Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice
Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice
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DOI：
10.1137/s0036139996312703
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发表时间：
1998-12-08
影响因子：
1.9
通讯作者：
Van Vleck, ES
中科院分区：
文献类型：
--
作者：
Cahn, JW;Mallet-Paret, J;Van Vleck, ES
我们考虑二维整数格上的无穷常微分方程组，在每一点上给定一个标量常微分方程，格点之间具有最近邻耦合。对于一类理想非线性项，我们得到了在每个方向e（i θ）上的行波解，并探讨了非线性项的波速度c、角度θ和失谐参数a之间的关系。特别感兴趣的是“传播失败”的现象，我们研究了临界值a = a*（θ）如何取决于θ，其中a*（θ）定义为传播失败（即波速c = 0）发生时的参数a的值。我们证明a*：R --> R在tan θ是无理数的每一点θ处是连续的，并且在tan θ是有理数或无穷大的地方是不连续的。
We consider infinite systems of ODEs on the two-dimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction e(i theta), and we explore the relation between the wave speed c, the angle theta, and the detuning parameter a of the nonlinearity. Of particular interest is the phenomenon of "propagation failure," and we study how the critical value a = a*(theta) depends on theta, where a*(theta) is defined as the value of the parameter a at which propagation failure (that is, wave speed c = 0) occurs. We show that a* : R --> R is continuous at each point theta where tan theta is irrational, and is discontinuous where tan theta is rational or infinite.