# 1D-spiral

### Patterns and Simulations

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The picture is a space-time plot with time going downward of a chemical concentration in a homogeneous one-dimensional medium. The arising pattern has been called a 'one-dimensional spiral': a self-organized source (in the homogeneous medium!) sends out pulses to left and right in an alternating fashion.

Parameters: $$a=0.84, b=0.15, 1/\epsilon=10.8$$, domain length=400, Neumann boundary conditions. Numerics is first order finite differences with explicit Euler in time.

The underlying equation is the two-component reaction-diffusion system

\begin{eqnarray} u_t &=& \frac 1 \epsilon u (u-1) \left(u- \frac{b+v}a\right) + u_{xx}\ v_t &=& f(u) -v \end{eqnarray}

where

$$f(u) = \cases{0,\quad0 \leq u < 1/3, 1-6.75 u (u-1)^2, \quad1/3 \leq u \leq 1, 1,\quad 1 < u}$$

which has been derived to model aspects of CO-oxidation on a platinum surface, see [M. Bär et al, J. Chem. Phys. 100 (1994) 1202]. In the picture, red corresponds to low and blue to high concentration of the v-component.

 Author Institutional Affiliation Centre for Mathematics and Computer Science (CWI), Dept. Modelling Analysis and Simulation (MAS) Author Email rademach@cwi.nl Author Postal Mail Kruislaan 413, 1098 SJ Amsterdam, the Netherlands Notes This simulation result has not been published. Similar results can be found in the work of M. Bär et al in the 90s. Keywords reaction-diffusion system, one-dimensional spiral, self-replication