The Inverse Spectral Problem for Periodic Conservative Multi-peakon Solutions of the Camassa–Holm Equation

Abstract

We solve the inverse spectral problem associated with periodic conservative multi-peakon solutions of the Camassa-Holm equation. The corresponding isospectral sets can be identified with finite dimensional tori.

A numerical study of variational discretizations of the Camassa–Holm equation

“…This system is Hamiltonian, and one of its hallmarks is that for initial data satisfying q i = q j for i = j and all p i having the same sign, one can find an explicit Lax pair, meaning the discrete system is in fact integrable. The Lax pair also serves as a starting point for studying general conservative multipeakon solutions with the help of spectral theory, see [27,28]. System (8) is however not suited as a numerical method for extending solutions beyond the collision of particles, which for instance occurs for the two peakon initial data with q 1 < q 2 , and p 2 < 0 < p 1 .…”

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

“…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.…”

“…Optimizing principal eigenvalues with indefinite measures -Proof of Theorem 1.3. Let us first consider optimization problem(12).…”

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

“…It has been shown in [1,2,4,5] that in this case, the coefficient ω in (1.5) can be recovered from the associated spectral data, which led to a representation of periodic multi-peakon solutions in terms of theta functions. However, a complete solution of the corresponding inverse spectral problem in this situation has been given only recently in [24] by considering the generalized differential equation (1.3). We point out again that this modification of the isospectral problem is in full accordance with the notion of global conservative solutions of the Camassa-Holm equation; see [27,29].…”