49,940 research outputs found
Chaos in the one-dimensional wave equation
AbstractThis paper deals with the chaotic behavior of the solutions of a mixed problem for the one-dimensional wave equation with a quadratic boundary condition. This behavior is studied through the connection between the energy function and quadratic discrete dynamical systems
Characteristics of chaos evolution in one-dimensional disordered nonlinear lattices
We numerically investigate the characteristics of chaos evolution during wave
packet spreading in two typical one-dimensional nonlinear disordered lattices:
the Klein-Gordon system and the discrete nonlinear Schr\"{o}dinger equation
model. Completing previous investigations \cite{SGF13} we verify that chaotic
dynamics is slowing down both for the so-called `weak' and `strong chaos'
dynamical regimes encountered in these systems, without showing any signs of a
crossover to regular dynamics. The value of the finite-time maximum Lyapunov
exponent decays in time as , with being different from the
value observed in cases of regular motion. In particular,
(weak chaos) and
(strong chaos) for both models, indicating the dynamical differences of the two
regimes and the generality of the underlying chaotic mechanisms. The
spatiotemporal evolution of the deviation vector associated with
reveals the meandering of chaotic seeds inside the wave packet, which is needed
for obtaining the chaotization of the lattice's excited part.Comment: 11 pages, 10 figure
Wave chaos as signature for depletion of a Bose-Einstein condensate
We study the expansion of repulsively interacting Bose-Einstein condensates
(BECs) in shallow one-dimensional potentials. We show for these systems that
the onset of wave chaos in the Gross-Pitaevskii equation (GPE), i.e. the onset
of exponential separation in Hilbert space of two nearby condensate wave
functions, can be used as indication for the onset of depletion of the BEC and
the occupation of excited modes within a many-body description. Comparison
between the multiconfigurational time-dependent Hartree for bosons (MCTDHB)
method and the GPE reveals a close correspondence between the many-body effect
of depletion and the mean-field effect of wave chaos for a wide range of
single-particle external potentials. In the regime of wave chaos the GPE fails
to account for the fine-scale quantum fluctuations because many-body effects
beyond the validity of the GPE are non-negligible. Surprisingly, despite the
failure of the GPE to account for the depletion, coarse grained expectation
values of the single-particle density such as the overall width of the atomic
cloud agree very well with the many-body simulations. The time dependent
depletion of the condensate could be investigated experimentally, e.g., via
decay of coherence of the expanding atom cloud.Comment: 12 pages, 10 figure
A new characterization of nonisotropic chaotic vibrations of the one-dimensional linear wave equation with a van der Pol boundary condition
AbstractThe one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations (Trans. Amer. Math. Soc. 350 (1998) 4265–4311, Internat. J. Bifur. Chaos 8 (1998) 423–445, Internat. J. Bifur. Chaos 8 (1998) 447–470, J. Math. Phys. 39 (1998) 6459–6489, Internat. J. Bifur. Chaos 12 (2002) 535–559). In this paper, we characterize nonisotropic chaotic vibration by means of the total variation theory. We obtain the classification results on the growth of the total variation of the snapshots on the spatial interval in the long-time horizon with respect to two parameters entering different regimes in R2
Spatial chaos in weakly dispersive and viscous media: a nonperturbative theory of the driven KdV-Burgers equation
The asymptotic travelling wave solution of the KdV-Burgers equation driven by
the long scale periodic driver is constructed. The solution represents a
shock-train in which the quasi-periodic sequence of dispersive shocks or
soliton chains is interspersed by smoothly varying regions. It is shown that
the periodic solution which has the spatial driver period undergoes period
doublings as the governing parameter changes. Two types of chaotic behavior are
considered. The first type is a weak chaos, where only a small chaotic
deviation from the periodic solution occurs. The second type corresponds to the
developed chaos where the solution ``ignores'' the driver period and represents
a random sequence of uncorrelated shocks. In the case of weak chaos the shock
coordinate being repeatedly mapped over the driver period moves on a chaotic
attractor, while in the case of developed chaos it moves on a repellor. Both
solutions depend on a parameter indicating the reference shock position in the
shock-train. The structure of a one dimensional set to which this parameter
belongs is investigated. This set contains measure one intervals around the
fixed points in the case of periodic or weakly chaotic solutions and it becomes
a fractal in the case of strong chaos. The capacity dimension of this set is
