Spectral theory for non-self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the focusing nonlinear Schrödinger equation with periodic boundary conditions

TL;DR

The paper addresses inverse spectral theory for a non-self-adjoint Dirac (Zakharov-Shabat) operator with $L$-periodic potentials and applies a Riemann-Hilbert problem formulation to solve the focusing NLS IVP under periodic boundary conditions. The approach develops a time-evolving, RH-based IST that relies on spectral data composed of the main spectrum and a single Dirichlet spectrum, and establishes a uniqueness result ensuring reconstruction of the potential from this data for both finite and infinite genus. A key contribution is the introduction of a regularizing matrix $B(z)$ that yields a well-posed RH problem and a reconstruction formula $Q(x)= lim_{z o fty} i z [ ext{commutator of }oldsymbol{ Phi} ext{ with }B^{-1}]$, together with explicit treatment of Dirichlet eigenvalues inside spectral bands. The framework supports numerical computation and sets the stage for future work on soliton gases, Deift-Zhou asymptotics, and connections to finite-genus RHPs, broadening the analytic and computational toolkit for the focusing NLS with periodic boundary conditions.

Abstract

We formulate the inverse spectral theory for a non-self-adjoint one-dimensional Dirac operator associated periodic potentials via a Riemann-Hilbert problem approach. We use the resulting formalism to solve the initial value problem for the focusing nonlinear Schrödinger equation. We establish a uniqueness theorem for the solutions of the Riemann-Hilbert problem, which provides a new method for obtaining the potential from the spectral data. The formalism applies for both finite- and infinite-genus potentials. As in the defocusing case, the formalism shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the focusing NLS equation.

Figures (4)

• Figure 1: Schematic diagram showing a particular realization of the analytic arcs $\Gamma^\pm_{n,k}$, non-degenerate main spectrum and the choice of branch cuts for $r_{1/2}(z)$ (in red) and for $r_{1/4}(z)$ (in blue).
• Figure .2: Left: The Lax spectrum (in blue) for $q(x)=\mathop{\rm dn}\nolimits(x,0)=1$. The periodic and antiperiodic eigenvalues are highlighted in magenta and red, respectively, and the Dirichlet eigenvalues are shown as large green circles outlined in black. Here the period $L$ was chosen to correspond to the period of $\mathop{\rm dn}\nolimits(x,0)$, namely $L=\pi$, and the base point was $x_0=0$. Right: Same, but for $q(x)=\mathrm{e}^{\text{i}\pi/3}$. Note how the Dirichlet eigenvalue is located at $z= \text{i}/2$, in agreement with \ref{['ytildeconstant']}.
• Figure .3: Plots of $q(x)=\mathop{\rm dn}\nolimits(x,m)$ for $m = 0.8$ and various choice of base points. Top left: $x_0=0$. Top right: $x_0=1$. Bottom left: $x_0=L/2$. Bottom right: $x_0=4$. Here $L=2K(m)$, where $K(m)$ is the complete elliptic integral of the first kind.
• Figure .4: Numerically calculated movable Dirichlet eigenvalues along the imaginary axis as a function of the base point $x_o$ for the potential $q(x) = \mathop{\rm dn}\nolimits(x,m)$ for $m=0.8$ (left) and $m=0.99$ (right).

Theorems & Definitions (50)

• Proposition 2.1
• Proposition 2.2
• Theorem 2.3
• Lemma 2.5
• proof
• Definition 2.6
• Proposition 2.7
• proof
• Definition 2.8
• Definition 2.9