scholarly journals Nonlinear Pulse Equipartition in Weakly Coupled Ordered Granular Chains With No Precompression

2013 ◽  
Vol 8 (3) ◽  
Author(s):  
Yuli Starosvetsky ◽  
M. Arif Hasan ◽  
Alexander F. Vakakis
Keyword(s):  
Nonlinear Dynamics ◽  
Solitary Waves ◽  
Practical Significance ◽  
Drastic Reduction ◽  
Strongly Nonlinear ◽  
Localized State ◽  
Weakly Coupled ◽  
Granular Chains ◽  
Initial Impulse ◽  
Primary Pulse

We report on the strongly nonlinear dynamics of an array of weakly coupled, noncompressed, parallel granular chains subject to a local initial impulse. The motion of the granules in each chain is constrained to be in one direction that coincides with the orientation of the chain. We show that in spite of the fact that the applied impulse is applied to one of the granular chains, the resulting pulse that initially propagates only in the excited chain gets gradually equipartitioned between its neighboring chains and eventually in all chains of the array. In particular, the initially strongly localized state of energy distribution evolves towards a final stationary state of formation of identical solitary waves that propagate in each one of the chains. These solitary waves are synchronized and have identical speeds. We show that the phenomenon of primary pulse equipartition between the weakly coupled granular chains can be fully reproduced in coupled binary models that constitute a significantly simpler model that captures the main qualitative features of the dynamics of the granular array. The results reported herein are of major practical significance since it indicates that the weakly coupled array of granular chains is a medium in which an initially localized excitation gets gradually defocused, resulting in drastic reduction of propagating pulses as they are equipartitioned among all chains.

Strongly Nonlinear Beat Phenomena and Energy Exchanges in Weakly Coupled Granular Chains on Elastic Foundations

2012 ◽  
Vol 72 (1) ◽  
pp. 337-361 ◽  
Author(s):  
Yuli Starosvetsky ◽  
M. Arif Hasan ◽  
Alexander F. Vakakis ◽  
Leonid I. Manevitch
Keyword(s):  
Elastic Foundations ◽  
Strongly Nonlinear ◽  
Weakly Coupled ◽  
Granular Chains ◽  
Energy Exchanges

Nonlinear Dynamics of Granular Chains

2010 ◽  
Author(s):  
Yuli Starosvetsky ◽  
Alexander F. Vakakis
Keyword(s):  
Nonlinear Dynamics ◽  
Traveling Waves ◽  
Equations Of Motion ◽  
Traveling Wave Solutions ◽  
Strongly Nonlinear ◽  
Nonlinear Energy Transfer ◽  
Granular Chains ◽  
Spatial Periodicity ◽  
Analytical Approaches ◽  
Nonlinear Energy

We study strongly nonlinear traveling waves in one-dimensional granular chains with no pre-compression. We directly study the discrete, strongly nonlinear governing equations of motion of these media without resorting to continuum approximations or homogenization, which enables us to compute families of stable multi-hump traveling wave solutions with arbitrary wavelengths. We develop systematic semi–analytical approaches for computing different families of nonlinear traveling waves parametrized by spatial periodicity (wavenumber) and energy. Our findings indicate that homogeneous granular chains possess complex nonlinear dynamics, including the capacity for intrinsic nonlinear energy transfer.

A regularized model for strongly nonlinear internal solitary waves

2009 ◽  
Vol 629 ◽  
pp. 73-85 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
RICARDO BARROS ◽  
TAE-CHANG JO
Keyword(s):  
Stability Analysis ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Wave Solution ◽  
Wave Amplitude ◽  
Solitary Wave Solution ◽  
Shear Instability ◽  
Internal Solitary Waves ◽  
Strongly Nonlinear ◽  
Regularized Model

