Problem: Find a self-similarity solution (soliton behavior) for $\partial_t u = \partial_{xx} u + u (4 - u^2)$ that connects the $u = 0$ solution in the $t \rightarrow -\infty$ limit to $u = 2$ in the $t \rightarrow \infty$ limit, subject to the boundary condition of $u(0, 0) = 1$. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants.
Solution: $$\boxed{u(x, t) = 2(1 + e^{-\sqrt{2}(x + 3\sqrt{2}t)})^{-1}}$$