Continuous dependence and estimates of eigenvalues for periodic generalized Camassa-Holm equations (Q2187195)

This article is concerned with the isospectral problem \[ \Phi' = \frac{1}{2} \begin{pmatrix} -1 & \lambda m \\ -\lambda m & 1 \end{pmatrix} \Phi \] for the modified Camassa-Holm equation, where \(m\) is a non-positive integrable function on a finite interval and \(\lambda\) is a spectral parameter. It is shown that eigenvalues depend continuously on the coefficient \(m\) with respect to a weak topology. The authors also establish a trace formula for the periodic spectrum, which yields a lower bound for the moduli of the periodic eigenvalues in terms of the \(L^1\) norm of \(m\).