# American Institute of Mathematical Sciences

March  2020, 19(3): 1609-1667. doi: 10.3934/cpaa.2020058

## Travelling corners for spatially discrete reaction-diffusion systems

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

* Corresponding author

Received  January 2019 Revised  September 2019 Published  November 2019

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.

Citation: H. J. Hupkes, L. Morelli. Travelling corners for spatially discrete reaction-diffusion systems. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1609-1667. doi: 10.3934/cpaa.2020058
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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
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
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 $\rho$
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
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
 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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