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.

arXiv comment16 pages. arXiv admin note: text overlap with arXiv:1801.04612

References in corpus (10)

• An integrable shallow water equation with peaked solitons
• The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's of Shallow Water and Dym Type
• An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation
• The inverse spectral problem for indefinite strings
• The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data
• Quadratic operator pencils associated with the conservative Camassa-Holm flow
• The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation
• Spectral Theory as Influenced by Fritz Gesztesy
• Real-valued algebro-geometric solutions of the two-component Camassa-Holm hierarchy
• A Lagrangian view on complete integrability of the two-component Camassa-Holm system

Cited by in corpus (3)

• A view of the peakon world through the lens of approximation theory
• Trace formulas and inverse spectral theory for generalized indefinite strings
• The conservative Camassa-Holm flow with step-like irregular initial data