Problem: Find a self-similarity solution (soliton behaviour) for the non-linear partial differential equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 5u(1-u)$$ with boundary condition: $\lim_{x \to -\infty} u(x,t) = 1,\lim_{x \to \infty} u(x,t) = 0$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ;.
Solution: $\boxed{u(x,t) =\frac{1}{1 + e^{\frac{\sqrt{5}}{2}(x - 2\sqrt{5}\, t)}}}$