Justification of the nonlinear Schroedinger equation for a resonant Boussinesq model Wolf-Patrick DuellGuido Schneider 35Q5535Q3535C2070K3076B15water wave modelmodulation equationapproximationnormal form The nonlinear Schroedinger equation formally describes slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet. It is the purpose of this paper to prove estimates between the formal approximation, obtained via the nonlinear Schroedinger equation, and true solutions of the original system. The method developed in an earlier paper in case of non-trivial quadratic resonances is improved to cover also the additional problem of a trivial resonance at the wavenumber $k = 0$ as it occurs for the water wave problem. For a Boussinesq equation, a formal and phenomenological model for surface water waves subject to gravity and surface tension, we establish the approximation property in case the formal NLS approximation is stable in the system for the three wave interaction associated to the resonance. Although we restrict ourselves to a Boussinesq equation, we believe that the result is also true for the full water wave problem. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2824 10.1512/iumj.2006.55.2824 en Indiana Univ. Math. J. 55 (2006) 1813 - 1834 state-of-the-art mathematics http://iumj.org/access/