Bright and dark breathers of the Benjamin–Ono equation on the traveling periodic background

The Benjamin–Ono (BO) equation describes long internal waves of small amplitude in deep fluids. Compared to its counterpart for shallow fluids, the Korteweg–de Vries (KdV) equation, the BO equation admits exact solutions for the traveling periodic and solitary waves as well as their interactions expressed in elementary (trigonometric and polynomial) functions. Motivated by a recent progress for the KdV equation, we discover here two scenarios of the soliton-periodic wave interactions which result in the propagation of either elevation (bright) or depression (dark) breathers (periodic in time coherent structures). The existence of two different breathers is related to the band-gap spectrum of the Lax operator associated with the traveling periodic wave. Given a simple structure of the exact solutions in the BO equation, we obtain a closed-form expression for multi-solitons interacting with the traveling periodic wave. •New families of breathers on the background of a traveling periodic wave are obtained for the classical Benjamin–Ono equation.•Solution families of breathers are derived by the degeneration of the multi-periodic solutions.•The determinant form of breather solutions is illustrated in the simplest cases graphically.•Breather solutions are intermediate points between the families of multi-periodic and multi-soliton solutions.