 Frozen-recovery cubic front

ID: frozen-recovery-cubic-front

v_{t} = D v_{xx} - (v - r_{1}) (v - r_{2}) (v - r_{3})

with

D > 0

and

r_{1} < r_{2} < r_{3}

, an exact decreasing travelling wave is

V (z) = r_{1} + \frac{r _{3} - r _{1}}{1 + e x p [( r _{3} - r _{1} ) ( z - z _{0} ) / 2 D ]}, z = x - c t .

Indeed

V^{'} = - (V - r_{1}) (r_{3} - V) / 2 D

, and substitution into

D V^{''} + c V^{'} - (V - r_{1}) (V - r_{2}) (V - r_{3}) = 0

leaves the factor

(r_{1} + r_{3}) /2 - r_{2} - c / 2 D

w_{t} = β v - γ w

with exactly constant

w

v

to be constant.

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