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首页> 外文期刊>Memoirs of the American Mathematical Society >The Dynamics of Modulated Wave Trains
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The Dynamics of Modulated Wave Trains

机译：调制波列的动力学

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We investigate the dynamics of weakly-modulated nonlinear wave trains. Forreaction-diffusion systems and for the complex Ginzburg–Landau equation, we es-tablish rigorously that slowly varying modulations of wave trains are well approxi-mated by solutions to the Burgers equation over the natural time scale. In additionto the validity of the Burgers equation, we show that the viscous shock profilesin the Burgers equation for the wave number can be found as genuine modulatedwaves in the underlying reaction-diffusion system. In other words, we establish theexistence and stability of waves that are time-periodic in appropriately moving co-ordinate frames which separate regions in physical space that are occupied by wavetrains of different, but almost identical, wave number. The speed of these shocksis determined by the Rankine–Hugoniot condition where the flux is given by thenonlinear dispersion relation of the wave trains. The group velocities of the wavetrains in a frame moving with the interface are directed toward the interface. Us-ing pulse-interaction theory, we also consider similar shock profiles for wave trainswith large wave number, that is, for an infinite sequence of widely separated pulses.The results presented here are applied to the FitzHugh–Nagumo equation and tohydrodynamic stability problems.
机译：我们研究了弱调制非线性波串的动力学。对于反应扩散系统以及复杂的Ginzburg-Landau方程，我们严格地建立了一个模型，即在自然时间尺度上通过Burgers方程的解很好地近似了缓慢变化的波列调制。除了Burgers方程的有效性外，我们还证明了在Burns方程中波数的粘性激波分布可以作为潜在反应扩散系统中的真正调制波。换句话说，我们在适当移动的坐标系中建立了时间周期的波的存在性和稳定性，该坐标系将物理空间中被不同但几乎相同的波数所占据的区域分开。这些冲击的速度由兰金-休格尼特条件确定，其中通量由波列的非线性色散关系给出。与界面一起移动的框架中的波列的群速度指向界面。使用脉冲相互作用理论，我们还考虑了具有大波数的波列，即无限远的离散脉冲序列的相似冲击分布。此处给出的结果适用于FitzHugh-Nagumo方程和水动力稳定性问题。

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