Problem: Suppose we have the following Partial Differential Equation, $\partial_{xxx}u + \partial_{t}u(1-\partial_xu)= 0$, for some function $u(x,t)$. Please find a travelling wave solution that connects the steady states $u = 1$ in the limit $t \rightarrow -\infty $ and $u = 3$ in the limit $t \rightarrow +\infty$. Place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions please separate them with a ;.
Solution: $\boxed{u(x,t) = 2+\tanh(\frac{3}{2}(x-9t))}$