The strongly nonlinear long-wave model for large amplitude internal waves in a two-layer system is regularized to eliminate shear instability due to the wave-induced velocity jump across the interface. The model is written in terms of the horizontal velocities evaluated at the top and bottom boundaries instead of the depth-averaged velocities, and it is shown through local stability analysis that internal solitary waves are locally stable to perturbations of arbitrary wavelengths if the wave amplitudes are smaller than a critical value. For a wide range of depth and density ratios pertinent to oceanic conditions, the critical wave amplitude is close to the maximum wave amplitude and the regularized model is therefore expected to be applicable to the strongly nonlinear regime. The regularized model is solved numerically using a finite-difference method and its numerical solutions support the results of our linear stability analysis. It is also shown that the solitary wave solution of the regularized model, found numerically using a time-dependent numerical model, is close to the solitary wave solution of the original model, confirming that the two models are asymptotically equivalent.

scholarly journals Microtubules: A network for solitary waves

2017 ◽  
Vol 82 (5) ◽  
pp. 469-481 ◽  
Author(s):  
Slobodan Zdravkovic
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Solitary Waves ◽  
Discrete Approximations ◽  
Mathematical Methods ◽  
In Cells

In the present paper we deal with nonlinear dynamics of microtubules. The structure and role of microtubules in cells are explained as well as one of models explaining their dynamics. Solutions of the crucial nonlinear differential equation depend on used mathematical methods. Two commonly used procedures, continuum and semi-discrete approximations, are explained. These solutions are solitary waves usually called as kink solitons, breathers and bell-type solitons.

scholarly journals Nonlinear dynamics of coupled transverse-rotational waves in granular chains

2019 ◽  
Vol 100 (6) ◽  
Author(s):  
Qicheng Zhang ◽  
Olga Umnova ◽  
Rodolfo Venegas
Keyword(s):  
Nonlinear Dynamics ◽  
Granular Chains ◽  
Rotational Waves

Strongly Nonlinear Spatially Periodic Traveling Waves in Granular Dimer Chains

2012 ◽  
Author(s):  
K. R. Jayaprakash ◽  
Alexander F. Vakakis ◽  
Yuli Starosvetsky
Keyword(s):  
Traveling Waves ◽  
Periodic Boundary Conditions ◽  
Mass Ratio ◽  
Efficient Energy ◽  
Periodic Traveling Waves ◽  
Strongly Nonlinear ◽  
Spatial Periodicity ◽  
Spatially Periodic ◽  
New Formulation ◽  
Primary Pulse

In the present work we study the dynamics of spatially periodic traveling waves in granular 1:1 (each bead is followed and preceded by a bead of different mass and/or stiffness) dimer chain with no pre-compression. The dynamics of a 1:1 dimer chain is governed by a single parameter, the mass ratio of the two beads forming each dimer pair of the chain. In particular, we demonstrate numerically the formation of special families of traveling waves with spatially periodic waveforms that are realized in semi-infinite dimer chains with the application of an arbitrary impulse. These traveling waves were first observed in the form of oscillatory tails in the trail of the propagating primary pulse. The energy radiated by the propagating primary pulse manifests in the form of traveling waves of varying spatial periodicity depending on the mass ratio. These traveling waves depend only on the mass ratio and are rescalable with respect to any arbitrary applied energy. The dynamics of these families of traveling waves is systematically studied by considering finite dimer chains (termed the ‘reduced systems’) subject to periodic boundary conditions. We demonstrate that these waves may exhibit interesting bifurcations or loss of stability as the system parameter varies. In turn, these bifurcations and stability exchanges in infinite dimer chains are correlated to previous studies of pulse attenuation in finite dimer chains through efficient energy radiation from the propagating pulse to the far field, mainly in the form of traveling waves. Based on these results a new formulation of attenuation and propagation zones (stop and pass bands) in semi-infinite granular dimer chains is proposed.

scholarly journals The Nonlinear Dynamics of Time-Dependent Subcritical Baroclinic Currents

