Modulations of viscous fluid conduit periodic waves
Modulations of viscous fluid conduit periodic waves
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DOI：
10.1098/rspa.2016.0533
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发表时间：
2016-07
期刊：
影响因子：
--
通讯作者：
M. Maiden;M. Hoefer
中科院分区：
文献类型：
--
作者：
M. Maiden;M. Hoefer
利用非线性波调制理论和数值方法研究了粘性液体管道沿着的调制周期界面波。大振幅驻波调制（Whitham）理论并不需要基础模型方程的可积性，但通常要么可积方程被研究，要么Whitham理论的全部范围没有发展。用波数和振幅表征了非线性、色散、不可积管道方程的周期波解。在弱非线性区域，非线性薛定谔（NLS）方程的散焦和聚焦变量的推导，取决于载波波数。暗和亮包络孤子被发现持续在长时间的数值解的导管方程，强非线性，大振幅包络孤子的存在提供了数值证据。由于非凸色散，调制不稳定性的周期波以上的临界波数的预测和观察。在大振幅区域，计算了Whitham调制方程的结构性质，包括严格双曲性、真实非线性和线性退化。从NLS临界波数在零振幅分叉是一个振幅依赖的椭圆形区域内的Whitham方程的最大不稳定的周期波被确定。粘性流体管道系统是一个数学上易处理的，实验上可行的模型系统，广泛的非线性，色散波动力学。
Modulated periodic interfacial waves along a conduit of viscous liquid are explored using nonlinear wave modulation theory and numerical methods. Large-amplitude periodic-wave modulation (Whitham) theory does not require integrability of the underlying model equation, yet often either integrable equations are studied or the full extent of Whitham theory is not developed. Periodic wave solutions of the nonlinear, dispersive, non-integrable conduit equation are characterized by their wavenumber and amplitude. In the weakly nonlinear regime, both the defocusing and focusing variants of the nonlinear Schrödinger (NLS) equation are derived, depending on the carrier wavenumber. Dark and bright envelope solitons are found to persist in long-time numerical solutions of the conduit equation, providing numerical evidence for the existence of strongly nonlinear, large-amplitude envelope solitons. Due to non-convex dispersion, modulational instability for periodic waves above a critical wavenumber is predicted and observed. In the large-amplitude regime, structural properties of the Whitham modulation equations are computed, including strict hyperbolicity, genuine nonlinearity and linear degeneracy. Bifurcating from the NLS critical wavenumber at zero amplitude is an amplitude-dependent elliptic region for the Whitham equations within which a maximally unstable periodic wave is identified. The viscous fluid conduit system is a mathematically tractable, experimentally viable model system for wide-ranging nonlinear, dispersive wave dynamics.