## Abstract

In this paper we establish the existence and analyticity of periodic solutions of a classical free-boundary model of the evolution of three-dimensional, capillary–gravity waves on the surface of an ideal fluid. The result is achieved through the application of bifurcation theory to a boundary perturbation formulation of the problem, and it yields analyticity *jointly* with respect to the perturbation parameter and the spatial variables. The travelling waves we find can be interpreted as resulting from the (nonlinear) interaction of two two-dimensional wavetrains, giving rise to a periodic travelling pattern. Our analyticity theorem extends the most sophisticated results known to date in the absence of resonance; ‘short crested waves’, which result from the interaction of two wavetrains with unit amplitude ratio are realized as a special case. Our method of proof also sheds light on the convergence and conditioning properties of classical boundary perturbation methods for the numerical approximation of travelling surface waves. Indeed, we demonstrate that the rather unstable numerical behaviour of these approaches can be attributed to the strong but subtle cancellations in the formulas underlying their classical implementations. These observations motivate the derivation and use of an alternative, stable, formulation which, in addition to providing our method of proof, suggests new stabilized implementations of boundary perturbation algorithms.

## 1. Introduction

The stable and accurate numerical simulation of free-surface ocean dynamics is one of the central problems in computational fluid mechanics. From shoaling and breaking of waves over nearshore regions to energy, momentum, and scalar transport in the open ocean, the rapid and reliable approximation of the surface of a fluid is a necessary tool in problems of physical relevance. Surface waves that propagate with constant velocity and without change of form (the travelling waves) are a distinguished class of motions which are believed to be a fundamental building-block of surface ocean dynamics.

In this paper we take up the mathematical question of regularity properties of travelling wave solutions of the classical water wave model (see §2), which constitutes an accurate representation for the motion of the free surface of the ocean. In particular, we demonstrate that the water wave problem in *d*-dimensions admits *surfaces* of solutions, parameterized by (*d*−1) many parameters *ε*∈**R**^{d−1}, which are *jointly* analytic in the parametric and spatial variables. Our method of proof is perturbative in nature and general enough to encompass every case away from resonances (see §4).

The first rigorous existence theorems for travelling wave solutions to the water wave model date to the results in two space dimensions without surface tension by Levi-Civita (1925) (infinite depth) and Struik (1926) (finite depth) who used complex variables techniques. With the advent of the modern computer there was a resurgence of interest in the problem in the 1970s and 1980s as highly nonlinear waveforms could now be simulated (e.g. Schwartz, 1974; Roberts, 1983; Schwartz & Roberts, 1983; Marchant & Roberts 1987). This resurgence was also accompanied by new theoretical developments. For instance, Reeder & Shinbrot (1981*a*,*b*) studied the phenomena of Wilton ripples which arise in two-dimensional travelling capillary–gravity water waves. They showed existence and smoothness of branches of travelling wave solutions which exist in the presence of resonance in the linearized problem. Other important theoretical results in two space dimensions include those of Toland and collaborators, who used various integral formulations of the two-dimensional water wave problem coupled with variational techniques (e.g. minimizers, mountain pass). An important early result of Toland's (1978) established the *global* existence of the bifurcating branch of solutions all the way to the Stokes singularity. Jones & Toland (1986) also looked at surface tension effects in two dimensions, and subharmonic bifurcations in (Jones & Toland 1985). Subharmonic bifurcation was also the object of Buffoni *et al*. (2000).

In three dimensions, on the other hand, the most general results to date are those of Craig & Nicholls (2000) who, in the presence of non-zero surface tension, established existence of travelling capillary–gravity water waves with an arbitrary fundamental period. The theorem of Craig & Nicholls used the surface formulation of Zakharov (1968) and Craig & Sulem (1993), coupled with the Lyapunov–Schmidt procedure from bifurcation theory. Other existence results in three dimensions include that of Sun (1993), who viewed the travelling wave as generated by a surface pressure, and Groves & Mielke (2001) and Groves (2001), who have studied travelling waves using a ‘spatial dynamics’ approach. In this formulation, the direction of propagation, in the traveling wave equations, is considered the dynamical quantity; the transverse direction is typically considered to be periodic and then periodic (in propagation direction) solutions are sought.

