  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.
