Problem: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + u^2 (1-u), \lim_{x \to \infty} u(x, t) = 0 \quad \text{and} \quad \lim_{x \to -\infty} u(x, t) = 1 $$ in the limit . 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: $$ \boxed{ u(x, t) = \frac{1}{2} \left[ 1 - \tanh\left( \frac{x - t/\sqrt{2}}{2\sqrt{2}} \right) \right] } $$