LIMITS FOR THE KORTEWEG-DE-VRIES APPROXIMATION

A complex-valued Boussinesq-equation is considered as system, where a KdV-equation can be derived as formal approximation equation. It is sh own that in the case of complex-valued initial conditions the validity of the approximation equation is strongly limited. To obtain a weaken ed approximation property analytic initial conditions are necessary. W e use the method usually refered as using scale of Banach spaces which is also common in the existence theory for the wafer wave problem. Th us, our result allows us to extend a result of Kano and Nishida from 1 986 on the validity of the KdV-approximation for the water wave proble m to a longer time scale.