  The long wave-short wave model describes the interaction between the long
wave and the short wave. Exact higher-order rational solution expressed by
determinants is calculated via the Hirota's bilinear method and the KP
hierarchy reduction. It is found that the fundamental rogue wave for the short
wave can be classified into three different patterns: bright, intermediate and
dark ones, whereas the rogue wave for the long wave is always bright type. The
higher-order rogue waves correspond to the superposition of fundamental rogue
waves. The modulation instability analysis show that the condition of the
baseband modulation instability where an unstable continuous-wave background
corresponds to perturbations with infinitesimally small frequencies, coincides
with the condition for the existence of rogue-wave solutions.
