Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 71 / 1 / a / Solution

Use the linear dispersive Stokes equation with time variable $t$. Put $E_k=e^{-ikx+\omega (k)t}$. Expanding the required local relation givesChoose $X=q_{xx}+ik{q}_x+(1-k^2)q$. Direct differentiation then yieldsThe unwanted coefficient of $q$ cancels precisely when $\omega (k)=ik(1-k^2)$. HenceWith these choices, the local relation is exactly $E_k(q_t+q_x+q_{xxx})=0$ for every complex spectral parameter for a linear boundary value problem.