Some mixed soliton wave interaction patterns and stabilities for Rabi-coupled nonlocal Gross–Pitaevskii equations

Some mixed interactions of the soliton, breather and rogue wave(RW) formations and their dynamics are studied in Rabi-coupled Bose–Einstein condensates(BECs) with spatially varying dispersion and nonlinearity. We consider the 2-component inhomogeneous Rabi-coupled Gross–Pitaevskii (GP) equations through suitable three kinds of rotational and similarity transformations. The effects of inhomogeneity and optical lattice hyperbolic potentials of the RWs are investigated with two different forms of potential strengths, and some oscillating behaviors of dark–bright solitons, RWs and breather solitons with Rabi coupling terms are shown in 2-component condensates. We demonstrate creation of some RWs coexisting with dark–bright soliton part in second component of the 2-component GP equations. We show that some mixed interactions of vector soliton, breather and RW formations by employing parabolic cylinder modulations, and find a striking feature of Rabi coupling with spatial modulation. Further, the RWs can be converted into broad based zero background RW appearing on the top of a bright soliton by introducing spatial modulation in 2-component systems. Some dynamic behaviors of the RW solutions are investigated analytically with the external potentials, and the mixed waves, interaction patterns and stabilities for Rabi-coupled nonlocal GP systems are presented with modulation instability, which can be used to calculate nonautonomous mixed wave interactions and the potential applications for the RW phenomena.