t

. Put

E_{k} = e^{- ik x + ω (k) t}

. Expanding the required local relation gives

(E_{k} q)_{t} + (E_{k} X)_{x} = E_{k} (q_{t} + ω q + X_{x} - ik X) .

Choose

X = q_{xx} + ik q_{x} + (1 - k^{2}) q

. Direct differentiation then yields

X_{x} - ik X = q_{xxx} + q_{x} - ik (1 - k^{2}) q .

The unwanted coefficient of

q

cancels precisely when

ω (k) = ik (1 - k^{2})

. Hence

ω (k) = i (k - k^{3}), X = q_{xx} + ik q_{x} + (1 - k^{2}) q .

With these choices, the local relation is exactly

E_{k} (q_{t} + q_{x} + q_{xxx}) = 0

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