Stability of front solutions of the bidomain equation
Stability of front solutions of the bidomain equation
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双域方程前沿解的稳定性
DOI：
10.1002/cpa.21634
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Hiroshi Matano and Yoichiro Mori
中科院分区：
文献类型：
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作者：
Hiroshi Matano and Yoichiro Mori
比多曼模型是描述心脏电活动的标准模型。在这里，我们研究了具有双稳态非线性项的双稳态方程(即双稳态Allen-Cahn方程)在二维空间中平面前解的稳定性。在经典的Allen-Cahn方程中，用符号为二次齐次正有理函数的傅立叶乘子算子代替经典的Allen-Cahn方程中的拉普拉斯算子。平面波前的稳定性可能取决于给定的各向异性的比多曼算子的传播方向。我们建立了平面前锋在每个传播方向上的稳定性和不稳定性的各种判据。我们的分析表明，与经典的或各向异性的Allen-Cahn方程形成鲜明对比的是，Biomain Allen-Cahn方程中的平面前锋可能是不稳定的。我们确定了两种类型的不稳定性，一种是关于长波微扰的，另一种是关于中波微扰的。有趣的是，波前在长波长扰动下是稳定的还是不稳定的，并不取决于双稳态非线性，而完全由适当定义的Frank图的凸性决定。另一方面，在中波长扰动下的稳定性确实取决于双稳态非线性的选择。即使在Frank图是凸的情况下，只要比多曼算符不退化为拉普拉斯算子，也可能发生中波长不稳定性。我们还将给出一个平面前沿在各个方向上都不稳定的显著例子。
The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen‐Cahn equation) in two spatial dimensions. In the bidomain Allen‐Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen‐Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen‐Cahn equation in striking contrast to the classical or anisotropic Allen‐Cahn equations. We identify two types of instabilities, one with respect to long‐wavelength perturbations, the other with respect to medium‐wavelength perturbations. Interestingly, whether the front is stable or unstable under long‐wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate‐wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate‐wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions.© 2016 Wiley Periodicals, Inc.