Problem: Find a traveling-wave solution to the nonlinear partial differential equation $$\partial_t u + \partial_x u = -u^3 \partial_{xx}u + (x-t)^2 + 1$$ and determine its leading-order behavior as $|x-t| \to \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. There should be no words inside the $\boxed{}$ environment; only an expression that can be evaluated by subsituting in values of $x$ and $t$. If there are absolute values in your final answer, replace those with parentheses before putting the answer in the $\boxed{}$ environment.
Solution: $$\boxed{u(x,t) = \sqrt{2} (x-t) (\ln(|x-t|))^{\frac{1}{4}}}$$