Calculation of resonant short-crested waves in deep water

Abstract

Solving the problem of resonant short-crested waves is very challenging because the appearance of small divisors causes the classical perturbation methods to fail. From a numerical point of view, the case of resonant gravity short-crested waves has been studied, but as far as we know, there are very few results in the case of resonant capillary-gravity short-crested waves. In fact, to the best of our knowledge, the most related study which has been made is the one of Craig and Nicholls [SIAM J. Math. Anal. 32, 323 (2000)] who gave existence theorems for the case where the surface tension is supposed not to be too small. There is a need for such an investigation, and the work considered herein therefore provides a calculation technique and presents new results on resonant short-crested gravity-capillary waves. We overcome the technical problems associated with small divisors by using a method derived from Whitham’s variational formulation of the classical problem of short-crested waves. Whitham’s method is...