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

TL;DR

This paper develops a spectral theory for non-self-adjoint Dirac operators with periodic potentials and applies it to solve the focusing nonlinear Schrödinger equation using inverse scattering, providing a new method for potential reconstruction.

Contribution

It introduces a Riemann-Hilbert problem approach for inverse spectral theory of non-self-adjoint Dirac operators with periodic potentials, applicable to both finite- and infinite-genus cases.

Findings

Unique reconstruction of potentials from spectral data.

Solution of the initial value problem for focusing NLS.

Formalism applies to finite- and infinite-genus potentials.

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 Schr\"odinger 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.