Travelling corners for spatially discrete reaction-diffusion systems

H. J. Hupkes; L. Morelli

doi:10.3934/cpaa.2020058

Article Contents

2020, Volume 19, Issue 3: 1609-1667. Doi: 10.3934/cpaa.2020058

This issue Previous Article Study of semi-linear $ \sigma $-evolution equations with frictional and visco-elastic damping Next Article Homogenization of a locally periodic time-dependent domain

Travelling corners for spatially discrete reaction-diffusion systems

H. J. Hupkes and
L. Morelli^,

Mathematisch Instituut - Universiteit Leiden, P.O. Box 9512; 2300 RA Leiden; The Netherlands

^* Corresponding author
^* Corresponding author

Received: January 2019

Revised: September 2019

Published online: March 2020

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Abstract

We consider reaction-diffusion equations on the planar square lattice that admit spectrally stable planar travelling wave solutions. We show that these solutions can be continued into a branch of travelling corners. As an example, we consider the monochromatic and bichromatic Nagumo lattice differential equation and show that both systems exhibit interior and exterior corners.

Our result is valid in the setting where the group velocity is zero. In this case, the equations for the corner can be written as a difference equation posed on an appropriate Hilbert space. Using a non-standard global center manifold reduction, we recover a two-component difference equation that describes the behaviour of solutions that bifurcate off the planar travelling wave. The main technical complication is the lack of regularity caused by the spatial discreteness, which prevents the symmetry group from being factored out in a standard fashion.

Keywords:

Mathematics Subject Classification: 34A33, 35B36.

Citation:

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Figure 1. The blue curves in the left and right panels depict the interface of an interior respectively exterior corner. Both corners travel at the speed $ d_{\varphi_-} = d_{\varphi_+} $ and share the coordinate system $ (n,l) $ depicted in the center. Angles are positive when oriented counter-clockwise and negative otherwise. All speeds are positive

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Figure 2. Both panels contain polar plots of the $ \zeta \mapsto c_{\rho,\zeta} $ relation, for various values of $ \rho > 0 $. Since $ c \le 0 $ in this setting, we have actually plotted the points $ -c_{\rho,\zeta}(\cos\zeta,\sin\zeta) $ for $ 0 \le \zeta \le \frac{\pi}{2} $. Notice the extra minima that start to form in the directions $ \tan \zeta = 1 $ and subsequently $ \tan \zeta = \frac{2}{3} $ as $ \rho $ is decreased

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Figure 3. The left panel contains numerically computed values for $ -\kappa_d(\rho) $. The sharp spikes occur at the critical value $ \rho_*(\zeta) $ where pinning sets in. We note that sign changes appear for $ \zeta = \frac{\pi}{2} $ but not for $ \zeta = 0 $. In particular, the identity $ c_g \equiv 0 $ for these directions implies that interior and exterior corners can both occur for $ \zeta = \frac{\pi}{2} $, while the horizontal direction $ \zeta = 0 $ features interior corners only. The right panel contains numerically computed values for $ c_g(\rho) $. Notice the zero-crossings for $ \tan \zeta = \frac{3}{4} $ and $ \tan \zeta = \frac{4}{5} $, which indicates the presence of interior corners at these two critical values for $ \rho $

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Figure 4. The left panel contains polar plots of the $ \zeta \mapsto c_{\rho, \alpha,\zeta} $ relation, with fixed $ \rho = 0 $. In particular, the curves consist of the points $ c_{\rho,\alpha,\zeta}(\cos\zeta,\sin\zeta) $. The right panel depicts the directional dispersion $ d(\zeta) = \frac{c_{\rho,\alpha, \zeta}}{\cos (\zeta - \zeta_*) } $, with $ \zeta_* = 0 $ for the left column and $ \zeta_* = \frac{\pi}{4} $ for the right column, again with $ \rho = 0 $. These results strongly suggest that $ [\partial_{\zeta}^2 d(\zeta) ]_{\zeta = \zeta_*} $ can take both signs as the diffusion coefficient $ \alpha $ is varied. In particular, both the horizontal and diagonal directions can have interior and exterior corners

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Table 1. Summary of the fashion in which the various assumptions in Theorem 2.3 were verified for the examples in §2.1-2.2, together with the encountered corner types

	Monochromatic - §2.1			Bichromatic - §2.2
	$ \zeta \in \mathbb{Z} \frac{\pi}{2} $	$ \zeta \in \frac{\pi}{4} + \mathbb{Z} \frac{\pi}{2} $	$ \tan \zeta \in \{ \frac{3}{4}, \frac{4}{5} \} $	$ \zeta \in \mathbb{Z} \frac{\pi}{2} $	$ \zeta \in \frac{\pi}{4} + \mathbb{Z} \frac{\pi}{2} $
$ c \neq 0 $ in $ \mathrm{(H}\Phi\mathrm{)} $	analytic for $ \rho_*(\zeta)< \left\vert{\rho}\right\vert < 1 $			numeric	analytic for open set $ (\rho, \alpha) $}
(HS1)-(HS3)	analytic			analytic
$ c_g = 0 $	analytic		numeric	analytic
$ [\partial^2_z \lambda_z]_{z =0 } \neq 0 $	analytic		numeric	analytic
$ [\partial^2_\varphi d_\varphi]_{\varphi = 0 } \neq 0 $	numeric			visual
Corner types	interior	both	interior	both

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