Problem: Find a self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \partial_t u = \partial_{xx} u - \frac{(\partial_x u)^2}{u} + u \ln u \left(1 - (\ln u)^2\right) $$ that connects the stable state $u = e$ and $u = e^{-1}$ in the limit $|t| \rightarrow \infty$. Please place your final solution in a \boxed{} LaTeX environment. If there are multiple solutions please separate them with a ;.
Solution: $\boxed{u(x,t) = e^{\tanh\left(\frac{x}{\sqrt{2}}\right)} ; u(x,t) = e^{-\tanh\left(\frac{x}{\sqrt{2}}\right)}}$