52 research outputs found
Exact stationary solutions of the parametrically driven and damped nonlinear Dirac equation
Two exact stationary soliton solutions are found in the parametrically driven and damped nonlinear Dirac equation. The parametric force
considered is a complex ac force. The solutions appear when their frequencies are locked to half the frequency of the parametric force, and
their phases satisfy certain conditions depending on the force amplitude and on the damping coe cient. Explicit expressions for the charge,
the energy, and the momentum of these solutions are provided. Their stability is studied via a variational method using an ansatz with only
two collective coordinates. Numerical simulations con rm that one of the solutions is stable, while the other is an unstable saddle point.
Consequently, the stabilization of damped Dirac solitons can be achieved via time-periodic parametric excitations.Junta de Andalucía and Ministerio de Economía y Competitividad of Spain FIS2017-89349-PMinisterio de Ciencia, Innovación y Universidades of Spain PGC2018-093998-BI0
Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation
In this paper we study numerically existence and stability of exact dark waves on the (nonintegrable) discrete nonlinear Schrödinger equation for a finite one-dimensional lattice. These are solutions that bifurcate from stationary dark modes with constant background intensity and zero intensity at a site, and whose initial state translates exactly one site each period of the internal oscillations. We show that exact dark waves are characterized by an oscillatory background whose wavelength is closely related with the velocity. Faster dark waves require smaller wavelengths. For slow enough velocity dark waves are linearly stable, but when trying to continue numerically a solution towards higher velocities bifurcations appear, due to rearrangements in the oscillatory tail in order to make possible a decreasing of the wavelength. However, in principle, one might control the stability of an exact dark wave adjusting a phase factor which plays the role of a discreteness parameter. In addition, we also study the regimes of existence and stability for stationary discrete gray modes, which are exact solutions with phase-twisted constant-amplitude background and nonzero minimum intensity. Also such solutions develop envelope oscillations on top of the homogeneous background when continued into moving phase-twisted solutions.Ministerio de Ciencia y Tecnología of Spain under Grant No. BFM2003-0301
Hysteresis in vibrated granular media
Some general dynamical properties of models for compaction of granular media based on master equations are analyzed. In particular, a one-dimensional lattice model with short-ranged dynamical constraints is considered. The stationary state is consistent with Edward's theory of powders. The system is submitted to processes in which the tapping strength is monotonically increased and decreased. In such processes the behavior of the model resembles the reversible–irreversible branches which have been recently observed in experiments. This behavior is understood in terms of the general dynamical properties of the model, and related to the hysteresis cycles exhibited by structural glasses in thermal cycles. The existence of a “normal” solution, i.e., a special solution of the master equation which is monotonically approached by all the other solutions, plays a fundamental role in the understanding of the hysteresis effects.Dirección General de Investigación Científica y Técnica (Spain) through Grant No. PB98-112
Thermal equilibrium in Einstein's elevator
We report fully relativistic molecular-dynamics simulations that verify the
appearance of thermal equilibrium of a classical gas inside a uniformly
accelerated container. The numerical experiments confirm that the local
momentum distribution in this system is very well approximated by the J\"uttner
function -- originally derived for a flat spacetime -- via the Tolman-Ehrenfest
effect. Moreover, it is shown that when the acceleration or the container size
is large enough, the global momentum distribution can be described by the
so-called modified J\"uttner function, which was initially proposed as an
alternative to the J\"uttner function
Ratchet effect in solids: Defect transport driven by biharmonic forces
PósterInterstitials and vacancies, in one-dimensional lattices, are
point defects that can be modelled by means of kinks or antikinks in a
discrete Frenkel-Kontorova model, with a sine-Gordon on-site potential.
The properties of kinks and antikinks are the same if a harmonic interaction
potential is considered. The ratchet properties of these defects in
the above mentioned context has been studied by Zolotaryuk and Salerno
when the system is driven by a biharmonic field. The properties of these
solutions are strongly altered when an anharmonic interaction potential
is introduced in the model, as the Peierls-Nabarro barrier is higher for
antikinks than for kinks. The aim of this poster is to show the effects of
the anharmoniciy of the interaction potential in the properties of kinks
and antikinks focusing in the assymetry between the properties of these
two species of topological solitonsMinisterio de Educación y Ciencia FIS2006-27277-
Comment on "Soliton ratchets induced by excitation of internal modes"
Very recently Willis et al. [Phys. Rev. E {\bf 69}, 056612 (2004)] have used
a collective variable theory to explain the appearance of a nonzero energy
current in an ac driven, damped sine-Gordon equation. In this comment, we prove
rigorously that the time-averaged energy current in an ac driven nonlinear
Klein-Gordon system is strictly zero.Comment: 3 pages, 1 figure, Submitted to Phys. Rev.
