fsh : A Saddle-Node Connection.

This demo illustrates the computation of travelling wave front solutions to the Fisher equation,

$$\begin{array}{cl} & w_t = w_{xx} + f(w), \qquad -\infty < x < \infty, \quad t > 0, \\ & f(w) \equiv w(1-w) . \\ \end{array}$$ (18.1)

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

$$\begin{array}{cl} & u_1'(z) = u_2(z), \\ & u_2'(z) = c u_2(z) - f\bigl(u_1(z)\bigr). \\ \end{array}$$ (18.2)

Its fixed point $(0,0)$ has two positive eigenvalues when $c>2$. The other fixed point, $(1,0)$, is a saddle point. A family of orbits connecting the two fixed points requires one free parameter; see FrDo:91 FrDo:91. Here we take this parameter to be the wave speed $c$.

In the first run a starting connecting orbit is computed by continuation in the period $T$. This procedure can be used generally for time integration of an ODE with AUTO . Starting data in STPNT correspond to a point on the approximate stable manifold of $(1,0)$, with $T$ small. In this demo the ``free'' end point of the orbit necessary approaches the unstable fixed point $(0,0)$. A computed orbit with sufficiently large $T$ is then chosen as restart orbit in the second run, where, typically, one replaces $T$ by $c$ as continuation parameter. However, in the second run below, we also add a phase condition, and both $c$ and $T$ remain free.

Table 18.1: Commands for running demo fsh.

AUTO -COMMAND	ACTION
! mkdir fsh	create an empty work directory
cd fsh	change directory
demo('fsh')	copy the demo files to the work directory
run(c='fsh.1')	continuation in the period $T$, with $c$ fixed; no phase condition
sv('0')	save output-files as b.0, s.0, d.0
run(c='fsh.2',s='0')	continuation in $c$ and $T$, with active phase condition. Constants changed : IRS, ICP, NINT, DS
sv('fsh')	save output-files as b.fsh, s.fsh, d.fsh