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.