Problem: Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 9(u^3)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 1$.
Solution: $\boxed{u(x,t)=\frac{1}{\cosh(\frac{3}{\sqrt{2}}(x+\sqrt{\frac{11}{2}}t))}}$