Problem: Solve the following nonlinear partial differential equation $$\partial_t u + \frac{3}{2} \partial_x u = u^2 \partial_{xx}u + \frac{2}{3}\left(x-\frac{3}{2}t\right)^{-2}$$ by finding a travelling wave solution, and determine its leading-order behavior as $|x-\frac{3}{2}t| \to \infty$. 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: $\boxed{\left(2\ln\left(x-\frac{3}{2}t\right)\right)^{1/3}}$