Problem: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 2 u(1-u)(u-\frac{1}{4}), \quad \lim_{x \to -\infty} u(x,t) = 1, \quad \lim_{x \to \infty} u(x,t) = 0 $$ in the limit $t \rightarrow \infty $. Please place your final solution in a $\boxed{}$ LaTeX Environment.DONT USE \operatorname in your boxed solution. There should be a free parameter $v$. Set $v=0.5$ YOU MUST DO THIS. If there are multiple solutions please separate them with a ;.
Solution: $$ \boxed{ u(x, t) = \frac{1}{1 + e^{x - 0.5t}} } $$