= Solution
For <separation of variables> matching the boundary <Fourier mode>, take $\psi_0=X(x)e^{iky}$ with $X(0)=1$. Because $\partial_y^3e^{iky}=-ik^3e^{iky}$, the reduced <ordinary differential equation> is
$$
2X'+k^3X=0.
$$
Therefore
$$
\boxed{\psi_0(x,y)=e^{-k^3x/2}e^{iky}.}
$$
The <decay of a Fourier mode in a third-order spatial evolution equation> requires a sign check: this satisfies the requested limit at positive infinity precisely when $\boxed{k>0}$. For $k=0$ it remains identically one, while for $k<0$ its amplitude grows exponentially. Hence for those nonpositive values there is no separated solution meeting both prescribed <boundary conditions>; the printed assumption that $k$ is merely real needs this additional restriction for decay.
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