Problem: Find a self similarity solution (soliton behvaiour) for the non-linear partial differential equation $\partial_t u +\sqrt{u}\partial_x u + \partial_x^3 u=0, \lim_{|x|\to\infty} u(x,t)=0$ in the limit $t\rightarrow\infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment. There should be a free parameter $v$. Set $v=0.5$. If there are multiple solutions please separate them with a ;.
Solution: $\boxed{u(x,t) = \frac{225}{256} \sech^4( \frac{\sqrt{2}}{8} (x - 0.5t) )}$