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arXiv · 2505.04790

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

Gino Biondini·Gregor Kovačič·Alexander Tovbis·Zachery Wolski·Zechuan Zhang
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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.

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Gino Biondini, Gregor Kovačič, Alexander Tovbis, Zachery Wolski, Zechuan Zhang. 2025-05-07. 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. https://arxiv.org/abs/2505.04790

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