Problem: Approximate a self similar traveling wave solution $ \partial_t u = \partial_{xx} u + u(1-u)(u-\frac{1}{4}) + x \left(\partial_t u + \frac{\sqrt{2}}{4} \partial_x u\right)$ in the limit $t \to \infty$. Return one exact expression for u(x,t). Please place your final solution in a $\boxed{}$ LaTeX Environment.If there are multiple solutions please separate them with a ;.
Solution: $\boxed{u(x,t) = \frac{1}{1+\exp{\left(\frac{x}{\sqrt{2}}-\frac{t}{4}\right)}}}$