Validity of the Korteweg–de Vries approximation for the two‐dimensional water wave problem in the arc length formulation

Abstract

AbstractWe 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. © 2011 Wiley Periodicals, Inc.