= Spatial root splitting for the dispersive Stokes resolvent
{title2=$r^3+r+p=0$}
For a time <Laplace transform> parameter with $\operatorname{Re}p>0$, the spatial <characteristic roots> satisfy $r^3+r+p=0$. Exactly one root $r_-$ has negative <real part> and two have positive <real part>. Indeed an imaginary root would force $p=-i(k-k^3)$ to be imaginary. The count is therefore constant on the connected right half-plane; for real $p>0$, the cubic has one negative real root and a conjugate pair with positive real parts. Repeated roots would satisfy $3r^2+1=0$, which again puts $p$ on the imaginary axis. Thus $r_-(p)$ is a nonzero <holomorphic function> in the right half-plane. Only $e^{r_-x}$ decays at positive spatial infinity.
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