[Paper Review] Microscopic conservation laws for the derivative Nonlinear Schrödinger equation

Xingdong Tang, Guixiang Xu|arXiv (Cornell University)|Dec 9, 2020

Advanced Mathematical Physics Problems32 references4 citations

TL;DR

This paper establishes a one-parameter family of microscopic conservation laws for the derivative nonlinear Schrödinger equation (DNLS) with small mass, using logarithmic perturbation determinants and spectral analysis of the Lax pair. The key contribution is the derivation of a microscopic conservation law for the DNLS flow, which provides coercivity and enables local smoothing estimates, advancing global well-posedness analysis in low regularity spaces beyond the energy class.

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 $A(κ)$ flow and the DNLS flow, which is motivated by \cite{HKV:NLS,KV:KdV:AnnMath,KVZ:KdV:GAFA}.

Motivation & Objective

• To extend macroscopic conservation laws to microscopic form for the derivative nonlinear Schrödinger equation (DNLS) with small mass.
• To establish a one-parameter family of microscopic conservation laws for the $A(\kappa)$ flow and the DNLS flow using spectral theory.
• To leverage coercivity from microscopic laws to derive local smoothing effects for DNLS solutions in $H^s(\mathbb{R})$ with $s < 1/2$.
• To support global well-posedness analysis in low regularity spaces by strengthening the control of solution dynamics.

Proposed method

• Utilizes the logarithmic perturbation determinant from [34, 35] to analyze spectral properties of the Lax pair for DNLS.
• Applies Harrop-Griffiths-Killip-Visan’s framework [11, 19] to derive microscopic conservation laws from the $A(\kappa)$ flow.
• Employs the Lax pair representation $\frac{d}{dt}L(\varkappa) = [P_{H_{\text{DNLS}}}, L(\varkappa)]$ to derive time evolution of Green's functions and spectral data.
• Derives the density $\rho(\varkappa)$ and flux $j_{\text{DNLS}}(\varkappa)$ explicitly using $g_{12}(\varkappa)$, $g_{21}(\varkappa)$, and $\gamma(\varkappa)$, with $\gamma(\varkappa)$ defined via the perturbation determinant.
• Verifies the microscopic conservation law $\partial_t \rho(\varkappa) + \partial_x j_{\text{DNLS}}(\varkappa) = 0$ through direct computation of time derivatives and algebraic identities.
• Relies on functional calculus and Fréchet derivatives in the Hamiltonian framework to ensure consistency with the DNLS dynamics.

Experimental results

Research questions

• RQ1 Can microscopic conservation laws be derived for the DNLS flow using spectral and perturbative techniques?
• RQ2 How does the logarithmic perturbation determinant contribute to constructing one-parameter families of microscopic conservation laws?
• RQ3 What is the role of the $A(\kappa)$ flow in generating microscopic conservation laws for DNLS?
• RQ4 Can these microscopic laws provide coercive estimates that imply local smoothing for DNLS solutions in low regularity spaces?
• RQ5 How do the derived conservation laws compare to macroscopic ones in terms of structural and analytical utility?

Key findings

• The paper constructs a one-parameter family of microscopic conservation laws for the DNLS flow via the Lax pair and spectral data $g_{12}(\varkappa)$, $g_{21}(\varkappa)$, and $\gamma(\varkappa)$.
• The microscopic conservation law $\partial_t \rho(\varkappa) + \partial_x j_{\text{DNLS}}(\varkappa) = 0$ is rigorously derived for $i\varkappa^2 \in \mathbb{R} \setminus (-1,1)$, with explicit expressions for $\rho(\varkappa)$ and $j_{\text{DNLS}}(\varkappa)$.
• The time evolution of $g_{12}(\varkappa)$, $g_{21}(\varkappa)$, and $\gamma(\varkappa)$ is derived from the $H_{\text{DNLS}}$ flow, enabling the conservation law derivation.
• The microscopic laws exhibit coercivity, which is essential for proving local smoothing effects in $H^s(\mathbb{R})$ with $s < 1/2$, extending beyond the energy space.
• The method generalizes earlier macroscopic conservation laws [21] by incorporating logarithmic perturbation determinants and spectral theory.
• The result supports future global well-posedness analysis in low regularity spaces by providing stronger control mechanisms than macroscopic conservation laws.