Freezing Traveling and Rotating Waves in Second Order Evolution Equations
Freezing Traveling and Rotating Waves in Second Order Evolution Equations
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冻结二阶演化方程中的行波和旋转波
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
J. Rottmann
中科院分区：
文献类型：
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作者：
W. Beyn;Denny Otten;J. Rottmann
在本文中，我们研究了在一维和多维空间中二阶波动方程所谓的冻结方法的实现。该方法将给定的偏微分方程转换为偏微分代数方程，然后进行数值求解。重新制定的目的是将解决方案的运动分离为共同运动的框架和变化尽可能小的轮廓。数值例子证明了该方法对于具有足够阻尼的半线性波动方程的可行性。我们处理一个空间维度中的行波和二维空间维度中的旋转波的情况。此外，我们研究了任意空间维度上的点谱和通过对轮廓进行线性化获得的算子的本质谱，并指出了波的非线性稳定性的后果。
In this paper we investigate the implementation of the so-called freezing method for second order wave equations in one and several space dimensions. The method converts the given PDE into a partial differential algebraic equation which is then solved numerically. The reformulation aims at separating the motion of a solution into a co-moving frame and a profile which varies as little as possible. Numerical examples demonstrate the feasability of this approach for semilinear wave equations with sufficient damping. We treat the case of a traveling wave in one space dimension and of a rotating wave in two space dimensions. In addition, we investigate in arbitrary space dimensions the point spectrum and the essential spectrum of operators obtained by linearizing about the profile, and we indicate the consequences for the nonlinear stability of the wave.