Problem: Solve the following nonlinear partial differential equation $$ \partial_t u + u^2 \partial_x u = \partial_{xx}u - (x-t)^{-2} $$ by finding a travelling wave solution of the form $u(x,t) = U(x-t)$, and determine its leading-order behavior as $x-t \to \infty$. Your final answer should contain only the variables $x$ and $t$. In particular, do not define any new functions or constants. Leave your final answer in a $\boxed{}$ LaTeX environment, so that plugging in values of $x$ and $t$ yield a numerical answer. Replace any absolute value signs with the usual () brackets for easy evaluation.
Solution: $\boxed{u(x,t) = 1 + (x-t)^{-1/2}}$