Semirational solutions and baseband modulational instability of the AB system in fluid mechanics

Abstract

Under investigation in this paper is the AB system describing marginally unstable baroclinic wave packets in geophysical fluids. By means of the n -fold modified Darboux transformation, the semirational solutions in terms of the determinants of the AB system are derived. These solutions, which are a combination of rational and exponential functions, can be used to model the nonlinear superposition of the Akhmediev breathers (or the Kuznetsov-Ma breathers) and the rogue waves. The k -order rogue wave of the AB system is produced by the interaction between the l-order rogue wave with $\frac{1}{2}(k-l)(k+l+1)$ neighboring elements in the $(k-l)$ -order breathers $(0<l<k)$ . The proper values of eigenvalue $ \lambda$ and shift $ S_{0}$ are also the requirement for generating the higher-order rogue waves. The link between the baseband modulational instability and the existence condition of these rogue waves is revealed.