 Decay of a Fourier mode in a third-order spatial evolution equation

A separated Fourier mode with initial value $e^{iky}$ at $x=0$ obeys $X^{\prime }=-k^{3}X/2$ 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.