Regarding *spatial* analyticity of solitary travelling water waves, we mention the seminal work of Lewy (1952) who, using complex variables techniques, established that (in the presence of gravity alone) once the surface is known to be *C*^{1}, it is automatically analytic. Matei (2002) and Craig & Matei (2003) have extended this result to non-zero capillarity in two and three dimensions, respectively, using a partial hodograph transform. We also mention the broad generalization of these techniques in the recent work of Koch *et al*. (2004). Bona & Li (1997) established both spatial analyticity and decay (at infinity) estimates for travelling waves for a wide class of nonlinear, dispersive wave equations. Using a velocity potential/streamfunction formulation, they extended these theorems to travelling water waves in two dimensions.

The three-dimensional results most closely related to those we present herein are those of Reeder & Shinbrot (1981*c*), who demonstrate the existence and parametric analyticity of ‘short-crested’ capillary–gravity waves of sufficiently small amplitude; short-crested waves are typically defined (Dias & Kharif 1999) as the waves which result from the (nonlinear) interaction of two periodic wavetrains of equal amplitude, infinite extent and non-zero angle of interaction (as the angle of interaction approaches zero, the waves are typically referred to as ‘long-crested’). Akin to the method we adopt in §3, Reeder & Shinbrot also use a ‘domain flattening’ change of variables. Our results expand on those of (Reeder & Shinbrot 1981*c*) in two important directions: first, our derivations demonstrate that, in fact, the free boundary and velocity potential are *jointly* analytic in space and bifurcation parameter (a fact that, of course, does not follow from separate analytic dependence); and second, our developments allow for the interaction of wavetrains of *arbitrary* amplitude ratio, i.e. not necessarily short-crested waves. To attain the latter, our approach entails the use of *multi-dimensional* perturbation parameters.

In addition to establishing existence and analyticity of hypersurfaces of travelling water waves, our work also sheds light on the convergence and conditioning properties of classical boundary perturbation methods for the numerical simulation of travelling capillary–gravity water waves. In particular, we discuss the method of Stokes (1847) (which we term the method of ‘field expansions’ (FE)) that was further refined and carried out to high order by Roberts (1983), Schwartz & Roberts (1983), and Marchant & Roberts (1987) for three-dimensional travelling water waves in the absence of surface tension. As we show in §2*d*, this method produces *unstable* results as the perturbation order is increased due to subtle but significant *cancellations* which are present in the underlying recursions. As we explain, a further consequence of this observation is that these FE recursions cannot be used for a direct proof of existence or analyticity. However, as we anticipated above, a direct proof can be realized once a ‘domain flattening’ change of variables is effected, as this can be shown to implicitly account for all significant cancellations. This latter fact suggests a stabilized approach to numerical simulation, whose thorough investigation we defer to future work (see also Nicholls & Reitich 2001*a*,*b*, 2003, 2004*a*,*b*).

The remainder of the paper is organized as follows: first, in §2, we introduce the equations of motion and the classical FE approach to simulating travelling water waves; in particular, in §2d, we demonstrate how these classical recursions rely heavily on significant cancellations for their convergence. In §3, we introduce a change of variables which substantially ameliorates these cancellations and paves the way for the analyticity proof of §4; some auxiliary results necessary for this latter proof are collected in appendix A.

## 2. Preliminaries

In this section, we briefly review the equations of motion of the water wave problem (ideal fluid, free-surface flow) and outline the classical FE technique for perturbatively computing solutions. With the aid of a numerical implementation of this algorithm and several simulations, we illustrate how these recursions are inherently unstable at high orders owing to underlying *cancellations*.

### (a) Equations of motion

Consider a *d*-dimensional (*d*=2, 3) fluid (one vertical dimension specified by the variable *y* and (*d*−1) horizontal dimensions specified by *x*) bounded below by an impermeable bottom at *y*=−*h* (*h* possibly infinite) and above by an undetermined air/fluid interface, *y*=*η*(*x*, *t*), which occupies the domainIn the case of finite depth, no generality is lost if *h* is set to one, as this simply amounts to a rescaling of independent variables. Consider also the classical assumption that the fluid be periodic with respect to the lattice *Γ*⊂**R**^{d−1}, which defines a parallelogram of periodicity *P*(*Γ*). The equations of motion of an ideal fluid in such a domain under the effects of gravity and capillarity are (Lamb 1993)(2.1a)(2.1b)(2.1c)(2.1d)where *φ* is the velocity potential, *g* is the constant of gravity, *σ* is the constant of capillarity and *κ* is the curvature:As we stated, we shall be concerned with travelling waves translating uniformly with speed *c*∈**R**^{d−1}, which satisfy(2.2a)

