In 1968 V.E. Zakharov derived the Nonlinear Schrödinger equation for the twodimensional water wave problem in the absence of surface tension, that is, for the evolution of gravity driven surface water waves, in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. In this paper we give a rigorous proof that the wave packets in the two-dimensional water wave problem in a canal of finite depth can be approximated over a physically relevant timespan by solutions of the Nonlinear Schrödinger equation.

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Abstract. We consider a nonlinear Klein-Gordon equation with a quasilinear quadratic term. The Nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the quasilinear KleinGordon equation. It is the purpose of this paper to present a method which allows one to prove error estimates in Sobolev norms between exact solutions of the quasilinear Klein-Gordon equation and the formal approximation obtained via the NLS equation. The paper contains the first validity proof of the NLS approximation of a nonlinear hyperbolic equation with a quasilinear quadratic term by error estimates in Sobolev spaces. We expect that the method developed in the present paper will allow an answer to the relevant question of the validity of the NLS approximation for other quasilinear hyperbolic systems.

We consider the two-dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension. It has been proven by several authors that long-wavelength solutions to this problem can be approximated over a physically relevant timespan by solutions of the Korteweg-de Vries equation or, for certain values of the surface tension, by solutions of the Kawahara equation. These proofs are formulated either in Lagrangian or in Eulerian coordinates. In this paper, we provide a new proof, which is simpler, more elementary, and shorter. Moreover, the rigorous justification of the KdV approximation can be given for the cases with and without surface tension together by one proof. In our proof, we parametrize the free surface by arc length and use some geometrically and physically motivated variables with good regularity properties. This formulation of the water wave problem has already been of great usefulness for Ambrose and Masmoudi to simplify the proof of the local well-posedness of the water wave problem in Sobolev spaces.

We consider a nonlinear dispersive equation with a quasilinear quadratic term. We establish two results. First, we show that solutions to this equation with initial data of order O(ε) in Sobolev norms exist for a time span of order O(ε −2 ) for sufficiently small ε. Secondly, we derive the Nonlinear Schrödinger (NLS) equation as a formal approximation equation describing slow spatial and temporal modulations of the envelope of an underlying carrier wave, and justify this approximation with the help of error estimates in Sobolev norms between exact solutions of the quasilinear equation and the formal approximation obtained via the NLS equation. The proofs of both results rely on estimates of appropriate energies whose constructions are inspired by the method of normal-form transforms. To justify the NLS approximation, we have to overcome additional difficulties caused by the occurrence of resonances. We expect that the method developed in the present paper will also allow to prove the validity of the NLS approximation for a larger class of quasilinear dispersive systems with resonances.