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

2025/05/07 by Biondini, Gino, Kovačič, Gregor, Tovbis, Alexander +2
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2505.04790

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.

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