Problem: Find a self similarity solution for the non-linear partial differential equation $ \partial_t u = \partial_{xx} u + (2u-\sqrt{5})(1-u^2), \lim_{x \to -\infty} u(x,t) = -1, \lim_{x \to \infty} u(x,t) = 1 $ in the limit $t \rightarrow \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.
Solution: $ \boxed{ u(x,t)=\tanh(x-\sqrt{5}t) }$