Substitution $n=f(\xi)$, $\xi=x-ct$, gives $f''+cf'+F(f)=0$ with $f(-\infty)=0$, $f(\infty)=B$. Multiplying by $f'$ gives $-c\int(f')^2d\xi=\int_0^BF(s)ds=B^4(1-2A/B)/12$. The populated state invades the empty state when $c<0$, equivalently $B>2A$; this is the condition preventing eventual extinction.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 6C / b / Solution

 Solution (source code)