Problem: Find a self-similarity solution for the non-linear partial differential equation: $d_t u = d_{xx}u + u(u - \alpha)(1 - u)$ where $\alpha$ is a constant such that $0 < \alpha < 1/2$. The solution connects $u=1$ ($t \rightarrow -\infty$) to $u=0$ ($t \rightarrow +\infty$). Determine the wave speed $c$ in terms of $\alpha$. Then, find the explicit form of the solution $u(x,t)$ given the initial condition $u(0,0) = \frac{1}{1+e^{\sqrt{2}/2}}$ and assuming $\alpha = 1/4$.
Solution: $$ \boxed{c(\alpha) = \frac{1-2\alpha}{\sqrt{2}}} $$;$$ \boxed{u(x,t) = \frac{1}{1+e^{\frac{1}{\sqrt{2}}(-x + \frac{\sqrt{2}}{4}t + 1)}}} $$