Semiclassical WKB Problem for the non-self-adjoint Dirac operator

TL;DR

This paper reviews recent rigorous results on the semiclassical behavior of the scattering data for a non-self-adjoint Dirac operator with oscillatory potential, using WKB methods to understand implications for the focusing cubic NLS equation.

Contribution

It provides a detailed analysis of the semiclassical scattering data for a non-self-adjoint Dirac operator using WKB techniques, connecting spectral analysis to nonlinear Schrödinger equation solutions.

Findings

Rigorous semiclassical analysis of scattering data

Application of exact WKB and Olver's WKB theory

Insights into the inverse scattering for NLS

Abstract

We review some recent rigorous results on the semiclassical behavior ($\epsilon\downarrow0$) of the scattering data of a non-self-adjoint Dirac operator with potential $A\exp\{iS/\epsilon\}$ where both $A$ and $S$ are differentiable functions tending to constants as $x \to \pm \infty$. We have either employed the so-called exact WKB method, or the older WKB theory of Olver. Our analysis is motivated by the need to understand the semiclassical behaviour of the focusing cubic NLS equation with initial data $A\exp\{iS/\epsilon\}$, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables us to obtain the solution of the NLS equation via inverse scattering theory.