One of the important open questions in the theory of free--surface ideal fluid flows is the dynamic stability of traveling wave solutions. In a spectral stability analysis, the first variation of the governing Euler equations is required which raises both theoretical and numerical issues. With Zakharov and Craig and Sulem's formulation of the Euler equations in mind, this paper addresses the question of analyticity properties of first (and higher) variations of the Dirichlet--Neumann operator. This analysis will have consequences not only for theoretical investigations, but also for numerical simulations of spectral stability of traveling water waves.
"Parametric analyticity of functional variations of Dirichlet-Neumann operators." Differential Integral Equations 21 (5-6) 541 - 574, 2008.