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Register a new userTrace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow
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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.
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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.
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.
We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.
We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic function for example). It is shown that the eigenvalues of H+V have asymptotics of the form lambda_n(H+V)=lambda_n(H)+W(sqrt n)n^{-1/4}+O(n^{-1/2}ln(n)) as nto+infty, where W is a quasi-periodic function which can be defined explicitly in terms of V.
Considered herein are the generalized Camassa-Holm and Degasperis-Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also determined. Finally, geometric descriptions of these two integrable equations from non-stretching invariant curve flows in centro-equiaffine geometries, pseudo-spherical surfaces and affine surfaces are given.
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