Problem: Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 2v^2 u^2 (1 - u), \quad \lim_{x \to \infty} u(x, t) = 0, \quad \lim_{x \to -\infty} u(x, t) = 1$$ in the limit $t \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ";".
Solution: $\boxed{u(x, t) = \frac{1}{1 + e^{0.5(x - 0.5 t)}}}$