= Solution
TODO: biological interpretations and organism-specific propagation are user-deferred. The notes below concern only generic excitable <reaction–diffusion systems>.
A <front solution> joins different asymptotic states in space. In a <reaction–diffusion system>, local nonlinear dynamics select possible states while <diffusion> couples neighboring points. A <travelling wave> with profile $u(x,t)=U(\xi)$, $\xi=x-ct$, converts a <partial differential equation> into a <boundary value problem> for the shape and speed. A pulse instead returns to the same resting state on both sides and can contain leading and trailing fronts.
A useful nondimensional <cubic fast-slow reaction-diffusion model>, in a convention of the <FitzHugh-Nagumo model>, is
$$
u_t=D u_{xx}+f(u)-w+I,\qquad
w_t=\varepsilon(u-\beta w),\qquad
f(u)=u(1-u)(u-\alpha),\quad0<\alpha<1,
$$
where $D>0$, $\varepsilon\ll1$, and $\beta>0$. Here $u$ is a fast activation variable, $w$ a slow negative-feedback variable and $I$ a generic drive; no physiological variable is assigned. Without spatial coupling, <nullclines> are $w=f(u)+I$ and $w=u/\beta$. At $I=0$, the resting state $(0,0)$ has <Jacobian matrix>
$$
\mathsf J=\begin{pmatrix}-\alpha&-1\\\varepsilon&-\varepsilon\beta\end{pmatrix},\qquad
\operatorname{tr}\mathsf J=-\alpha-\varepsilon\beta<0,\quad
\det\mathsf J=\varepsilon(1+\alpha\beta)>0.
$$
It has <linear stability>. With appropriate parameter choices a sufficiently large perturbation makes a fast excursion to the other outer branch of the cubic, followed by slow feedback and return. Thus a stable rest state can be excitable without being linearly unstable. Other parameters or applied drive can yield multiple equilibria or oscillations; these behaviours should not be asserted for every parameter set.
The fast-subsystem front can be obtained explicitly by freezing $w=I=0$. For the <bistable cubic reaction-diffusion equation>, $u=0,1$ are stable and $u=\alpha$ is unstable. The decreasing <logistic front of a bistable cubic equation> is
$$
\boxed{U(\xi)=\frac1{1+\exp[\xi/\sqrt{2D}]},\qquad
c=\sqrt{D/2}(1-2\alpha).}
$$
To check it, $U'=-U(1-U)/\sqrt{2D}$ and substitution into $DU''+cU'+f(U)=0$ cancels every term. For $\alpha<1/2$, the high state advances to the right; for $\alpha>1/2$, the direction reverses. At $\alpha=1/2$ the front is stationary. More generally, multiplying the travelling-wave equation by $U'$ and integrating yields
$$
c\int_{-\infty}^{\infty}(U')^2d\xi=\int_0^1f(u)du=\frac{1-2\alpha}{12}.
$$
This links propagation direction to the bias between the two stable states and recovers <Maxwell balance for a scalar reaction-diffusion front> at zero speed. It is a bistable selected-speed problem, rather than a universal Fisher-type minimum-speed formula.
For the coupled fast-slow system, travelling profiles satisfy
$$
DU''+cU'+f(U)-W+I=0,\qquad
cW'+\varepsilon(U-\beta W)=0.
$$
A slow recovery wake can support a return front behind the advancing fast front, producing a travelling pulse for suitable parameters. Its width reflects the slow recovery time and propagation speed, while each sharp edge has the faster <diffusion>–reaction scale. The frozen-recovery speed above is a controlled subsystem result, not an exact speed for every coupled pulse. A definite pulse speed requires parameters and connecting <boundary condition>; none are specified by this brief-note question.
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2012/paper-1-excitable-fronts.png]
{title=Generic excitable fronts}
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