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We consider the two-dimensional water wave problem in an infinitely longcanal of finite depth both with and without surface tension. In order todescribe the evolution of the envelopes of small oscillating wave packet-likesolutions to this problem the Nonlinear Schr\"odinger equation can be derivedas a formal approximation equation. In recent years, the validity of thisapproximation has been proven by several authors for the case without surfacetension. In this paper, we rigorously justify the Nonlinear Schr\"odingerapproximation for the cases with and without surface tension by proving errorestimates over a physically relevant timespan in the arc length formulation ofthe two-dimensional water wave problem. The error estimates are uniform withrespect to the strength of the surface tension, as the height of the wavepacket and the surface tension go to zero.

archive for rational mechanics and analysis

Validity of the Nonlinear Schr\"odinger Approximation for the  Two-Dimensional Water Wave Problem With and Without Surface Tension in the  Arc Length Formulation