Problem: Suppose we have the following Reaction-Diffusion type Partial Differential Equation, $ \partial_t u = \partial_{xx} u - \alpha (u - \frac{1}{5}) + \beta (u - \frac{1}{5})^3 $, for some function $u(x,t)$ where $\alpha,\beta > 0$. Please find a self similarity solution (solition behavior) that connects the state $u = \frac{1}{5}$ in the limit $t \rightarrow -\infty $ to $u = \frac{1}{5}$ in the limit $t \rightarrow +\infty$. Use the substitution $z=x-vt$ and express the answer in terms of $z$. Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.
Solution: $\boxed{u(x,t) = \sqrt{\frac{2 \alpha}{\beta}}\text{sech}(\sqrt{\alpha} z) + \frac{1}{5}; u(x,t) = -\sqrt{\frac{2\alpha}{\beta}}\text{sech}(\sqrt{\alpha} z) + \frac{1}{5}}$