Schnakenberg方程中的模式选择：从正常扩散到反常扩散
Pattern selection in the Schnakenberg equations: From normal to anomalous diffusion
摘要：
研究了经典和分数阶Schnakenberg方程中的斑图形成，以理解反常扩散的非局部效应。从线性稳定性分析入手，我们发现，如果激活剂和抑制剂具有相同的扩散功率，则图灵不稳定性空间只取决于扩散系数之比$κ_1/κ_2$ .然而，较小的扩散功率可能会引入更大的不稳定波数和更宽的频带，这意味着图案可能是更混乱的分数的情况下。然后，我们应用弱非线性分析来预测图灵空间中的斑点，条纹和混合模式的参数制度。我们的数值模拟证实了分析结果，并展示了正常和异常扩散对斑图形成的差异。我们发现，在超扩散的存在下，图案表现出多尺度结构。扩散功率越小，不稳定波数越大，斑图尺度越小。
英文摘要：
Pattern formation in the classical and fractional Schnakenberg equations is studied to understand the nonlocal effects of anomalous diffusion. Starting with linear stability analysis, we find that if the activator and inhibitor have the same diffusion power, the Turing instability space depends only on the ratio of diffusion coefficients $κ_1/κ_2$. However, the smaller diffusive powers might introduce larger unstable wave numbers with wider band, implying that the patterns may be more chaotic in the fractional cases. We then apply a weakly nonlinear analysis to predict the parameter regimes for spot, stripe, and mixed patterns in the Turing space. Our numerical simulations confirm the analytical results and demonstrate the differences of normal and anomalous diffusion on pattern formation. We find that in the presence of superdiffusion the patterns exhibit multiscale structures. The smaller the diffusion powers, the larger the unstable wave numbers and the smaller the pattern scales.
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