Problem: Find a solution to the nonlinear partial differential equation $ \partial_t u + \partial_x u = -\left(u + \frac{1}{u}\right) \partial_{xx}u + (x-t)^2 + \frac{1}{x-t}$ for its leading-order behavior as $|x-t| \to \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.If there are multiple solutions, please separate them with a ;. 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 $t$.
Solution: $\boxed{u(x,t) \sim \frac{1}{\sqrt{2}} (x-t)^2}$