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Hexagonal spike clusters for some PDE's in 2D

  • * Corresponding author: Juncheng Wei

    * Corresponding author: Juncheng Wei 

The authors are supported by NSERC discovery grants

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  • We study hexagonal spike cluster patterns for Gierer-Meinhardt reaction-diffusion system with a precursor on all of $ \mathbb R^2 $. These clusters consist of $ N $ spikes which form a nearly hexagonal lattice of a finite size. The lattice density is locally nearly constant, but globally non-uniform. We also characterize a similar hexagonal spike cluster steady state for a simple elliptic PDE $ 0 = \Delta u - u +u^2 + \varepsilon |x|^2 $ with a small "confinement well" $ \varepsilon |x|^2 $. The key idea is to explicitly exploit the local hexagonality structure to asymptotically approximate the solution using certain lattice sums. In the limit of many spikes, we derive the effective spike density as well as the cluster radius. This effective density is a solution to a certain separable first-order ODE coupled to an integral boundary condition.

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


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  • Figure 1.  Cluster steady-state solution to (1) consisting of 20 spikes. Contour plot of $ a $ and $ h $ are shown in (a) and (b) respectively. Parameter values are $ \varepsilon = 0.15 $ and $ \mu(x) = 1+0.02\left\vert x\right\vert ^{2}. $ Computational domain was taken to be $ x\in(-15,15)^{2} $; increasing the computational domain did not change spike locations. (c): Centers of spikes from the PDE simulation compared with centers generated by the reduced system (15). Dashed line denotes spike boundary computed asymptotically from (17). (d): Spike height $ h(x_{j}) $ versus $ \left\vert x_{j}\right\vert . $ Comparison between full numerical simulation, the reduced system (15) and theoretical prediction (17)

    Figure 4.  LEFT: Steady state for (15) with $ N = 500, $ $ \mu(x) = 1+0.025x^{2} $ and $ \varepsilon = 0.08.\ $Dots represent the steady state $ x_{j}; $ their size and colour are proportional to $ H_{j}. $ Dashed line represents the theoretical boundary of the steady state in the continuum limit $ N\gg1. $ MIDDLE: scatter plot of the average distance $ u(x_{j}) $ from a point to any of its neighbours, as a function of $ \left\vert x_{j}\right\vert . $ Solid curve is the analytical prediction of the continuum limit as given by (17). RIGHT: Scatter plot of the $ H_{j} $ as a function of $ \left\vert x_{j}\right\vert $ and comparison to theory

    Figure 2.  Left: steady state solution to the one-dimensional equation (5). Right: inter-spike spacing, comparison between asymptotics (10) and the steady state of (5) computed numerically. Parameters are $ a = 0.1 $ and $ N = 50$

    Figure 3.  LEFT: steady state for (13) with $ N = 500 $ and $ a = 0.1. $ Dots represent the steady state $ x_{j}. $ Dashed line represents the theoretical boundary of the steady state in the continuum limit $ N\gg1. $ RIGHT: scatter plot of the average distance $ u(x_{j}) $ from a point to any of its neighbours, as a function of $ \left\vert x_{j}\right\vert . $ Solid curve is the analytical prediction of the continuum limit as given by equations (14)

    Figure 5.  u and f

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