This paper outlines the theoretical background of a new approach towards an accurate and well-conditioned perturbative calculation of Dirichlet{Neumann operators (DNOs) on domains that are perturbations of simple geometries. Previous work on the analyticity of DNOs has produced formulae that, as we have found, are very ill-conditioned. We show how a simple change of variables can lead to recursions that satisfy analyticity estimates without relying on subtle cancellation properties at the heart of previous formulae.

This paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom. The object is to describe the character of wave propagation in a long wave asymptotic regime, extending the results of R. Rosales & G. Papanicolaou [RP]. We take the point of view of perturbation of a Hamiltonian system dependent on a small scaling parameter, with the starting point being V.E. Zakharov's Hamiltonian [Z] for the Euler equations for water waves. We consider bottom topography which is periodic in horizontal variables on a short length scale, with the amplitude of variation being of the same order as the fluid depth. The bottom may also exhibit slow variations at the same length scale as, or longer than, the order of the wavelength of the surface waves.In the two dimensional case of waves in a channel, we give an alternate derivation of the effective KdV equation that is obtained in [RP]. In addition, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves, in cases in which the bottom possesses both short and long scale variations. In certain cases we also obtain unidirectional equations that are similar to the KdV equation. In three dimensions we obtain effective three dimensional long wave equations in a Boussinesq scaling regime, and again in certain cases an effective KP system in the appropriate unidirectional limit.The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators. In the present case this involves the DirichletNeumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium.

Abstract. The main results of this paper are existence theorems for traveling gravity and capillary gravity water waves in two dimensions, and capillary gravity water waves in three dimensions, for any periodic fundamental domain. This is a problem in bifurcation theory, yielding curves in the two dimensional case and bifurcation surfaces in the three dimensional case. In order to address the presence of resonances, the proof is based on a variational formulation and a topological argument, which is related to the resonant Lyapunov center theorem. 1. Introduction. Nonlinear periodic traveling waves on the free surface of an ideal fluid tend to form hexagonal patterns. This phenomenon is the focus of a number of recent papers on the subject of water waves, and it is the topic of the present article. In previous work, various approximations to the evolution equations for free surface waves are used, in particular the KP system by J. Hammack, N. Scheffner, and H. Segur [11], and J. Hammack, D. McCallister, N. Scheffner, and H. Segur [12], and alternatively with certain formal shallow water expansions of the Euler equations by P. Milewski and J.B. Keller [16]. A natural question is whether similar patterns can be shown to occur in solutions of the full Euler equations themselves. This is the focus of a series of papers by the present authors. In [18] and in [20] we report on hexagonal wave patterns and other phenomena in numerical computations of solutions, which are shown to satisfy spectral criteria for numerical convergence to solutions of Euler's equations. In the present article we describe rigorous existence results for periodic traveling wave solutions in free surfaces. The goal is to prove the existence of nontrivial traveling wave solutions to the water wave problem for gravity and capillary gravity waves in two and three dimensions. In two dimensions this is proven for both gravity and capillary gravity water waves, constituting a new and relatively straightforward approach to the theorems of T. Levi-Civita and D. Struik. In three dimensions we prove the existence of traveling capillary gravity water waves. However, the problem of gravity waves in three dimensions exhibits the phenomena of small divisors, and it remains open. The theorem that we prove is given below.

In this paper we present results on the stability of perturbation methods for the evaluation of Dirichlet-Neumann operators (DNO) defined on domains that are viewed as complex deformations of a basic, simple geometry. In such cases, geometric perturbation methods, based on variations of the spatial domains of definition, have long been recognized to constitute efficient and accurate means for the approximation of DNO and, in fact, several numerical implementations have been previously proposed. Inspired by our recent analytical work, here we demonstrate that the convergence of these algorithms is, quite generally, limited by numerical instability. Indeed, we show that these standard perturbative methods for the calculation of DNO suffer from significant ill-conditioning which is manifest even for very smooth boundaries, and whose severity increases with boundary roughness. Moreover, and again motivated by our previous work, we introduce an alternative perturbative approach that we show to be numerically stable. This approach can be interpreted as a reformulation of classical perturbative algorithms (in suitable independent variables), and thus it allows for both direct comparison and the possibility of analytic continuation of the perturbation series. It can also be related to classical (preconditioned) spectral approaches and, as such, it retains, in finite arithmetic, the spectral convergence properties of classical perturbative methods, albeit at a higher computational cost (as it does not take advantage of possible dimensional reductions). Still, as we show, an alternative approach such as the one we propose may be mandated in cases where substantial information is contained in high-order harmonics and /or perturbation coefficients of the solution.

The analytic dependence of Dirichlet-Neumann operators (DNO) with respect to variations of their domain of definition has been successfully used to devise diverse computational strategies for their estimation. These strategies have historically proven very competitive when dealing with small deviations from exactly solvable geometries, as in this case the perturbation series of the DNO can be easily and recursively evaluated. In this paper we introduce a scheme for the enhancement of the domain of applicability of these approaches that is based on techniques of analytic continuation. We show that, in fact, DNO depend analytically on variations of arbitrary smooth domains. In particular, this implies that they generally remain analytic beyond the disk of convergence of their power series representations about a canonical separable geometry. And this, in turn, guarantees that alternative summation mechanisms, such as Padé approximation, can be effectively used to numerically access this extended domain of analyticity. Our method of proof is motivated by our recent development of stable recursions for the coefficients of the perturbation series. Here, we again utilize this recursion as we compare and contrast the performance of our new algorithms with that of previously advanced perturbative methods. The numerical results clearly demonstrate the beneficial effect of incorporating analytic continuation procedures into boundary perturbation methods. Moreover, the results also establish the superior accuracy and applicability of our new approach which, as we show, allows for precise calculations corresponding to very large perturbations of a basic geometry.