Breathers in inhomogeneous nonlinear lattices: an analysis via centre manifold reduction
We consider an infinite chain of particles linearly coupled to their nearest
neighbours and subject to an anharmonic local potential. The chain is assumed
weakly inhomogeneous. We look for small amplitude discrete breathers. The
problem is reformulated as a nonautonomous recurrence in a space of
time-periodic functions, where the dynamics is considered along the discrete
spatial coordinate. We show that small amplitude oscillations are determined by
finite-dimensional nonautonomous mappings, whose dimension depends on the
solutions frequency. We consider the case of two-dimensional reduced mappings,
which occurs for frequencies close to the edges of the phonon band. For an
homogeneous chain, the reduced map is autonomous and reversible, and
bifurcations of reversible homoclinics or heteroclinic solutions are found for
appropriate parameter values. These orbits correspond respectively to discrete
breathers, or dark breathers superposed on a spatially extended standing wave.
Breather existence is shown in some cases for any value of the coupling
constant, which generalizes an existence result obtained by MacKay and Aubry at
small coupling. For an inhomogeneous chain the study of the nonautonomous
reduced map is in general far more involved. For the principal part of the
reduced recurrence, using the assumption of weak inhomogeneity, we show that
homoclinics to 0 exist when the image of the unstable manifold under a linear
transformation intersects the stable manifold. This provides a geometrical
understanding of tangent bifurcations of discrete breathers. The case of a mass
impurity is studied in detail, and our geometrical analysis is successfully
compared with direct numerical simulations
Regular and chaotic transport of discrete solitons in asymmetric potentials
Ratchet dynamics of topological solitons of the forced and damped discrete double sine-Gordon system are studied. Directed transport occurring both in regular and in chaotic regions of the phase space and its dependence on damping, amplitude, and frequency of the driving, asymmetry parameter, and coupling constant, has been extensively investigated. We show that the passage from ratchet phase-locked regime to chaotic ratchets occurs via a period doubling route to chaos and that, quite surprisingly, pinned states can exist inside phase locking and chaotic transport regions for intermediate values of the coupling constant. The possibility to control chaotic discrete soliton ratchets by means of both small subharmonic signals and more general periodic drivings, has also been investigated.MICINN (Project No. FIS2008-04848)MICINN (Project No. FIS2008-02873
Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity
The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter κ is analyzed, when the external force is periodic in space and given by f(x) = r cos(Kx), both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force λ = 2π/K, and the soliton width ls, which is a function of the nonlinearity κ as well as the initial frequency ω0 of the solitary wave, there is a transition (at fixed ω0) from trapped to unbound behavior of the soliton, which depends on the parameters r and K of the external force and the nonlinearity parameter κ. We previously studied this phenomena when κ = 1 (2019 J. Phys. A: Math. Theor. 52 285201) where we showed that for λ ≫ ls the soliton oscillates in an effective potential, while for λ ≪ ls it moves uniformly as a free particle. In this paper we focus on the κ dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of r and K when κ = 1/2,1,2 at fixed value of the frequency ω0. Since the solitary wave gets narrower for fixed ω0 as a function of κ, we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase κ.Ministerio de Economía y Competitividad of Spain FIS2017-89349-PMinisterio de Ciencia, Innovación y Universidades of Spain PGC2018-093998-B-I0
Soliton ratchet induced by random transitions among symmetric sine-Gordon potentials
The generation of net solitonmotion induced by randomtransitions among N symmetric phase-shifted sine-Gordon potentials is investigated,
in the absence of any external force and without any thermal noise. The phase shifts of the potentials and the damping coe cients depend
on a stationary Markov process. Necessary conditions for the existence of transport are obtained by an exhaustive study of the symmetries
of the stochastic system and of the soliton velocity. It is shown that transport is generated by unequal transfer rates among the phase-shifted
potentials or by unequal friction coe cients or by a properly devised combination of potentials (N > 2). Net motion and inversions of the
currents, predicted by the symmetry analysis, are observed in simulations as well as in the solutions of a collective coordinate theory. A model
with high e cient solitonmotion is designed by usingmultistate phase-shifted potentials and by breaking the symmetrieswith unequal transfer
rates.Junta de Andalucía / Ministerio de Economía y Competitividad of Spain FIS2017-86478-PJunta de Andalucía / Ministerio de Economía y Competitividad of Spain FIS2017-89349-PConsejería de Conocimiento, Investigación y Universidad, Junta de Andalucía, and European Regional Development Fund (ERDF) SOMM17/6105/UG
- …