Guillaume James

Nonlinear analysis of the dynamics of DNA breathing.

The base pairs that encode the genetic information in DNA show large amplitude localized excitations called DNA breathing. We discuss the experimental observations of this phenomenon and its theoretical analysis. Starting from a model introduced to study the thermal denaturation of DNA, we show that it can qualitatively describe DNA breathing but is quantitatively not satisfactory. We show how the model can be modified to be quantitatively correct. This defines a nonlinear lattice model, which is interesting in itself because it has nonlinear localized excitations, forming a new class of discrete breather.

Breathers in oscillator chains with Hertzian interactions

We prove nonexistence of breathers (spatially localized and time-periodic oscilla- tions) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertzs contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newtons cradle under the effect of gravity. Using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schrodinger (DpS) equation, we show the existence of discrete breathers and study their spectral prop- erties and mobility. Due to the fully nonlinear character of Hertzian interactions, breathers are found to be much more localized than in classical nonlinear lattices and their motion occurs with less dispersion. In addition, we study numerically the excitation of a traveling breather after an impact at one end of a semi-infinite chain. This case is well described by the DpS equation when local oscillations are faster than binary collisions, a situation occuring e.g. in chains of stiff cantilevers decorated by spherical beads. When a hard anharmonic part is added to the local potential, a new type of traveling breather emerges, showing spontaneous direction-reversing in a spatially homogeneous system. Finally, the interaction of a moving breather with a point defect is also considered in the cradle system. Almost total breather reflec- tions are observed at sufficiently high defect sizes, suggesting potential applications of such systems as shock wave reflectors.

Modelling DNA at the mesoscale : a challenge for nonlinear science?

A vehicle which is capable of recognizing shapes in a predetermined area, comprising: a plurality of ultrasonic sensors, an encoder, map drawing means for sequentially and continuously drawing a map of the prescribed area determined by information received from the encoder and ultrasonic sensors, memory means for storing the map drawn by the map drawing means and control means for instructing rectilinear movement, starting stopping and turning of the vehicle so as to move in a serpentine fashion, wherein the ultrasonic sensors, encoder and map drawing means are operated by the control means, the memory means writes and stores a history of its own movements in the area and as information is received remembers detected information from the encoder, and a change of direction of the vehicle which is instructed by the control means is determined by information on the map as well as the areas through which the vehicle has previously passed.

Gaussian solitary waves and compactons in Fermi–Pasta–Ulam lattices with Hertzian potentials

We consider a class of fully nonlinear Fermi–Pasta–Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order α>1. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyse the propagation of localized waves when α is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic Korteweg–de Vries (KdV) equation and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with Hölder-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When , we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile.

Periodic Travelling Waves and Compactons in Granular Chains

We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz’s contact forces. Each bead periodically undergoes a compression phase followed by free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behavior, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wave number, we observe fast instabilities of the periodic travelling waves, leading to a disordered regime.

Approximation of solitons in the discrete NLS equation

Abstract We study four different approximations for finding the profile of discrete solitons in the one-dimensional Discrete Nonlinear Schrödinger (DNLS) Equation. Three of them are discrete approximations (namely, a variational approach, an approximation to homoclinic orbits and a Green-function approach), and the other one is a quasi-continuum approximation. All the results are compared with numerical computations.

Breather Solutions of the Discrete p-Schrödinger Equation

We consider the discrete p-Schrodinger (DpS) equation, which approximates small amplitude oscillations in chains of oscillators with fully-nonlinear nearest-neighbors interactions of order \(\alpha = p - 1> 1\). Using a mapping approach, we prove the existence of breather solutions of the DpS equation with even- or odd-parity reflectional symmetries. We derive in addition analytical approximations for the breather profiles and the corresponding intersecting stable and unstable manifolds, valid on a whole range of nonlinearity orders α. In the limit of weak nonlinearity (α → 1+), we introduce a continuum limit connecting the stationary DpS and logarithmic nonlinear Schrodinger equations. In this limit, breathers correspond asymptotically to Gaussian homoclinic solutions. We numerically analyze the stability properties of breather solutions depending on their even- or odd-parity symmetry. A perturbation of an unstable breather generally results in a translational motion (traveling breather) when α is close to unity, whereas pinning becomes predominant for larger values of α.

Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing

We study the existence of discrete breathers (time-periodic and spatially localized oscillations) in a chain of coupled nonlinear oscillators modelling the breathing of DNA. We consider a modification of the Peyrard–Bishop model introduced by Peyrard et al. [Nonlinear analysis of the dynamics of DNA breathing, J. Biol. Phys. 35 (2009), 73–89], in which the reclosing of base pairs is hindered by an energy barrier. Using a new kind of continuation from infinity, we prove for weak couplings the existence of large amplitude and low frequency breathers oscillating around a localized equilibrium, for breather frequencies lying outside resonance zones. These results are completed by numerical continuation. For resonant frequencies (with one multiple belonging to the phonon band) we numerically obtain discrete breathers superposed on a small oscillatory tail.

Floquet analysis of Kuznetsov-Ma breathers: A path towards spectral stability of rogue waves

In the present work, we aim at taking a step towards the spectral stability analysis of Peregrine solitons, i.e., wave structures that are used to emulate extreme wave events. Given the space-time localized nature of Peregrine solitons, this is a priori a nontrivial task. Our main tool in this effort will be the study of the spectral stability of the periodic generalization of the Peregrine soliton in the evolution variable, namely the Kuznetsov-Ma breather. Given the periodic structure of the latter, we compute the corresponding Floquet multipliers, and examine them in the limit where the period of the orbit tends to infinity. This way, we extrapolate towards the stability of the limiting structure, namely the Peregrine soliton. We find that multiple unstable modes of the background are enhanced, yet no additional unstable eigenmodes arise as the Peregrine limit is approached. We explore the instability evolution also in direct numerical simulations.

Breathers in FPU systems, near and far from the phonon band

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