Problem: Find a self-similar solution (soliton behaviour) for the non-linear partial differential equation $\partial_t u - \frac{10}{\sqrt{30}} \, \partial_x u = \frac{2}{5} \, \partial_x^2 u + 2 u (1 - u), \quad \lim_{x \to -\infty} u(x,t) = 1, \quad \lim_{x \to \infty} u(x,t) = 0$ in the limit $t \rightarrow \infty$. Please write the full equation and don't introduce new variables. Place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ;.
Solution: $\boxed{u(x,t) = \frac{1}{\left(1 + \exp\left[\sqrt{\frac{5}{6}}\left(x + \frac{10}{\sqrt{30}} t \right)\right] \right)^2}}$