Problem: Find a self similarity solution (soliton behavior) for the non-linear partial differential equation $ \partial_t u + 2u^{1/2} \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 $v=0.5$. If there are multiple solutions please separate them with a ;.
Solution: $$\boxed{u(x,t) = \frac{225}{1024} \sech^4\left( \frac{\sqrt{2}}{8}(x - 0.5t) \right) }$$