## Journals

- Advances in Mathematics of Communications
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### Open Access Journals

DCDS

Existence of large-amplitude time-periodic breathers localized near
a single site is proved for the discrete Klein--Gordon equation, in the case
when the derivative of the on-site potential has a compact support.
Breathers are obtained at small coupling
between oscillators and under nonresonance conditions.
Our method is different from the classical anti-continuum limit
developed by MacKay and Aubry, and yields in general
branches of breather solutions that cannot
be captured with this approach.
When the coupling constant goes to zero,
the amplitude and period of oscillations
at the excited site go to infinity.
Our method is based on near-identity transformations, analysis of
singular limits in nonlinear oscillator equations, and
fixed-point arguments.

DCDS-S

This paper is a first attempt to derive a qualitatively simple
model coupling the dynamics of a charged biopolymer and its
diffuse cloud of counterions. We consider here the case of a single actin filament.
A zig-zag chain model
introduced by Zolotaryuk et al [28] is used to represent the actin helix,
and calibrated using
experimental data on the stiffness constant of actin.
Starting from the continuum drift-diffusion model describing
counterion dynamics, we derive a discrete damped diffusion equation for
the quantity of ionic charges in a one-dimensional grid along actin.
The actin and ionic cloud models are coupled via electrostatic effects.
Numerical simulations of the coupled system show that mechanical waves propagating along
the polymer can generate charge density waves with intensities in the $pA$ range,
in agreement with experimental measurements of ionic currents along actin.

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