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arXiv · 1810.07818

A Riemann--Hilbert Problem Approach to Periodic Infinite Gap Hill's Operators and the Korteweg--de Vries Equation

Kenneth T-R. McLaughlin·Patrik V. Nabelek
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Abstract

We formulate the inverse spectral theory of infinite gap Hill's operators with bounded periodic potential as a Riemann--Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann--Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potential evolves according to the KdV equation, we use integrability to derive an associated Riemann--Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann--Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann--Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation.

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Kenneth T-R. McLaughlin, Patrik V. Nabelek. 2019-12-02. A Riemann--Hilbert Problem Approach to Periodic Infinite Gap Hill's Operators and the Korteweg--de Vries Equation. https://arxiv.org/abs/1810.07818

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