599 research outputs found
A class of equations with peakon and pulson solutions (with an Appendix by Harry Braden and John Byatt-Smith)
We consider a family of integro-differential equations depending upon a
parameter as well as a symmetric integral kernel . When and
is the peakon kernel (i.e. up to rescaling) the
dispersionless Camassa-Holm equation results, while the Degasperis-Procesi
equation is obtained from the peakon kernel with . Although these two
cases are integrable, generically the corresponding integro-PDE is
non-integrable. However,for the family restricts to the pulson family of
Fringer & Holm, which is Hamiltonian and numerically displays elastic
scattering of pulses. On the other hand, for arbitrary it is still possible
to construct a nonlocal Hamiltonian structure provided that is the peakon
kernel or one of its degenerations: we present a proof of this fact using an
associated functional equation for the skew-symmetric antiderivative of .
The nonlocal bracket reduces to a non-canonical Poisson bracket for the peakon
dynamical system, for any value of .Comment: Contribution to volume of Journal of Nonlinear Mathematical Physics
in honour of Francesco Caloger
Two-component generalizations of the Camassa-Holm equation
A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered
Inverse problems associated with integrable equations of Camassa-Holm type; explicit formulas on the real axis, I
The inverse problem which arises in the Camassa--Holm equation is revisited
for the class of discrete densities. The method of solution relies on the use
of orthogonal polynomials. The explicit formulas are obtained directly from the
analysis on the real axis without any additional transformation to a "string"
type boundary value problem known from prior works
Bridge Hopping on Conducting Polymers in Solution
Configurational fluctuations of conducting polymers in solution can bring
into proximity monomers which are distant from each other along the backbone.
Electrons can hop between these monomers across the "bridges" so formed. We
show how this can lead to (i) a collapse transition for metallic polymers, and
(ii) to the observed dramatic efficiency of acceptor molecules for quenching
fluorescence in semiconducting polymers.Comment: RevTeX 12 pages + 2 Postscript figure
Formation energy and interaction of point defects in two-dimensional colloidal crystals
The manipulation of individual colloidal particles using optical tweezers has
allowed vacancies to be created in two-dimensional (2d) colloidal crystals,
with unprecedented possibility of real-time monitoring the dynamics of such
defects (Nature {\bf 413}, 147 (2001)). In this Letter, we employ molecular
dynamics (MD) simulations to calculate the formation energy of single defects
and the binding energy between pairs of defects in a 2d colloidal crystal. In
the light of our results, experimental observations of vacancies could be
explained and then compared to simulation results for the interstitial defects.
We see a remarkable similarity between our results for a 2d colloidal crystal
and the 2d Wigner crystal (Phys. Rev. Lett. {\bf 86}, 492 (2001)). The results
show that the formation energy to create a single interstitial is
lower than that of the vacancy. Because the pair binding energies of the
defects are strongly attractive for short distances, the ground state should
correspond to bound pairs with the interstitial bound pairs being the most
probable.Comment: 5 pages, 2 figure
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