Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod
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Publication:1384761
DOI10.1007/BF01170373zbMath0910.73036OpenAlexW1995410519MaRDI QIDQ1384761
Publication date: 20 April 1998
Published in: Acta Mechanica (Search for Journal in Brave)
Full work available at URL: https://doi.org/10.1007/bf01170373
existence of travelling wavesfar-field solutionfinite-amplitude finite-wavelength casefinite-amplitude long-wave caselinearized case
Rods (beams, columns, shafts, arches, rings, etc.) (74K10) Nonlinear elasticity (74B20) Vibrations in dynamical problems in solid mechanics (74H45)
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II. 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Cites Work
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Nonlinear axisymmetric waves in compressible hyperelastic rods: Long finite amplitude waves
- On waves in slender elastic rods
- Numerical schemes for a model for nonlinear dispersive waves
- Numerical study of the regularized long-wave equation. I: Numerical methods
- Nonlinear Waves in a Rod: Results for Incompressible Elastic Materials
- Solitary waves on nonlinear elastic rods. I
- A Spectral Method for Solving the RLW Equation
- Solitary-Wave Interactions in Elastic Rods
- On the theory of rods. I. Derivations from the three-dimensional equations
- On the theory of rods II. Developments by direct approach
- Euler operators and conservation laws of the BBM equation
- Model equations for long waves in nonlinear dispersive systems
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