Traveling waves and localized modes in one-dimensional homogeneous granular chains with no precompression

Yuli Starosvetsky, Alexander F Vakakis

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Abstract

We study a class of strongly nonlinear traveling waves and localized modes in one-dimensional homogeneous granular chains with no precompression. Until now the only traveling-wave solutions known for this class of systems were the single-hump solitary waves studied by Nesterenko in the continuum approximation limit. Instead, 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 multihump traveling-wave solutions with arbitrary wavelengths. We develop systematic semianalytical approaches for computing different families of nonlinear traveling waves parametrized by spatial periodicity (wave number) and energy, and show that in a certain asymptotic limit, these wave families converge to the known single-hump solitary wave studied by Nesterenko. In addition, we demonstrate the existence of an additional class of stable strongly localized out-of-phase standing waves in perfectly homogeneous granular chains with no precompression or disorder. Until now such localized solutions were known to exist only in granular chains with strong precompression. Our findings indicate that homogeneous granular chains possess complex intrinsic nonlinear dynamics, including intrinsic nonlinear energy transfer and localization phenomena.

Original languageEnglish (US)
Article number026603
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume82
Issue number2
DOIs
StatePublished - Aug 20 2010

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traveling waves
Traveling Wave
Nonlinear Waves
Solitary Waves
Traveling Wave Solutions
Continuum
solitary waves
continuums
Asymptotic Limit
Standing Wave
Energy Transfer
Approximation
homogenizing
approximation
standing waves
Homogenization
Periodicity
Nonlinear Dynamics
nonlinear equations
Disorder

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics

Cite this

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abstract = "We study a class of strongly nonlinear traveling waves and localized modes in one-dimensional homogeneous granular chains with no precompression. Until now the only traveling-wave solutions known for this class of systems were the single-hump solitary waves studied by Nesterenko in the continuum approximation limit. Instead, 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 multihump traveling-wave solutions with arbitrary wavelengths. We develop systematic semianalytical approaches for computing different families of nonlinear traveling waves parametrized by spatial periodicity (wave number) and energy, and show that in a certain asymptotic limit, these wave families converge to the known single-hump solitary wave studied by Nesterenko. In addition, we demonstrate the existence of an additional class of stable strongly localized out-of-phase standing waves in perfectly homogeneous granular chains with no precompression or disorder. Until now such localized solutions were known to exist only in granular chains with strong precompression. Our findings indicate that homogeneous granular chains possess complex intrinsic nonlinear dynamics, including intrinsic nonlinear energy transfer and localization phenomena.",
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