Problem: Find a localized self similarity solution (soliton behvaiour) for the non-linear partial differential equation $u_{tt}-u_{xx}-3\bigl(u^{2}\bigr)_{xx}-u_{xxxx}=0$ with $u(0, 0) = 1/4$ as the maximum value and $\partial_t u(-2, 1) < 0$. Localization means $u(x,t)$ and its derivatives vanish at $t= \pm \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment.
Solution: $\boxed{u(x, t) = \frac{1}{4\cosh^2(\frac{x-\sqrt{\frac{3}{2}}t}{2\sqrt{2}})}}$