Summary: By a nonlinear change of variables from the original one, we derive a ”small steepness full dispersion” system for surface water waves which is consistent with the water wave system. This system is symmetrizable and we prove that the Cauchy problem is well-posed on large time of order 1/{\(\epsilon\)} where {\(\epsilon\)} is the steepness coefficient, implying (together with the results of D. Lannes) its full rigorous justification as an asymptotic model to the full Euler equations with free surface.(To Peter Constantin with friendship and admiration.){
©2012 American Institute of Physics}