make use of the logarithmic perturbation determinant \(A(\kappa )\) introduced in Rybkin (in: Topics in Operator Theory, Birkhäuser Verlag, Basel, 2010), Simon (in: Trace ideals and their applications, American Mathematical Society, Providence, 2005) to show one-parameter family of microscopic conservation laws of the \(A(\kappa )\) flow and the DNLS flow, which is motivated by Harrop-Griffiths et al. (Sharp well-poseness for the cubic NLS and MKdV in \(H^s\). arXiv:2003.05011), Killip and Visan (Ann Math 190:249–305, 2019), Killip et al. (Geom Funct Anal 28:1062–1090, 2018).