Problem: Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \partial_t u + u^2 \partial_x u + \partial_x^3 u = 0 $ in the limit $t\to\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=6$. If there are multiple solutions please separate them with a ;.
Solution: $$\boxed{u(x,t) = 6 \sech{\left( \sqrt{6} (x - 6t) \right)}}$$