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The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect

  • * Corresponding author: Shin-Ichiro Ei

    * Corresponding author: Shin-Ichiro Ei 
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  • 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.

    Mathematics Subject Classification: Primary: 35B36, 37L10; Secondary: 35K57.


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  • Figure 1.  The image of an integral kernel $ K $ with the Mexican hat profile on $ {\mathbb{R}} $

    Figure 2.  The images of localized patterns. (a) Pulse solution when $ P^{+} = P^{-} = \mathit{\boldsymbol{0}} $. (b) Front solution when $ n = 1 $ and $ P^{+}>P^{-} $

    Figure 3.  The image of $ P(z;\mathit{\boldsymbol{h}}) $ when $ N = 2 $

    Figure 4.  The image of $ P(z;\mathit{\boldsymbol{h}}) $. (a) $ (N^+, N^-) = (1,1) $. (b) $ (N^+, N^-) = (2,1) $

    Figure 5.  The image of the interaction of two standing fronts, when the kernel is a non-negative function

    Figure 6.  (a) The graph of (30) when $ \varepsilon = 0.01,\ q_1 = 1.0,\ q_2 = 2.0 $. (b) The graph of $ G(\lambda) $ when $ d = 1.0 $, $ f'(1) = -1 $ and the integral kernel is same as Figure 6 (a)

    Figure 7.  (a) The numerical solution of $ (2) $ on the interval $ (0,40) $ when $ t = 100.0 $, where $ f(u) = \frac{1}{2}u(1-u^2) $ and the other parameters are same as that in >Figure 6. (b) The graph of $ \log|u(t,x)-1| $ on the interval $ (20,35) $ when $ t = 100.0 $, where $ u(t,x) $ is the numerical solution of $ (2) $

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