14 research outputs found
Gap and out-gap breathers in a binary modulated discrete nonlinear Schr\"odinger model
We consider a modulated discrete nonlinear Schr\"odinger (DNLS) model with
alternating on-site potential, having a linear spectrum with two branches
separated by a 'forbidden' gap. Nonlinear localized time-periodic solutions
with frequencies in the gap and near the gap -- discrete gap and out-gap
breathers (DGBs and DOGBs) -- are investigated. Their linear stability is
studied varying the system parameters from the continuous to the
anti-continuous limit, and different types of oscillatory and real
instabilities are revealed. It is shown, that generally DGBs in infinite
modulated DNLS chains with hard (soft) nonlinearity do not possess any
oscillatory instabilities for breather frequencies in the lower (upper) half of
the gap. Regimes of 'exchange of stability' between symmetric and antisymmetric
DGBs are observed, where an increased breather mobility is expected. The
transformation from DGBs to DOGBs when the breather frequency enters the linear
spectrum is studied, and the general bifurcation picture for DOGBs with tails
of different wave numbers is described. Close to the anti-continuous limit, the
localized linear eigenmodes and their corresponding eigenfrequencies are
calculated analytically for several gap/out-gap breather configurations,
yielding explicit proof of their linear stability or instability close to this
limit.Comment: 17 pages, 12 figures, submitted to Eur. Phys. J.
Standing wave instabilities in a chain of nonlinear coupled oscillators
We consider existence and stability properties of nonlinear spatially
periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of
coupled anharmonic oscillators. Specifically, we consider Klein-Gordon (KG)
chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site
potentials, as well as discrete nonlinear Schroedinger (DNLS) chains
approximating the small-amplitude dynamics of KG chains with weak inter-site
coupling. The SWs are constructed as exact time-periodic multibreather
solutions from the anticontinuous limit of uncoupled oscillators. In the
validity regime of the DNLS approximation these solutions can be continued into
the linear phonon band, where they merge into standard harmonic SWs. For SWs
with incommensurate wave vectors, this continuation is associated with an
inverse transition by breaking of analyticity. When the DNLS approximation is
not valid, the continuation may be interrupted by bifurcations associated with
resonances with higher harmonics of the SW. Concerning the stability, we
identify one class of SWs which are always linearly stable close to the
anticontinuous limit. However, approaching the linear limit all SWs with
nontrivial wave vectors become unstable through oscillatory instabilities,
persisting for arbitrarily small amplitudes in infinite lattices. Investigating
the dynamics resulting from these instabilities, we find two qualitatively
different regimes for wave vectors smaller than or larger than pi/2,
respectively. In one regime persisting breathers are found, while in the other
regime the system rapidly thermalizes.Comment: 57 pages, 21 figures, to be published in Physica D. Revised version:
Figs. 5 and 12 (f) replaced, some new results added to Sec. 5, Sec.7
(Conclusions) extended, 3 references adde
Asymptotic analysis of breather modes in a two-dimensional mechanical lattice
We consider a two-dimensional square lattice in which each node is restricted to the plane of the lattice, but is permitted to move in both directions of the lattice. We assume nodes are connected to nearest neighbours along the lattice directions with nonlinear springs, and to diagonal neighbours with linear springs. We consider a generalised Klein-Gordon system, that is, where there is an onsite potential at each node in addition to the (nonlinear) nearest-neighbour interactions. We derive the equations of motion for the displacements from the Hamiltonian. We use asymp-totic techniques to derive the form of small amplitude breather solutions, and find necessary conditions required for their existence. We find two types of mode, which we term 'optical' and 'acoustic', based on the analysis of other lattices which support dispersion relations with multiple branches. In addition to the usual inequality on the sign of the nonlinearity in order for the NLS to be of the focusing type, we obtain an additional ellipticity constraint, that is a restriction in the two-dimensional wavenumber space, required for the spatial differential operator to be elliptic. Highlights • we consider a 2D square lattice with in-plane motion of nodes • we use a weakly nonlinear asymptotic expansion to derive envelope equation • we find breather solutions of an associated 2D NLS • two conditions for breathers: usual focusing, additional ellipticity constrain
Discrete Breathers
Nonlinear classical Hamiltonian lattices exhibit generic solutions in the
form of discrete breathers. These solutions are time-periodic and (typically
exponentially) localized in space. The lattices exhibit discrete translational
symmetry. Discrete breathers are not confined to certain lattice dimensions.