(2.2b)

(2.2c)

(2.2d)

### (b) Bifurcation theory

Adopting a bifurcation theoretic approach, we seek solutions of equation (2.2) near the quiescent state (*φ*=*η*=0 and *any* velocity *c*) which forms a ‘trivial’ family of solutions. Bifurcation theory requires the analysis of the linearization of equation (2.2) about these trivial solutions which leads to consideration of the problem(2.3a)(2.3b)(2.3c)(2.3d)The periodic boundary conditions (2.3*a*) and (2.3*b*) imply thatEquations (2.3*c*) and (2.3*d*) become(2.4)for every *k*∈*Γ*′, *k*≠0. Thus a non-trivial solution can exist only if the matrix *A*(*c*_{0}, *k*) is singular for some *k*∈*Γ*′, that is, if the determinant(2.5)vanishes. In this case, if a pair (*c*_{0}, *k*) satisfies , then a non-trivial solution of equation (2.3) is(2.6a)(2.6b)where *α*_{k}∈**C** is an arbitrary constant.

Our approach to finding non-trivial solutions of equation (2.3) when *d*=2 is to choose a wavenumber *κ*_{1}∈*Γ*′(*κ*_{1}≠0) and solve for the corresponding *c*_{0} such that , i.e.(2.7)without loss of generality, we can always select the positive root. With *c*_{0} chosen in this way, we can write whereand we have used the fact that tanh(*k*)/*k* is even to drop the absolute value.

Clearly, when *k*=0, ±*κ*_{1}, *Λ*_{σ} is zero and the null space of the linearized operator is at least three-dimensional. An important question is whether other wavenumbers will produce zeros of *Λ*_{σ} resulting in a higher dimensional null space; this scenario is one of *resonance* and is left outside the scope of our current theory. However, one can easily make some general statements concerning the possibility of resonance. In the case of zero surface tension (*σ*=0)which (for *k*>0) is strictly decreasing, implying that *ψ*(*k*)=*ψ*(*κ*_{1}) if and only if *k*=±*κ*_{1}. Thus, in this case, there can be no resonance. However, for *σ*>0, the derivative of *ψ* may vanish at a point if *σ*/*g* is sufficiently small. However, even in this case, the existence of an additional *integer* root of *ψ*(*k*)=*ψ*(*κ*_{1}) will not occur generically.

Our approach to finding non-trivial solutions of equation (2.3) when *d*>2 is to choose (*d*−1) many wavenumbers and solve the corresponding set of (*d*−1) equations , i.e.(2.8)where *K*∈**R**^{(d−1)×(d−1)} has rows *κ*_{1}, *κ*_{2}, …, *κ*_{d−1}, and *R*∈**R**^{d−1} has *j*th entry(2.9)Among the 2^{d−1} choices for the vector *R*, we will always choose the one such that *R*_{j}>0. When *d*>2 there is always the possibility, though rare, that additional ‘resonant’ wavenumbers *κ*_{d}, …, *κ*_{p} may exist such that *Λ*(*c*_{0}, *κ*_{j})=0 for *j*=*d*, …, *p*. In fact, when *σ*=0 the number *p* can be infinite; see e.g. (Craig & Nicholls 2000) for a more complete discussion of these issues.

For general *d*≥2, in the non-resonant case (*p*=*d*−1), the first-order solution will be of the form:Clearly, by varying *ϵ*_{j}, we lose no generality by setting *ρ*_{j}=1. Furthermore, by fixing the crest of the linear solution at zero we may choose *θ*_{j}=0. Therefore our solution surface will be parameterized by (*d*−1) many parameters, *ϵ*_{j}. We note here that, in particular, our construction in §4 will deliver a unique solution even in the *degenerate* case where one or more of the *ϵ*_{j} is set to vanish identically. For instance, when *d*=3 if we set *ϵ*_{2}=0, then the linear solution becomesand *κ*_{2} plays no role; in fact, the presence of *κ*_{2} is purely artificial. As a consequence of this, our unique solutions comprise, in this situation, only a one-dimensional family within the two-dimensional (Stokes) manifold of solutions corresponding to *κ*_{1} (Craig & Nicholls 2002). Interestingly, however, this manifold can be completely recovered from our solutions by adding appropriate velocity components. More precisely, if our unique solution is (*η*(*ϵ*_{1},0), *ϕ*(*ϵ*_{1},0), *c*(*ϵ*_{1},0)), then all solutions can be parameterized by (*ϵ*_{1}, *δ*)→(*η*(*ϵ*_{1},0), *ϕ*(*ϵ*_{1},0), *c*(*ϵ*_{1},0)+*δ*c*), where *c**.*κ*_{1}=0. Similar remarks apply to the case *ϵ*_{1}=0 and *ϵ*_{1}=*ϵ*_{2}=0 (trivial branch); in this latter case the parameterization is .

