Rogue-wave solutions for an inhomogeneous nonlinear system in a geophysical fluid or inhomogeneous optical medium

• We investigate an inhomogeneous nonlinear system.
• We construct the generalized Darboux transformation.
• We derive the first- and second-order bright and dark rogue-wave solutions.
• Properties of the first- and second-order bright and dark rogue waves are discussed.

Under investigation in this paper is an inhomogeneous nonlinear system, which describes the marginally-unstable baroclinic wave packets in a geophysical fluid or ultra-short pulses in nonlinear optics with certain inhomogeneous medium existing. By virtue of a kind of the Darboux transformation, under the Painlevé integrable condition, the first- and second-order bright and dark rogue-wave solutions are derived. Properties of the first- and second-order bright and dark rogue waves with α(t), which measures the state of the basic flow, and β(t), representing the interaction of the wave packet and mean flow, are graphically presented and analyzed: α(t) and β(t) have no influence on the wave packet, but affect the correction of the basic flow. When we choose α(t) as a constant and linear function, respectively, the shapes of the first- and second-order dark rogue waves change, and the peak heights and widths of them alter with the value of β(t) changing.