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Volume 20, Issue 1
New Expressions of Periodic Waves and a Novel Phenomenon in a Compressible Hyperelastic Rod

Zhengrong Liu & Bengong Zhang

J. Part. Diff. Eq., 20 (2007), pp. 80-96.

Published online: 2007-02

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  • Abstract

A In this paper, we employ both bifurcation method of dynamical systems and numerical exploration of differential equations to investigate the periodic waves of a general compressible hyperelastic rod equation u_t+3uu_x-u_{xxt}-ϒ(2u_xu_{xx}+uu_{xxx})=0, with parameter ϒ ‹ 0. New expressions including explicit expressions and implicit expressions are obtained. Some previous results are extended. Specially, a new phenomenon is found: when the initial value tends to a certain number, the periodic shock wave suddenly changes into a smooth periodic wave. In dynamical systems, this represents that one of orbits can pass through the singular line. The coherency of numerical simulation and theoretical derivation implies the correctness of our results.

  • Keywords

Hyperelastic rod bifurcation method numerical exploration periodic waves

  • AMS Subject Headings

34A20 34C25 35B65 58F05 76B25.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-20-80, author = {Zhengrong Liu and Bengong Zhang }, title = {New Expressions of Periodic Waves and a Novel Phenomenon in a Compressible Hyperelastic Rod}, journal = {Journal of Partial Differential Equations}, year = {2007}, volume = {20}, number = {1}, pages = {80--96}, abstract = {

A In this paper, we employ both bifurcation method of dynamical systems and numerical exploration of differential equations to investigate the periodic waves of a general compressible hyperelastic rod equation u_t+3uu_x-u_{xxt}-ϒ(2u_xu_{xx}+uu_{xxx})=0, with parameter ϒ ‹ 0. New expressions including explicit expressions and implicit expressions are obtained. Some previous results are extended. Specially, a new phenomenon is found: when the initial value tends to a certain number, the periodic shock wave suddenly changes into a smooth periodic wave. In dynamical systems, this represents that one of orbits can pass through the singular line. The coherency of numerical simulation and theoretical derivation implies the correctness of our results.

}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5295.html} }
TY - JOUR T1 - New Expressions of Periodic Waves and a Novel Phenomenon in a Compressible Hyperelastic Rod AU - Zhengrong Liu & Bengong Zhang JO - Journal of Partial Differential Equations VL - 1 SP - 80 EP - 96 PY - 2007 DA - 2007/02 SN - 20 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5295.html KW - Hyperelastic rod KW - bifurcation method KW - numerical exploration KW - periodic waves AB -

A In this paper, we employ both bifurcation method of dynamical systems and numerical exploration of differential equations to investigate the periodic waves of a general compressible hyperelastic rod equation u_t+3uu_x-u_{xxt}-ϒ(2u_xu_{xx}+uu_{xxx})=0, with parameter ϒ ‹ 0. New expressions including explicit expressions and implicit expressions are obtained. Some previous results are extended. Specially, a new phenomenon is found: when the initial value tends to a certain number, the periodic shock wave suddenly changes into a smooth periodic wave. In dynamical systems, this represents that one of orbits can pass through the singular line. The coherency of numerical simulation and theoretical derivation implies the correctness of our results.

Zhengrong Liu and Bengong Zhang . (2007). New Expressions of Periodic Waves and a Novel Phenomenon in a Compressible Hyperelastic Rod. Journal of Partial Differential Equations. 20 (1). 80-96. doi:
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