The Inverse Spectral Transform for the Conservative Camassa–Holm Flow with Decaying Initial Data

Abstract

We establish the inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data. In particular, it is employed to prove the existence of weak solutions for the corresponding Cauchy problem.

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

“…where z is a complex spectral parameter. Of course, this differential equation has to be understood in a distributional way in general; compare [17,23,26,33]. All of the facts stated without proofs in this section may be found in [17, Appendix A], [18, Appendices A and B], [22].…”

“…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.…”

A Lagrangian View on Complete Integrability of the Two-Component Camassa–Holm System

“…where z is a complex spectral parameter. Moreover, there are good reasons (see [18,19] as well as Section 4) to expect that it also serves as an isospectral problem for global conservative solutions of the two-component Camassa-Holm system [25]. Of course, due to the low regularity of the coefficients, the differential equation (3.3) has to be understood in a distributional sense; cf.…”

“…Especially the last property attracted a lot of attention and led to the construction of different types of global weak solutions via a generalized method of characteristics [6,7,32,33,26]. For conservative solutions, another possible approach is based on the solution of an inverse problem for an indefinite Sturm-Liouville problem [3,18,19,21]; the inverse spectral method. The aim of this note is to point out some connections between these two ways of describing weak conservative solutions.…”

The Classical Moment Problem and Generalized Indefinite Strings

“…Similarly to Krein strings, we shall call such a triple (L, ω, υ) a generalized indefinite string; see [10]. In this respect, let us also mention briefly that a lot of the interest in spectral problems of the form (1.4) stems from the fact that they arise as isospectral problems for the conservative Camassa-Holm flow [2,3,6,7,8,11,15].…”