Problem: Find a self similarity solution (traveling wave front) for the reaction-diffusion equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + 4u(1 - u^2) $$ that connects the stable state $u=-1$ as $x \to -\infty$ to the stable state $u=+1$ as $x \to +\infty$. The solution should satisfy $u(x,t) = 0$ when $x=ct$ (i.e., the center of the wave where $\xi=0$). Please place your final solution $u(x,t)$ in a \boxed{} LaTeX environment. If you have a free parameter, set it to 2.
Solution: $\boxed{u(x,t) = \tanh(\sqrt{2} x)}$