599 research outputs found
Degasperis-Procesi peakons and the discrete cubic string
We use an inverse scattering approach to study multi-peakon solutions of the
Degasperis-Procesi (DP) equation, an integrable PDE similar to the Camassa-Holm
shallow water equation. The spectral problem associated to the DP equation is
equivalent under a change of variables to what we call the cubic string
problem, which is a third order non-selfadjoint generalization of the
well-known equation describing the vibrational modes of an inhomogeneous string
attached at its ends. We give two proofs that the eigenvalues of the cubic
string are positive and simple; one using scattering properties of DP peakons,
and another using the Gantmacher-Krein theory of oscillatory kernels.
For the discrete cubic string (analogous to a string consisting of n point
masses) we solve explicitly the inverse spectral problem of reconstructing the
mass distribution from suitable spectral data, and this leads to explicit
formulas for the general n-peakon solution of the DP equation. Central to our
study of the inverse problem is a peculiar type of simultaneous rational
approximation of the two Weyl functions of the cubic string, similar to
classical Pade-Hermite approximation but with lower order of approximation and
an additional symmetry condition instead. The results obtained are intriguing
and nontrivial generalizations of classical facts from the theory of Stieltjes
continued fractions and orthogonal polynomials.Comment: 58 pages, LaTeX with AMS packages, to appear in International
Mathematics Research Paper
Dynamics of interlacing peakons (and shockpeakons) in the Geng-Xue equation
We consider multipeakon solutions, and to some extent also multishockpeakon
solutions, of a coupled two-component integrable PDE found by Geng and Xue as a
generalization of Novikov's cubically nonlinear Camassa-Holm type equation. In
order to make sense of such solutions, we find it necessary to assume that
there are no overlaps, meaning that a peakon or shockpeakon in one component is
not allowed to occupy the same position as a peakon or shockpeakon in the other
component. Therefore one can distinguish many inequivalent configurations,
depending on the order in which the peakons or shockpeakons in the two
components appear relative to each other. Here we are in particular interested
in the case of interlacing peakon solutions, where the peakons alternatingly
occur in one component and in the other. Based on explicit expressions for
these solutions in terms of elementary functions, we describe the general
features of the dynamics, and in particular the asymptotic large-time
behaviour. As far as the positions are concerned, interlacing Geng-Xue peakons
display the usual scattering phenomenon where the peakons asymptotically travel
with constant velocities, which are all distinct, except that the two fastest
peakons will have the same velocity. However, in contrast to many other peakon
equations, the amplitudes of the peakons will not in general tend to constant
values; instead they grow or decay exponentially. Thus the logarithms of the
amplitudes (as functions of time) will asymptotically behave like straight
lines, and comparing these lines for large positive and negative times, one
observes phase shifts similar to those seen for the positions of the peakons.
In addition to these K+K interlacing pure peakon solutions, we also investigate
1+1 shockpeakon solutions, and collisions leading to shock formation in a 2+2
peakon-antipeakon solution.Comment: 59 pages, 6 figures. pdfLaTeX + AMS packages + hyperref + TikZ.
Changes in v2: minor typos corrected, reference list updated and enhanced
with hyperlink
The Canada Day Theorem
The Canada Day Theorem is an identity involving sums of minors
of an arbitrary symmetric matrix. It was discovered as a
by-product of the work on so-called peakon solutions of an integrable nonlinear
partial differential equation proposed by V. Novikov. Here we present another
proof of this theorem, which explains the underlying mechanism in terms of the
orbits of a certain abelian group action on the set of all -edge matchings
of the complete bipartite graph .Comment: 16 pages. pdfLaTeX + AMS packages + TikZ. Fixed a hyperlink problem
and a few typo
- …