= Solution
For $z=ct-x$ the <chain rule> gives $\partial_t=c\,d/dz$, $\partial_x=-d/dz$, $\partial_{xx}=d^2/dz^2$. Thus
$$
\boxed{\epsilon c v_z=f(v)-w+\epsilon^2v_{zz},\qquad cw_z=\gamma v-w.}
$$
As $z$ increases from the material ahead of the wave towards that behind it, the singular pulse construction is $A\to B\to C\to D\to A$. The front $AB$ raises $v$ at approximately fixed $w=0$ from the low root to the high root. On the plateau $BC$, $f(v)\simeq w$ on the right attracting branch while $w$ rises slowly. The back $CD$ decreases $v$ at approximately fixed $w=w_0$ to the left attracting branch. The recovery tail then lowers $w$ back to zero on that branch, returning to $A=(0,0)$. Both sharp transitions have thickness $O(\epsilon)$ in $z$; the other segments have the slower recovery scale.
The figure labels all four points and the front/back orientations. This frozen-recovery construction needs $\epsilon\gamma\ll1$ as well as $\epsilon\ll1$ if $\gamma$ is large, so that the <recovery variable> remains nearly fixed in an inner transition. It also needs an admissible positive front speed for the chosen orientation, addressed below.
\Image[/past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2010-fhn-pulse.png]
{title=Matched front, plateau, back and recovery tail, with the inner front heteroclinic orbit}
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