Abstract
We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data of the form
$${\epsilon B(\epsilon \alpha)e^{ik \alpha}}$$
for k > 0, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrödinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times of order
$${O(\epsilon^{-2})}$$
provided the initial data differs from the wave packet by at most
$${O(\epsilon^{3/2})}$$
in Sobolev spaces. These results are obtained by directly applying modulational analysis to the evolution equation with no quadratic nonlinearity constructed in Wu (Invent Math 117(1):45–135, 2009) and by the energy method.
