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The Benjamin-Ono (BO) equation describes long internal waves of smallamplitude in deep fluids. Compared to its counterpart for shallow fluids, theKorteweg-de Vries (KdV) equation, the BO equation admits exact solutions forthe traveling periodic and solitary waves as well as their interactionsexpressed in elementary (trigonometric and polynomial) functions. Motivated bya recent progress for the KdV equation, we discover here two scenarios of thesoliton-periodic wave interactions which result in the propagation of eitherelevation (bright) or depression (dark) breathers (periodic in time coherentstructures). The existence of two different breathers is related to theband-gap spectrum of the Lax operator associated with the traveling periodicwave. Given a simple structure of the exact solutions in the BO equation, weobtain a closed-form expression for multi-solitons interacting with thetraveling periodic wave.

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