Long Wave Approximations for Water Waves
We derive a class of symmetric and conservative systems which describe the interaction of long water waves. We prove rigorously that the solutions of the water waves equations can be approximated in terms of the solutions to these symmetric systems. We use this fact to prove rigorously that all the systems of the wide class obtained by J.L. Bona et al.  (and in particular the historical Boussinesq system) provide a good approximation to the water waves problem. Our error estimates are better than that obtained in the case of the decoupled KdV-KdV approximation, are valid in the 2-D case for any existing solution to the Euler equations, and remain true in the periodic case.
KeywordsEuler Equation Water Wave Conservative System Wave Approximation Periodic Case
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