calculated.Comment: 32 pages, 12 PostScript figures, useses elsart.sty and boxedeps.tex,
fig.11 is not included and can be requested from <[email protected]
Extreme Events in Nonlinear Lattices
The spatiotemporal complexity induced by perturbed initial excitations
through the development of modulational instability in nonlinear lattices with
or without disorder, may lead to the formation of very high amplitude,
localized transient structures that can be named as extreme events. We analyze
the statistics of the appearance of these collective events in two different
universal lattice models; a one-dimensional nonlinear model that interpolates
between the integrable Ablowitz-Ladik (AL) equation and the nonintegrable
discrete nonlinear Schr\"odinger (DNLS) equation, and a two-dimensional
disordered DNLS equation. In both cases, extreme events arise in the form of
discrete rogue waves as a result of nonlinear interaction and rapid coalescence
between mobile discrete breathers. In the former model, we find power-law
dependence of the wave amplitude distribution and significant probability for
the appearance of extreme events close to the integrable limit. In the latter
model, more importantly, we find a transition in the the return time
probability of extreme events from exponential to power-law regime. Weak
nonlinearity and moderate levels of disorder, corresponding to weak chaos
regime, favour the appearance of extreme events in that case.Comment: Invited Chapter in a Special Volume, World Scientific. 19 pages, 9
figure
Fully 3-wave model to study the hard transition to chaotic dynamics in alfven wave-fronts
The derivative nonlinear Schrödinger (DNLS) equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. No matter how small the growth rate of the unstable wave, the four-dimensional flow for the three wave amplitudes and a relative phase, with both resistive damping and linear Landau damping, exhibits chaotic relaxation oscillations that are absent for zero growth-rate. This hard transition in phase-space behavior occurs for left-hand (LH) polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable. The parameter domain developing chaos is much broader than the corresponding domain in a reduced 3-wave model that assumes equal dampings
of the daughter wave
Chaotic shock waves of a Bose-Einstein condensate
It is demonstrated that the well-known Smale-horseshoe chaos exists in the
time evolution of the one-dimensional Bose-Einstein condensate (BEC) driven by
the time-periodic harmonic or inverted-harmonic potential. A formally exact
solution of the time-dependent Gross-Pitaevskii equation is constructed, which
describes the matter shock waves with chaotic or periodic amplitudes and
phases. When the periodic driving is switched off and the number of condensed
atoms is conserved, we obtained the exact stationary states and non-stationary
states. The former contains the stable non-propagated shock wave, and in the
latter the shock wave alternately collapses and grows for the harmonic trapping
or propagates with exponentially increased shock-front speed for the
antitrapping. It is revealed that existence of chaos play a role for
suppressing the blast of matter wave. The results suggest a method for
preparing the exponentially accelerated BEC shock waves or the stable
stationary states.Comment: 5 pages, 1 figure
Nonlinear Lattice Waves in Random Potentials
Localization of waves by disorder is a fundamental physical problem
encompassing a diverse spectrum of theoretical, experimental and numerical
studies in the context of metal-insulator transition, quantum Hall effect,
light propagation in photonic crystals, and dynamics of ultra-cold atoms in
optical arrays. Large intensity light can induce nonlinear response, ultracold
atomic gases can be tuned into an interacting regime, which leads again to
nonlinear wave equations on a mean field level. The interplay between disorder
and nonlinearity, their localizing and delocalizing effects is currently an
intriguing and challenging issue in the field. We will discuss recent advances
in the dynamics of nonlinear lattice waves in random potentials. In the absence
of nonlinear terms in the wave equations, Anderson localization is leading to a
halt of wave packet spreading.
Nonlinearity couples localized eigenstates and, potentially, enables
spreading and destruction of Anderson localization due to nonintegrability,
chaos and decoherence. The spreading process is characterized by universal
subdiffusive laws due to nonlinear diffusion. We review extensive computational
studies for one- and two-dimensional systems with tunable nonlinearity power.
We also briefly discuss extensions to other cases where the linear wave
equation features localization: Aubry-Andre localization with quasiperiodic
potentials, Wannier-Stark localization with dc fields, and dynamical
localization in momentum space with kicked rotors.Comment: 45 pages, 19 figure
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