Problem: Find a self-similarity solution 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) - \delta \partial_x u $ where $\delta$ is a real constant 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 semicolon.
Solution: $\boxed{u(x,t) = e^{\tanh\left(\frac{x-\delta t}{\sqrt{2}}\right)}; u(x,t) = e^{-\tanh\left(\frac{x-\delta t}{\sqrt{2}}\right)}}$