Microscopic conservation laws for the derivative Nonlinear Schrödinger equation

Abstract

Compared with macroscopic conservation law for the solution of the derivative nonlinear Schrödingger equation (DNLS) with small mass in \cite{KlausS:DNLS}, 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 introduced in \cite{Rybkin:KdV:Cons Law, Simon:Trace} to show one-parameter family of microscopic conservation laws of the $A(\kappa )$ flow and the DNLS flow, which is motivated by \cite{HKV:NLS,KV:KdV:AnnMath,KVZ:KdV:GAFA}.

arXiv comment25 pages. Comments are welcome

References in corpus (4)

• Global well-posedness for Schrödinger equation with derivative in $H^{1/2}(\mathbb{R})$
• $H^s$ Bounds for the Derivative Nonlinear Schrödinger Equation
• Global well-posedness for the derivative nonlinear Schrödinger equation
• Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schrödinger equation in the critical frequency case

Cited by in corpus (2)

• On the well-posedness problem for the derivative nonlinear Schrödinger equation
• On a priori estimates and existence of periodic solutions to the modified Benjamin-Ono equation below $H^{1/2}(\mathbb{T})$