  We consider instability and localized patterns arising from long wave-short
wave (LWSW) resonance in the non-integrable regime numerically. We study the
stability and instability of elliptic-function periodic waves with respect to
subharmonic perturbations, whose period is a multiple of the period of the
elliptic waves. We thus find the modulational instability (MI) of the
corresponding dnoidal waves. Upon varying parameters of dnoidal waves,
spectrally unstable ones can be transformed into stable states via the
Hamiltonian Hopf bifurcation. For snoidal waves, we find a transition of the
dominant instability scenario between the MI and instability with a bubble-like
spectrum. For cnoidal waves, we produce three variants of the MI. Evolution of
the unstable states is also considered, leading to formation of rogue waves on
top of the elliptic-wave and continuous-wave backgrounds.
