53 research outputs found
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
Non-interlacing peakon solutions of the Geng-Xue equation
The aim of the present paper is to derive explicit formulas for arbitrary
peakon solutions of the Geng-Xue equation, a two-component generalization of
Novikov's cubically nonlinear Camassa-Holm type equation. By performing
limiting procedures on the previosly known formulas for so-called interlacing
peakon solutions, where the peakons in the two component occur alternatingly,
we turn some of the peakons into zero-amplitude "ghostpeakons", in such a way
that the remaining ordinary peakons occur in any desired configuration. We also
study the large-time asymptotics of these solutions.Comment: 133 pages, 25 figures. pdfLaTeX + AMS packages + hyperref + Tik
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
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
Quasi-Lagrangian Systems of Newton Equations
Systems of Newton equations of the form
with an integral of motion quadratic in velocities are studied. These equations
generalize the potential case (when A=I, the identity matrix) and they admit a
curious quasi-Lagrangian formulation which differs from the standard Lagrange
equations by the plus sign between terms. A theory of such quasi-Lagrangian
Newton (qLN) systems having two functionally independent integrals of motion is
developed with focus on two-dimensional systems. Such systems admit a
bi-Hamiltonian formulation and are proved to be completely integrable by
embedding into five-dimensional integrable systems. They are characterized by a
linear, second-order PDE which we call the fundamental equation. Fundamental
equations are classified through linear pencils of matrices associated with qLN
systems. The theory is illustrated by two classes of systems: separable
potential systems and driven systems. New separation variables for driven
systems are found. These variables are based on sets of non-confocal conics. An
effective criterion for existence of a qLN formulation of a given system is
formulated and applied to dynamical systems of the Henon-Heiles type.Comment: 50 pages including 9 figures. Uses epsfig package. To appear in J.
Math. Phy
Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation
Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm
equation, admitting peaked soliton (peakon) solutions, which has nonlinear
terms that are cubic, rather than quadratic. In this paper, the explicit
formulas for multipeakon solutions of Novikov's cubically nonlinear equation
are calculated, using the matrix Lax pair found by Hone and Wang. By a
transformation of Liouville type, the associated spectral problem is related to
a cubic string equation, which is dual to the cubic string that was previously
found in the work of Lundmark and Szmigielski on the multipeakons of the
Degasperis-Procesi equation.Comment: 41 pages, LaTeX + AMS packages + pstrick
Ghostpeakons and Characteristic Curves for the Camassa-Holm, Degasperis-Procesi and Novikov Equations
We derive explicit formulas for the characteristic curves associated with the
multipeakon solutions of the Camassa-Holm, Degasperis-Procesi and Novikov
equations. Such a curve traces the path of a fluid particle whose instantaneous
velocity equals the elevation of the wave at that point (or the square of the
elevation, in the Novikov case). The peakons themselves follow characteristic
curves, and the remaining characteristic curves can be viewed as paths of
'ghostpeakons' with zero amplitude; hence, they can be obtained as solutions of
the ODEs governing the dynamics of multipeakon solutions. The previously known
solution formulas for multipeakons only cover the case when all amplitudes are
nonzero, since they are based upon inverse spectral methods unable to detect
the ghostpeakons. We show how to overcome this problem by taking a suitable
limit in terms of spectral data, in order to force a selected peakon amplitude
to become zero. Moreover, we use direct integration to compute the
characteristic curves for the solution of the Degasperis-Procesi equation where
a shockpeakon forms at a peakon-antipeakon collision. In addition to the
theoretical interest in knowing the characteristic curves, they are also useful
for plotting multipeakon solutions, as we illustrate in several examples
The inverse spectral problem for the discrete cubic string
Given a measure on the real line or a finite interval, the "cubic string"
is the third order ODE where is a spectral parameter. If
equipped with Dirichlet-like boundary conditions this is a nonselfadjoint
boundary value problem which has recently been shown to have a connection to
the Degasperis-Procesi nonlinear water wave equation. In this paper we study
the spectral and inverse spectral problem for the case of Neumann-like boundary
conditions which appear in a high-frequency limit of the Degasperis--Procesi
equation. We solve the spectral and inverse spectral problem for the case of
being a finite positive discrete measure. In particular, explicit
determinantal formulas for the measure are given. These formulas generalize
Stieltjes' formulas used by Krein in his study of the corresponding second
order ODE .Comment: 24 pages. LaTeX + iopart, xypic, amsthm. To appear in Inverse
Problems (http://www.iop.org/EJ/journal/IP
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