An Isospectral Problem for Global Conservative Multi-Peakon Solutions of the Camassa–Holm Equation

Abstract

Abstract. We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the conservative Camassa-Holm equation is integrable by the inverse spectral transform in the multi-peakon case.

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

“…for every Borel set B ⊆ R. Due to the low regularity of the coefficients, this differential equation has to be understood in a distributional sense in general (see Section 2 below for details). The relevance of the spectral problem (1.3) stems from the fact that it serves as an isospectral problem for the conservative Camassa-Holm flow; see [17,20,21]. Virtually the entire literature dedicated to the periodic isospectral problem for the conservative Camassa-Holm flow deals with the case when the measure υ vanishes identically and the coefficient ω obeys additional smoothness or positivity restrictions.…”

Quadratic operator pencils associated with the conservative Camassa-Holm flow

“…The significance of this rather specific spectral problem stems from the fact that it arises as the isospectral problem of a particular completely integrable nonlinear wave equation. More precisely, it has been identified as an isospectral problem for the two-component Camassa-Holm system [11,32] and it turned out recently [22] that it also serves as an isospectral problem for global conservative solutions of the Camassa-Holm equation [9,29,31].…”

“…Since the coefficient ω is allowed to change sign and because of the presence of the measure υ, spectral theory for (1.1) is outside of most standard theory for Sturm-Liouville problems and requires distinct methods to deal with it. In particular, direct and inverse spectral theory for (1.1) is still not sufficiently developed for applications to the Camassa-Holm flow (but see [3,5,6,7,12,14,21,22,23,27]). Moreover, except for [22], all of these references only deal with the case when the measure υ is not present at all.…”

“…In particular, direct and inverse spectral theory for (1.1) is still not sufficiently developed for applications to the Camassa-Holm flow (but see [3,5,6,7,12,14,21,22,23,27]). Moreover, except for [22], all of these references only deal with the case when the measure υ is not present at all. However, let us also mention that problems similar to (1.1) have been studied in [42,43,44,45] in the context of indefinite strings, where the authors dealt with the spectral problem in a Krein space setting.…”

The Schur algorithm for an indefinite Stieltjes moment problem

“…In particular, they noticed that in the indefinite case, the inverse spectral problem for multi‐peakon solutions of the Camassa–Holm equation is not always solvable in the class of such continued fractions. In it was shown that such inverse spectral problem is solvable in the class of continued fractions of the form with polynomials of formal degree 1 (, ). This result is in the full correspondence with the description of solutions of the Stieltjes indefinite moment problem given in .…”