Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow

Abstract

This article is concerned with the isospectral problem \[ -f'' + \frac{1}{4} f = zωf + z^2 \upsilon f \] for the periodic conservative Camassa-Holm flow, where $\omega$ is a periodic real distribution in $H_{\mathrm{loc}}^{-1}(\mathbb{R})$ and $\upsilon$ is a periodic non-negative Borel measure on $\mathbb{R}$. We develop basic Floquet theory for this problem, derive trace formulas for the associated spectra and establish continuous dependence of these spectra on the coefficients with respect to a weak${}^{\ast }$ topology.

16 pages. arXiv admin note: text overlap with arXiv:1801.04612