Breather continuation from infinity in nonlinear oscillator chains
Guillaume James Dmitry Pelinovsky
Discrete & Continuous Dynamical Systems - A 2012, 32(5): 1775-1799 doi: 10.3934/dcds.2012.32.1775
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.
keywords: nonlocal bifurcations Discrete Klein--Gordon equation discrete breathers fixed-point arguments. asymptotic methods
Nonlinear lattice models for biopolymers: Dynamical coupling to a ionic cloud and application to actin filaments
Cynthia Ferreira Guillaume James Michel Peyrard
Discrete & Continuous Dynamical Systems - S 2011, 4(5): 1147-1166 doi: 10.3934/dcdss.2011.4.1147
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.
keywords: soliton propagation nonlinear lattice Biopolymer dynamics actin filament drift-diffusion model charge density waves. dynamical coupling to condensed ions

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