  We consider the $L^2$ well-posedness of third order Benjamin-Ono equation. We
show that by means of a normal form and a gauge transformation, the equation
can be changed into an Airy-type equation. A second goal of this work is to
establish that the solutions to the nonlinear third order Benjamin-Ono equation
problem exhibit a dispersive decay estimate analogue to the corresponding
linear associated problem. The key ingredient is the use of a nonlinear vector
field method.
