Problem: Find a nontrivial self similarity solution (soliton behaviour) for the non-linear partial differential equation $$ \partial_t u + \left(\frac{45}{16} u^{1/2} - \frac{3}{2} u\right)\partial_x u + \partial_x^3 u=0 $$ The wave is assumed to travel with velocity $v=0.5$. The boundary condition is $\lim_{|x|\to\infty} u(x,t)=0$. Consider this in the limit "$t\rightarrow\infty$". Please place your final solution in a $\boxed{}$ LaTeX Environment. If there are multiple solutions, please separate them with a ";". If there exist forms in terms of exponential for functions in the output, then you should use them.
Solution: $\boxed{u(x,t) = \frac{16}{\left(6 + e^{\frac{x-0.5t}{2\sqrt{2}}} + e^{-\left(\frac{x-0.5t}{2\sqrt{2}}\right)}\right)^2}}$