  Different from statistical considerations on stochastic wave fields, this
paper aims to contribute to the understanding of (some of) the underlying
physical phenomena that may give rise to the occurrence of extreme, rogue,
waves. To that end a specific deterministic wavefield is investigated that
develops extreme waves from a uniform background. For this explicitly described
nonlinear extension of the Benjamin-Feir instability, the soliton on finite
background of the NLS equation, the global down-stream evolving distortions,
the time signal of the extreme waves, and the local evolution near the extreme
position are investigated. As part of the search for conditions to obtain
extreme waves, we show that the extreme wave has a specific optimization
property for the physical energy, and comment on the possible validity for more
realistic situations.
