J. Chen and D.E. Pelinovsky
Bright and dark breathers of the Benjamin-Ono equation on the traveling periodic background
Wave Motion 126 (2024) 103263 (10 pages)
Abstract:
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.
Keywords:
Benjamin-Ono equation; traveling periodic wave; multi-periodic solutions; multi-soliton solutions; Lax spectrum; breathers.