Target patterns in a 2D array of oscillators with nonlocal coupling
Target patterns in a 2D array of oscillators with nonlocal coupling
复制标题
DOI：
10.1088/1361-6544/aac9a6
复制
发表时间：
2017-06
期刊：
影响因子：
1.7
通讯作者：
Gabriela Jaramillo;S. Venkataramani
中科院分区：
文献类型：
--
作者：
Gabriela Jaramillo;S. Venkataramani
我们分析的效果，添加一个弱的，本地化的，不均匀的二维阵列的振荡器与非本地耦合。我们提出并证明了在这个系统中的相动力学模型。我们的模型是一个广义的粘性程函方程，这是已知的反应扩散系统中的行波的相位调制。我们表明存在一个分支的目标模式的解决方案，从空间均匀的状态分叉时，强度的不均匀性，是非零的，我们还表明，这些目标模式有一个渐近波数，是小的超越所有订单。我们的证明策略是对解的一个近似形式给出一个很好的证明，并利用隐函数定理证明其附近解的存在性。分析提出了两个挑战。首先，关于齐次状态的线性化是扩散型卷积算子，因此在通常的Sobolev空间上是不可逆的。第二，一个正规的摄动展开并没有提供一个很好的ananalgesia应用隐函数定理，因为非线性发挥了重要作用，在确定相关的近似，这也需要是“正确的"所有订单。我们克服了这两点证明Fredholm性质的线性化在适当的Kondratiev空间和使用的近似解，这是使用匹配的渐近性得到一个精致的anastrophic。
We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction–diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when , the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in . The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be ‘correct’ to all orders in . We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution which was obtained using matched asymptotics.