47 research outputs found
New Solutions for Slow Moving Kinks in a Forced Frenkel-Kontorova Chain
We construct new traveling wave solutions of moving kink type for a modified, driven, dynamic Frenkel-Kontorova model, representing dislocation motion under stress. Formal solutions known so far are inadmissible for velocities below a thresh- old value. The new solutions fill the gap left by this loss of admissibility. Analytical and numerical evidence is presented for their existence; however, dynamic simula- tions suggest that they are probably unstable
An energy-based stability criterion for solitary traveling waves in Hamiltonian lattices
In this work, we revisit a criterion, originally proposed in [Nonlinearity
{\bf 17}, 207 (2004)], for the stability of solitary traveling waves in
Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the
implications of this criterion from the point of view of stability theory, both
at the level of the spectral analysis of the advance-delay differential
equations in the co-traveling frame, as well as at that of the Floquet problem
arising when considering the traveling wave as a periodic orbit modulo a shift.
We establish the correspondence of these perspectives for the pertinent
eigenvalue and Floquet multiplier and provide explicit expressions for their
dependence on the velocity of the traveling wave in the vicinity of the
critical point. Numerical results are used to corroborate the relevant
predictions in two different models, where the stability may change twice. Some
extensions, generalizations and future directions of this investigation are
also discussed
Nucleation and propagation of phase mixtures in a bistable chain
This item is copyrighted by the American physical society, and can be found electronically from http://journals.aps.org/prb/abstract/10.1103/PhysRevB.79.144123.We consider a prototypical discrete model of phase transitions. The model consists of a chain of particles, each interacting with its nearest and next-to-nearest neighbors. The long-range interaction between next-to-nearest neighbors is assumed to be harmonic, while the nearest-neighbor interactions are nonlinear and bistable. We consider overdamped dynamics of the chain and after suitable rescaling obtain a discrete reaction-diffusion equation with a negative diffusion coefficient. Using a biquadratic nearest-neighbor interaction potential and introducing new variables, we construct and study traveling-wave-like solutions that describe dynamics of phase mixtures in the lattice. Depending on the value of the applied force, phase mixtures either get trapped in one of the multiple equilibrium states or propagate through the chain at a constant speed. At low velocities near the depinning threshold, the motion is of stick-slip type. Numerical results for smoother potentials also suggest the existence and stability of the steady motion in a certain range of applied loads
Universality classes of transition fronts in the FPU model
Steady transition fronts in nonlinear lattices are among the most important
dynamic coherent structures. We use the Fermi-Pasta-Ulam model with piecewise
linear nonlinearity to show that there are exactly three distinct classes of
such fronts which differ fundamentally in how (and whether) they produce and
transport oscillations. To make this Hamiltonian problem analytically
transparent, we construct a quasicontinuum approximation generating all three
types of fronts and then show that the interconnection between different
classes of fronts in the original discrete model is the same as in the
quasicontinuum model. The proposed framework unifies previous attempts to
classify the transition fronts as radiative, dispersive, topological or
compressive and categorizes them instead as different types of dynamic defects
Nonlinear Waves in a Strongly Nonlinear Resonant Granular Chain
We explore a recently proposed locally resonant granular system bearing harmonic internal resonators in a chain of beads interacting via Hertzian elastic contacts. In this system, we propose the existence of two types of configurations: (a) small-amplitude periodic traveling waves and (b) dark-breather solutions, i.e., exponentially localized, time periodic states mounted on top of a non-vanishing background. We also identify conditions under which the system admits long-lived bright breather solutions. Our results are obtained by means of an asymptotic reduction to a suitably modified version of the so-called discrete p-Schrödinger (DpS) equation, which is established as controllably approximating the solutions of the original system for large but finite times (under suitable assumptions on the solution amplitude and the resonator mass). The findings are also corroborated by detailed numerical computations. A remarkable feature distinguishing our results from other settings where dark breathers are observed is the complete absence of precompression in the system, i.e., the absence of a linear spectral band
Moving discrete breathers in a β-FPU lattice revisited
In this work we revisit the existence, stability and dynamical properties of moving discrete breathers in -FPU lattices. On the existence side, we propose a numerical procedure, based on a continuation along a sequence of velocities, that allows us to systematically construct breathers traveling more than one lattice site per period. On the stability side, we explore the stability spectrum of the obtained waveforms via Floquet analysis and connect it to the energy-frequency bifurcation diagrams. We illustrate in this context examples of the energy being a multivalued function of the frequency, showcasing the coexistence of different moving breathers at the same frequency. Finally, we probe the moving breather dynamics and observe how the associated instabilities change their speed, typically slowing them down over long-time simulations
Discrete breathers in a mechanical metamaterial
We consider a previously experimentally realized discrete model that describes a mechanical metamaterial consisting of a chain of pairs of rigid units connected by flexible hinges. Upon analyzing the linear band structure of the model, we identify parameter regimes in which this system may possess discrete breather solutions with frequencies inside the gap between optical and acoustic dispersion bands. We compute numerically exact solutions of this type for several different parameter regimes and investigate their properties and stability. Our findings demonstrate that upon appropriate parameter tuning within experimentally tractable ranges, the system exhibits a plethora of discrete breathers, with multiple branches of solutions that feature period-doubling and symmetry-breaking bifurcations, in addition to other mechanisms of stability change such as saddle-center and Hamiltonian Hopf bifurcations. The relevant stability analysis is corroborated by direct numerical computations examining the dynamical properties of the system and paving the way for potential further experimental exploration of this rich nonlinear dynamical lattice setting
Nonlinear waves in a strongly nonlinear resonant granular chain
We explore a recently proposed locally resonant granular system bearing harmonic internal resonators in a chain of beads interacting via Hertzian elastic contacts. In this system, we propose the existence of two types of configurations: (a) small-amplitude periodic traveling waves and (b) dark-breather solutions, i.e., exponentially localized, time periodic states mounted on top of a non-vanishing background. We also identify conditions under which the system admits long-lived bright breather solutions. Our results are obtained by means of an asymptotic reduction to a suitably modified version of the so-called discrete p-Schrödinger (DpS) equation, which is established as controllably approximating the solutions of the original system for large but finite times (under suitable assumptions on the solution amplitude and the resonator mass). The findings are also corroborated by detailed numerical computations. A remarkable feature distinguishing our results from other settings where dark breathers are observed is the complete absence of precompression in the system, i.e., the absence of a linear spectral band
Beyond Kinetic Relations
We introduce the concept of kinetic equations representing a natural
extension of the more conventional notion of a kinetic relation. Algebraic
kinetic relations, widely used to model dynamics of dislocations, cracks and
phase boundaries, link the instantaneous value of the velocity of a defect with
an instantaneous value of the driving force. The new approach generalizes
kinetic relations by implying a relation between the velocity and the driving
force which is nonlocal in time. To make this relations explicit one needs to
integrate the system of kinetic equations. We illustrate the difference between
kinetic relation and kinetic equations by working out in full detail a
prototypical model of an overdamped defect in a one-dimensional discrete
lattice. We show that the minimal nonlocal kinetic description containing now
an internal time scale is furnished by a system of two ordinary differential
equations coupling the spatial location of defect with another internal
parameter that describes configuration of the core region.Comment: Revised version, 33 pages, 9 figure