In this paper, we analyze the interaction of localized patterns such as traveling wave solutions for reaction-diffusion systems with nonlocal effect in one space dimension. We consider the case that a nonlocal effect is given by the convolution with a suitable integral kernel. At first, we deduce the equation describing the movement of interacting localized patterns in a mathematically rigorous way, assuming that there exists a linearly stable localized solution for general reaction-diffusion systems with nonlocal effect. When the distances between localized patterns are sufficiently large, the motion of localized patterns can be reduced to the equation for the distances between them. Finally, using this equation, we analyze the interaction of front solutions to some nonlocal scalar equation. Under some assumptions, we can show that the front solutions are interacting attractively for a large class of integral kernels.

中文翻译：

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具有非局部效应的反应扩散系统的弱相互作用局部模式的运动

在本文中，我们分析了一维空间中具有非局部效应的反应扩散系统的局部模式（例如行波解）的相互作用。我们考虑这样一种情况，即通过使用适当的积分核进行卷积来给出非局部效应。首先，我们假设数学模型严格地推导了描述相互作用的局部模式的运动的方程式，假定存在具有非局部效应的一般反应扩散系统的线性稳定局部化解。当局部图案之间的距离足够大时，可以将局部图案的运动减小为它们之间的距离的等式。最后，使用该方程，我们分析了一些非局部标量方程的前沿解的相互作用。在某些假设下，