nag : A Saddle-Saddle Connection.

This demo illustrates the computation of traveling wave front solutions to Nagumo's equation,

$$\begin{array}{cl} & w_t = w_{xx} + f(w,a), \qquad -\infty < x... ...0, \\ & f(w,a) \equiv w(1-w)(w-a), \qquad 0<a<1. \\ \end{array}$$ (18.3)

We look for solutions of the form $w(x,t)=u(x+ct)$, where $c$ is the wave speed. This gives the first order system

$$\begin{array}{cl} & u_1'(z) = u_2(z), \\ & u_2'(z) = c u_2(z) - f\bigl(u_1(z),a\bigr), \\ \end{array}$$ (18.4)

where $z=x+ct$, and $' = d/dz$. If $a=1/2$ and $c=0$ then there are two analytically known heteroclinic connections, one of which is given by

$$u_1(z) = \frac{ {e^{\frac{1}{2} \sqrt{2} z}} }{ {1 + e^{\frac{1 }{ 2} \sqrt{2} z}} }, \qquad u_2(z) = u_1'(z), \qquad -\infty < z < \infty.$$

The second heteroclinic connection is obtained by reflecting the phase plane representation of the first with respect to the $u_1$-axis. In fact, the two connections together constitute a heteroclinic cycle. One of the exact solutions is used below as starting orbit. To start from the second exact solution, change SIGN=-1 in the routine STPNT in nag.f and repeat the computations below; see also FrDo:91 FrDo:91.

Table 18.2: Commands for running demo nag.

AUTO -COMMAND	ACTION
! mkdir nag	create an empty work directory
cd nag	change directory
demo('nag')	copy the demo files to the work directory
run(c='nag.1')	compute part of first family of heteroclinic orbits
sv('nag')	save output-files as b.nag, s.nag, d.nag
run(c='nag.2',s='nag')	compute first family in opposite direction. Constants changed : DS
ap('nag')	append output-files to b.nag, s.nag, d.nag