Necessary ingredients for their occurence are the existence of upper bounds on
the phonon spectrum (of small fluctuations around the groundstate) of the
system as well as the nonlinearity in the differential equations. We will
present existence proofs, formulate necessary existence conditions, and discuss
structural stability of discrete breathers. The following results will be also
discussed: the creation of breathers through tangent bifurcation of band edge
plane waves; dynamical stability; details of the spatial decay; numerical
methods of obtaining breathers; interaction of breathers with phonons and
electrons; movability; influence of the lattice dimension on discrete breather
properties; quantum lattices - quantum breathers. Finally we will formulate a
new conceptual aproach capable of predicting whether discrete breather exist
for a given system or not, without actually solving for the breather. We
discuss potential applications in lattice dynamics of solids (especially
molecular crystals), selective bond excitations in large molecules, dynamical
properties of coupled arrays of Josephson junctions, and localization of
electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see
also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm
Discrete Breathers in One- and Two-Dimensional Lattices
Discrete breathers are time-periodic and spatially localised exact
solutions in translationally invariant nonlinear lattices. They
are generic solutions, since only moderate conditions are required
for their existence. Closed analytic forms for breather solutions
are generally not known. We use asymptotic methods to determine
both the properties and the approximate form of discrete breather
solutions in various lattices.
We find the conditions for which the one-dimensional FPU chain
admits breather solutions, generalising a known result for
stationary breathers to include moving breathers. These
conditions are verified by numerical simulations. We show that the
FPU chain with quartic interaction potential supports long-lived
waveforms which are combinations of a breather and a kink. The
amplitude of classical monotone kinks is shown to have a nonzero
minimum, whereas the amplitude of breathing-kinks can be
arbitrarily small.
We consider a two-dimensional FPU lattice with square rotational
symmetry. An analysis to third-order in the wave amplitude is
inadequate, since this leads to a partial differential equation
which does not admit stable soliton solutions for the breather
envelope. We overcome this by extending the analysis to
higher-order, obtaining a modified partial differential equation
which includes known stabilising terms. From this, we determine
regions of parameter space where breather solutions are expected.
Our analytic results are supported by extensive numerical
simulations, which suggest that the two-dimensional square FPU
lattice supports long-lived stationary and moving breather modes.
We find no restriction upon the direction in which breathers can
travel through the lattice. Asymptotic estimates for the breather
energy confirm that there is a minimum threshold energy which must
be exceeded for breathers to exist in the two-dimensional lattice.
We find similar results for a two-dimensional FPU lattice with
hexagonal rotational symmetry
Superdiffusive Transport and Energy Localization in Disordered Granular Crystals
We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to be fundamentally different from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder: an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements), and for two families of initial conditions: displacement perturbations and velocity perturbations. We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics strongly depends on the initial condition. In particular, for displacement perturbations, the long-time asymptotic behavior of the second moment ˜m2 has oscillations that depend on the type of disorder, with a complex trend that is markedly different from a power law and which is particularly evident for an Anderson-like disorder. By contrast, for velocity perturbations, we find that a standard scaling ˜m2 ∼ t γ (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, ˜m2 and the inverse participation ratio P −1 satisfy scaling relations ˜m2 ∼ t γ and P −1 ∼ t −η , and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with η \u3c 0.1) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of the wave. For an Anderson-like disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly γ ≈ 1.7 and η ≈ 1) for all three types of disorder and for both types of initial conditions
Complex Systems: Nonlinearity and Structural Complexity in spatially extended and discrete systems
Resumen Esta Tesis doctoral aborda el estudio de sistemas de muchos elementos (sistemas discretos) interactuantes. La fenomenología presente en estos sistemas esta dada por la presencia de dos ingredientes fundamentales: (i) Complejidad dinámica: Las ecuaciones del movimiento que rigen la evolución de los constituyentes son no lineales de manera que raramente podremos encontrar soluciones analíticas. En el espacio de fases de estos sistemas pueden coexistir diferentes tipos de trayectorias dinámicas (multiestabilidad) y su topología puede variar enormemente dependiendo de dos parámetros usados en las ecuaciones. La conjunción de dinámica no lineal y sistemas de muchos grados de libertad (como los que aquí se estudian) da lugar a propiedades emergentes como la existencia de soluciones localizadas en el espacio, sincronización, caos espacio-temporal, formación de patrones, etc... (ii) Complejidad estructural: Se refiere a la existencia de un alto grado de aleatoriedad en el patrón de las interacciones entre los componentes. En la mayoría de los sistemas estudiados esta aleatoriedad se presenta de forma que la descripción de la influencia del entorno sobre un único elemento del sistema no puede describirse mediante una aproximación de campo medio. El estudio de estos dos ingredientes en sistemas extendidos se realizará de forma separada (Partes I y II de esta Tesis) y conjunta (Parte III). Si bien en los dos primeros casos la fenomenología introducida por cada fuente de complejidad viene siendo objeto de amplios estudios independientes a lo largo de los últimos años, la conjunción de ambas da lugar a un campo abierto y enormemente prometedor, donde la interdisciplinariedad concerniente a los campos de aplicación implica un amplio esfuerzo de diversas comunidades científicas. En particular, este es el caso del estudio de la dinámica en sistemas biológicos cuyo análisis es difícil de abordar con técnicas exclusivas de la Bioquímica, la Física Estadística o la Física Matemática. En definitiva, el objetivo marcado en esta Tesis es estudiar por separado dos fuentes de complejidad inherentes a muchos sistemas de interés para, finalmente, estar en disposición de atacar con nuevas perspectivas problemas relevantes para la Física de procesos celulares, la Neurociencia, Dinámica Evolutiva, etc..