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

Abstract

This article is concerned with the isospectral problemfor the periodic conservative Camassa-Holm flow, where ω is a periodic real distribution in H −1 loc (R) and υ is a periodic non-negative Borel measure on 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 * topology.

On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures

“…(48)On the other hand, one hassup (a,b,α,β)∈D A,B F (a, b, α, β) = +∞. (49)By (42) and (49), one has triviallyM(A, B) = sup ρ∈M A,B λ prin (ρ) ≥ sup (a,b,α,β)∈D A,B F (a, b, α, β) = +∞.Hence M(A, B) = +∞, the desired result(13).Next we consider the infimum value L(A, B). According to (42) and (47), one hasL(A, B) = inf ρ∈M A,B λ prin (ρ) ≥ inf (a,b,α,β)∈D A,B F (a, b, α, β) = 4A B 2 −A 2 .…”

“…For example, problem ( 7)-(N ) possesses a sequence of eigenvalues τ 0 (ν) < τ 1 (ν) < • • • → +∞, where τ 0 (ν) is called the zeroth Neumann eigenvalue. Spectral theory of MDEs and some types of GODEs has been extensively studied in recent works like [11,12,13,16,26,37]. In [37], by considering a nonzero nonnegative measure ρ ∈ M 0 (I) as a weight, for the (weighted) eigenvalue problem…”

“…These results on second-order MDEs, together with the so-called completely continuous dependence of eigenvalues on measures, have been extended in [16] to some class of third-order symmetric MDEs. Motivated by the study on the Camassa-Holm equation [8,9] and general strings, some spectral and inverse spectral problems on second-order GODEs with indefinite distributions as weights have been developed in [11,12,13] and are still under developing. In order to solve problems (4) and ( 5) and for the theoretical purpose, in this paper, by taking indefinite measures ρ ∈ M 0 (I), we will give a relatively complete study on the (positive) principal eigenvalues of MDEs ( 8 As before, by a principal eigenvalue λ of ( 8)-(N ), we always refer λ to a positive principal eigenvalue.…”

A view of the peakon world through the lens of approximation theory

“…The periodic peakon problem has also been studied more recently by Eckhardt and Kostenko [102], and in fact the whole subject of forward and inverse spectral problems related to the CH equation has been greatly enriched in the last decade by their work toghether with Teschl and other collaborators [104,100,101,103,97,98]. In the work perhaps most relevant for this article, they present a very compelling interpretation of the mechanism of the peakon-antipeakon collisions [99].…”

Minimization of lowest positive periodic eigenvalue for the Camassa–Holm equation with indefinite potential

“…When the potential m is indefinite, inverse spectral theory for (1.2) has not been sufficiently developed up to now (but see [17,19,21]). In [21], the inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation was studied by considering suitable weak solutions of (1.2) due to the lack of regularity of m. We refer to [17,18,19,22,23] for recent important results on isospectral problems and inverse spectral problems for the Camassa-Holm equation with different boundary conditions.…”