The Benjamin-Ono equation
is a nonlinear partial integro-differential equation
. It was introduced by Benjamin (1967) and Ono (1975) as an equation describing one-dimensional internal waves
in deep water.
The Benjamin-Ono equation has a form
is the Hilbert transform
It is a soliton
equation and can be solved by an inverse scattering transform
-  T. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-562.
-  H. Ono, Algebraic solitary waves in stratified fluids, J. Phys.Soc. Japan 39 (1975), 1082-1091.