 Spatial root splitting for the dispersive Stokes resolvent

For a time Laplace transform parameter with $\mathrm{Re}p>0$, 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.