281 research outputs found
A note on the Painleve analysis of a (2+1) dimensional Camassa-Holm equation
We investigate the Painleve analysis for a (2+1) dimensional Camassa-Holm
equation. Our results show that it admits only weak Painleve expansions. This
then confirms the limitations of the Painleve test as a test for complete
integrability when applied to non-semilinear partial differential equations.Comment: Chaos, Solitons and Fractals (Accepted for publication
An integrable shallow water equation with peaked solitons
We derive a new completely integrable dispersive shallow water equation that
is biHamiltonian and thus possesses an infinite number of conservation laws in
involution. The equation is obtained by using an asymptotic expansion directly
in the Hamiltonian for Euler's equations in the shallow water regime. The
soliton solution for this equation has a limiting form that has a discontinuity
in the first derivative at its peak.Comment: LaTeX file. Figure available from authors upon reques
Tauberian identities and the connection to Wile E. Coyote physics
The application of the motion of a vertically suspended mass-spring system
released under tension is studied focusing upon the delay timescale for the
bottom mass as a function of the spring constants and masses. This
``hang-time", reminiscent of the Coyote and Road Runner cartoons, is quantified
using the far-field asymptotic expansion of the bottom mass' Laplace transform.
These asymptotics are connected to the short time mass dynamics through
Tauberian identities and explicit residue calculations. It is shown, perhaps
paradoxically, that this delay timescale is maximized in the large mass limit
of the top ``boulder". Experiments are presented and compared with the
theoretical predictions. This system is an exciting example for the teaching of
mass-spring dynamics in classes on Ordinary Differential Equations, and does
not require any normal mode calculations for these predictions
Vortex Induced Oscillations of Cylinders
This article submitted to the APS-DFD 2008 conference, accompanies the fluid
dynamics video depicting the various orientational dynamics of a hinged
cylinder suspended in a flow tank. The different behaviors displayed by the
cylinder range from steady orientation to periodic oscillation and even
autorotation. We illustrate these features using a phase diagram which captures
the observed phenomena as a function of Reynolds number and reduced inertia. A
hydrogen bubble flow visualization technique is also used to show vortex
shedding structure in the cylinder's wake which results in these oscillations.Comment: 3 page
Parametric Representation for the Multisoliton Solution of the Camassa-Holm Equation
The parametric representation is given to the multisoliton solution of the
Camassa-Holm equation. It has a simple structure expressed in terms of
determinants. The proof of the solution is carried out by an elementary theory
of determinanats. The large time asymptotic of the solution is derived with the
fomula for the phase shift. The latter reveals a new feature when compared with
the one for the typical soliton solutions. The peakon limit of the phase shift
ia also considered, showing that it reproduces the known result.Comment: 14 page
Peakons, R-Matrix and Toda-Lattice
The integrability of a family of hamiltonian systems, describing in a
particular case the motionof N ``peakons" (special solutions of the so-called
Camassa-Holm equation) is established in the framework of the -matrix
approach, starting from its Lax representation. In the general case, the
-matrix is a dynamical one and has an interesting though complicated
structure. However, for a particular choice of the relevant parameters in the
hamiltonian (the one corresponding to the pure ``peakons" case), the -matrix
becomes essentially constant, and reduces to the one pertaining to the finite
(non-periodic) Toda lattice. Intriguing consequences of such property are
discussed and an integrable time discretisation is derived.Comment: 12 plain tex page
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