The NLS Approximation Makes Wrong Predictions for the Water Wave Problem in Case of Small Surface Tension and Spatially Periodic Boundary Conditions

The nonlinear Schrodinger (NLS) equation describes small modulations in time and space of a spatially and temporally oscillating wave packet advancing in a laboratory frame. It has first been derived for the so called water wave problem in 1968 and the proof that it makes correct predictions has been recently the subject of intensive research. It is the purpose of this paper to show that in general this approximation is not valid without further assumptions. We construct a counter example showing that the NLS approximation makes wrong predictions for the water wave problem in case of small surface tension and spatially periodic boundary conditions.