2007 ◽  
Vol 37 (4) ◽  
pp. 1001-1021 ◽  
Author(s):  
G. R. Flierl ◽  
J. Pedlosky
Keyword(s):  
Nonlinear Dynamics ◽  
Small Degree ◽  
Mean Value ◽  
Time Dependent ◽  
Unstable Wave ◽  
Amplitude Dependence ◽  
Strongly Nonlinear ◽  
System A ◽  
Turbulent Characteristics ◽  
Baroclinic Currents

Abstract The nonlinear dynamics of baroclinically unstable waves in a time-dependent zonal shear flow is considered in the framework of the two-layer Phillips model on the beta plane. In most cases considered in this study the amplitude of the shear is well below the critical value of the steady shear version of the model. Nevertheless, the time-dependent problem in which the shear oscillates periodically is unstable, and the unstable waves grow to substantial amplitudes, in some cases with strongly nonlinear and turbulent characteristics. For very small values of the shear amplitude in the presence of dissipation an analytical, asymptotic theory predicts a self-sustained wave whose amplitude undergoes a nonlinear oscillation whose period is amplitude dependent. There is a sensitive amplitude dependence of the wave on the frequency of the oscillating shear when the shear amplitude is small. This behavior is also found in a truncated model of the dynamics, and that model is used to examine larger shear amplitudes. When there is a mean value of the shear in addition to the oscillating component, but such that the total shear is still subcritical, the resulting nonlinear states exhibit a rectified horizontal buoyancy flux with a nonzero time average as a result of the instability of the oscillating shear. For higher, still subcritical, values of the shear, a symmetry breaking is detected in which a second cross-stream mode is generated through an instability of the unstable wave although this second mode would by itself be stable on the basic time-dependent current. For shear values that are substantially subcritical but of order of the critical shear, calculations with a full quasigeostrophic numerical model reveal a turbulent flow generated by the instability. If the beta effect is disregarded, the inviscid, linear problem is formally stable. However, calculations show that a small degree of nonlinearity is enough to destabilize the flow, leading to large amplitude vacillations and turbulence. When the most unstable wave is not the longest wave in the system, a cascade up scale to longer waves is observed. Indeed, this classically subcritical flow shows most of the qualitative character of a strongly supercritical flow. This result supports previous suggestions of the important role of background time dependence in maintaining the atmospheric and oceanic synoptic eddy field.

Traveling and solitary waves in monodisperse and dimer granular chains

2017 ◽  
Vol 31 (10) ◽  
pp. 1742001 ◽  
Author(s):  
Yuli Starosvetsky ◽  
K. R. Jayaprakash ◽  
Alexander F. Vakakis
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Traveling Waves ◽  
Granular Media ◽  
Equations Of Motion ◽  
Continuum Approximation ◽  
Granular Chains ◽  
Solitary Pulse ◽  
Countably Infinite ◽  
Dimer Granular Chains

We provide a review of propagating traveling waves and solitary pulses in uncompressed one-dimensional ([Formula: see text]) ordered granular media. The first such solution in homogeneous granular media was discovered by Nesterenko in the form of a single-hump solitary pulse with energy-dependent profile and velocity. Considering directly the discrete, strongly nonlinear governing equations of motion of these media (i.e., without resorting to continuum approximation or homogenization), we show the existence of countably infinite families of stable multi-hump propagating traveling waves with arbitrary wavelengths. A semi-analytical approach is used to study the dependence of these waves on spatial periodicity (wavenumber) and energy, and to show that in a certain asymptotic limit, these families converge to the single-hump Nesterenko solitary wave. Then the study is extended in dimer granular chains composed of alternating “heavy” and “light” beads. For a set of specific mass ratios between the light and heavy beads, we show the existence of multi-hump solitary waves that propagate faster than the Nesterenko solitary wave in the corresponding homogeneous granular chain composed of only heavy beads. The existence of these waves has interesting implications in energy transmission in ordered granular chains.

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