Problem: Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 6 (u^2)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = \frac{1}{2}$.
Solution: $\boxed{u(x,t) = \frac{1}{2} \text{ sech}^2(\frac{1}{\sqrt{2}} (x - \sqrt{3}t)); u(x,t) = \frac{1}{2} \text{ sech}^2(\frac{1}{\sqrt{2}} (x + \sqrt{3}t))}$