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

Setsuro Fujiié, Nicholas Hatzizisis, Spyridon Kamvissis

Abstract

We review some recent rigorous results on the semiclassical behavior ($ε\downarrow0$) of the scattering data of a non-self-adjoint Dirac operator with potential $A\exp\{iS/ε\}$ 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 $A\exp\{iS/ε\}$, 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.

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

Abstract

We review some recent rigorous results on the semiclassical behavior () of the scattering data of a non-self-adjoint Dirac operator with potential where both and are differentiable functions tending to constants as . 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 , 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.
Paper Structure (10 sections, 17 theorems, 92 equations, 1 figure)

This paper contains 10 sections, 17 theorems, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The case of zero initial phase
  3. The case of non-analytic data
  4. The case of non-trivial initial phase
  5. The general method
  6. The exact WKB method
  7. Connection around a simple turning point
  8. Eigenvalues near a generic point on the asymptotic spectral arc
  9. The geometry of a particular case
  10. Eigenvalues near zero

Key Result

Theorem 2.1

Assume (A1). Then, for any $\delta>0$, there exists $\sigma>0$ independent of $\epsilon$ such that as $\epsilon\to 0$ uniformly for $\lambda\in (-\infty, \delta]\cup [\delta,\infty)$.

Figures (1)

  • Figure 1: Stokes lines near a simple turning point.

Theorems & Definitions (26)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Example 2.5
  • Example 2.6
  • Corollary 2.7
  • Corollary 2.8
  • Theorem 3.1
  • Corollary 3.2
  • ...and 16 more