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On the spectrum of the periodic focusing Zakharov-Shabat operator

Gino Biondini, Jeffrey Oregero, Alexander Tovbis

Abstract

The spectrum of the focusing Zakharov-Shabat operator on the circle is studied, and its explicit dependence on the presence of a semiclassical parameter is also considered. Several new results are obtained. In particular: (i) it is proved that the resolvent set is comprised of two connected components, (ii) new bounds on the location of the Floquet and Dirichlet spectra are obtained, some of which depend explicitly on the value of the semiclassical parameter, (iii) it is proved that the spectrum localizes to a "cross" in the spectral plane in the semiclassical limit. The results are illustrated by discussing several examples in which the spectrum is computed analytically or numerically.

On the spectrum of the periodic focusing Zakharov-Shabat operator

Abstract

The spectrum of the focusing Zakharov-Shabat operator on the circle is studied, and its explicit dependence on the presence of a semiclassical parameter is also considered. Several new results are obtained. In particular: (i) it is proved that the resolvent set is comprised of two connected components, (ii) new bounds on the location of the Floquet and Dirichlet spectra are obtained, some of which depend explicitly on the value of the semiclassical parameter, (iii) it is proved that the spectrum localizes to a "cross" in the spectral plane in the semiclassical limit. The results are illustrated by discussing several examples in which the spectrum is computed analytically or numerically.

Paper Structure

This paper contains 20 sections, 29 theorems, 201 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General properties of the spectrum
  4. Properties of the spectrum in the semiclassical limit
  5. Properties of the spectrum for real or symmetric potentials
  6. Proofs
  7. Proofs: General properties of the spectrum
  8. Proofs: Semiclassical limit
  9. Proofs: Real or symmetric potentials
  10. Examples
  11. Plane wave potentials
  12. Piecewise continuous potentials
  13. Real-valued periodic single-lobe potentials
  14. Periodic potentials with rapid phase variations
  15. Definitions
  16. ...and 5 more sections

Key Result

Theorem 2.1

(Floquet, Floquet_intFloquet) Consider the system of linear homogeneous ODEs given by where $A\in L^1_{{\rm loc}}({\mathbb R})$ is a $n \times n$ matrix-valued function such that $A(x+L) = A(x)$. Then any fundamental matrix solution $Y(x)$ of e:yprime can be written in the Floquet normal form where $\Psi(x+L) = \Psi(x)$, $\Psi$ is nonsingular, and $R$ is a constant matrix.

Figures (8)

  • Figure 1: Left: Schematic illustration of the Lax spectrum of the focusing Zakharov-Shabat scattering problem with a generic periodic potential. Right: Schematic illustation of the Lax spectrum of the focusing Zakharov-Shabat scattering problem with a real, even, or odd periodic potential. In these cases elements of the Floquet spectrum come in quartets, i.e., $\{z, \overline{z}, -z, -\overline{z} \}$ (see Lemma \ref{['l:symmetries']}).
  • Figure 2: Left: Schematic diagram of the spectrum for a finite-band potential. Right: Schematic diagram of the spectrum for an infinite-band potential
  • Figure 3: Left: Bounds on the Lax spectrum for $\epsilon=1$ (dark red). Right: Bounds on the Lax spectrum when $0<\epsilon<1$ (dark red).
  • Figure 4: Left: The $\delta$-neighborhood of $\Sigma_{\infty}$ (light gray). Right: Bounds on the Lax spectrum for $0 < \epsilon \leq \epsilon_{*}$ (dark red) with the $\delta$-neighborhood of $\Sigma_{\infty}$ (light gray).
  • Figure 5: Lax spectrum for the potential $q(x)=\mathop{\rm e}\nolimits^{\text{i} x}$ with minimal period $L=2\pi$ (blue). Contours $\Gamma=\{z\in{\mathbb C}:\mathop{\rm Im}\nolimits\Delta_\epsilon(z)=0\}$ (black dashed). The curve which bounds the imaginary component of elements in the spectrum $|\mathop{\rm Im}\nolimits z| = \mathop{\rm min}\nolimits\{\| q\|_{\infty},\, \epsilon\|q'\|_{\infty}/2|\mathop{\rm Re}\nolimits z|\}$ (red dashed). Left: $\epsilon=1$. Right: $\epsilon=0.2$.
  • ...and 3 more figures

Theorems & Definitions (30)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Corollary 3.6
  • Lemma 3.7
  • ...and 20 more