Problem: Find a localized travelling wave solution (soliton) for the nonlinear partial differential equation $u_{tt} - u_{xx} - 8(u^2)_{xx} - u_{xxxx}=0$ subject to the condition that the maximum value is $u(0,0) = 2$.
Solution: $\boxed{\frac{2}{\cosh^2(2\sqrt{\frac{2}{3}}(x-\sqrt{\frac{35}{3}}t))}}$