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Register a new userThe inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation
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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.
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This article is concerned with the isospectral problem [ -f + frac{1}{4} f = zomega f + z^2 upsilon f ] for the periodic conservative Camassa-Holm flow, where $omega$ is a periodic real distribution in $H^{-1}_{mathrm{loc}}(mathbb{R})$ and $upsilon$ is a periodic non-negative Borel measure on $mathbb{R}$. 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$^ast$ topology.
It is well-known that by requiring solutions of the Camassa-Holm equation to satisfy a particular local conservation law for the energy in the weak sense, one obtains what is known as conservative solutions. As conservative solutions preserve energy, one might be inclined to think that any solitary traveling wave is conservative. However, in this paper we prove that the traveling waves known as stumpons are not conservative. We illustrate this result by comparing the stumpon to simulations produced by a numerical scheme for conservative solutions, which has been recently developed by Galtung and Raynaud.
In this paper we review the recent progress in the (indefinite) string density problem and its applications to the Camassa--Holm equation.
The soliton solutions of the Camassa-Holm equation are derived by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by other methods.
In this paper, we study one of generalized Heisenberg ferromagnet equations with self-consistent sources, namely, the so-called M-CIV equation with self-consistent sources (M-CIVESCS). The Lax representation of the M-CIVESCS is presented. We have shown that the M-CIVESCS and the CH equation with self-consistent sources (CHESCS) is geometrically equivalent each to other. The gauge equivalence between these equations is proved. Soliton (peakon) and pseudo-spherical surfaces induced by these equations are considered. The one peakon solution of the M-CIVESCS is presented.
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