= Decay of a Fourier mode in a third-order spatial evolution equation
{title2=$2\psi_x+i\psi_{yyy}=0\ \Longrightarrow\ \psi=e^{-k^3x/2}e^{iky}$}
A separated <Fourier mode> with initial value $e^{iky}$ at $x=0$ obeys $X'=-k^3X/2$ and $X(0)=1$. Its amplitude decays for $x\to+\infty$ precisely when $k>0$. The zero mode stays constant and negative modes grow. Thus a decay requirement on a half-space imposes a sign condition on the real wave number; a formal separated expression alone does not establish compatibility of both <boundary conditions>.
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