Problem: Find a traveling wave solution to the Fisher's equation:$$\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2} + r u (1-u)$$that connects the stable states u=0 and u=1 on the boundary condition $ U(-\infty) = 1, \quad U(+\infty) = 0 $
Solution: $$\boxed{u(x, t) = \frac{1}{2} \left( 1 - \tanh\left( \sqrt{\frac{r}{8D}} \left( x - 3\sqrt{\frac{rD}{2}} t \right) \right) \right)}$$