Problem: Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 2 (u^3)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = \frac{\sqrt{3}}{3}$.
Solution: $\boxed{u(x,t) = \frac{\sqrt{3}}{3} \text{ sech}(\frac{1}{\sqrt{3}} (x - \frac{2}{\sqrt{3}}t)); u(x,t) = \frac{\sqrt{3}}{3} \text{ sech}(\frac{1}{\sqrt{3}} (x + \frac{2}{\sqrt{3}}t))}$