### (c) Field expansions

A classical approach to finding approximate solutions to equation (2.2) was devised by Stokes (1847) in the mid-1800s. It consists of the boundary perturbations philosophy we have termed ‘field expansions’ (FE) to distinguish it from the alternative ‘operator expansions’ approach, (e.g. Nicholls & Reitich 2004*a*,*b*) carried out to low (first or second) order. This method was expanded to higher orders by subsequent authors with the most recent attempts being those of Roberts (1983), Roberts & Schwartz (1983), and Marchant & Roberts (1987). For ease of comparison with the results contained in these papers we adopt their notation in the current exposition of the FE approach.

For simplicity, we consider equation (2.2) in the case of two dimensions (*d*=2), 2*π*-periodicity, zero capillarity (*σ*=0), and infinite depth (*h*=∞). Following Roberts (1983), we define the surface velocities:and expand(2.10a)(2.10b)We find that we must solve(2.11a)(2.11b)(2.11c)(2.11d)where

To solve these equations we note that, on account of equations (2.11*a*) and (2.11*b*), and the periodic boundary conditions, *η*_{n} and *φ*_{n} can be expressed as(2.12)To find forms for *U*_{n} and *V*_{n} we first writeAs can be easily verified, the coefficients *E*_{n} are polynomials in *γ* of degree *n* which can be recursively found from the relations(2.13)Then, we haveso that we can write where(2.14)Similarly, where(2.15)Finally, we can rewrite equation (2.11) as(2.16a)(2.16b)(2.16c)(2.16d)which, using equation (2.12), is equivalent to(2.17)whereandThe FE procedure consists of solving equation (2.17) recursively up to a specified order *n*=*N*, starting with relations of the form (2.6), where *k*=*κ*_{1} and *c*_{0} are chosen to satisfy *Λ*_{0}(*c*_{0}, *κ*_{1})=0. For example, choosing *κ*_{1}=*c*_{0}=1 and normalizing *g*=1 the linear part of the solution (taking *α* real) is (cf. equation (2.6))(2.18)

The procedure can be carried out to an arbitrarily high order *N* by recursively solving equation (2.17). Of course, by the choice of *c*_{0}, the matrix in equation (2.17) is singular at *k*=±*κ*_{1} and a compatibility condition is required to ensure solvability. This is provided by an appropriate choice of *c*_{n−1} in equation (2.17) which closes the system of equations.

### (d) Cancellations

It should be noted that this derivation of the FE recursions, equation (2.17), is purely formal in nature. Indeed, for instance, the recurrence entails spatial derivatives of the velocity potential of an increasingly high order (cf. equations (2.14) and (2.15)) whose growth should be controlled if the series in equation (2.10) are to be shown to converge. On the other hand, if such control is to be based on the relations (2.17), it will demand, for instance, that we bound *U*_{n}, *V*_{n} recursively from equations (2.14) and (2.15). Here, however, the only obvious bound will, by necessity, use the triangle inequality in the order *l* (cf. equations (2.14) and (2.15)). As we have shown in related applications of boundary perturbation approaches (Nicholls & Reitich 2001*a*,*b*, 2003, 2004*a*,*b*), these bounds will consistently fail to provide useful growth control on any norm of the solutions as they destroy significant cancellations that are present in the corresponding recurrences.

To substantiate this claim we next present a set of numerical experiments that demonstrate the existence of cancellations in equations (2.14) and (2.15) as well as their implications in attempts at numerically simulating travelling water waves with high-order versions of the FE scheme. We begin by choosing specific parameters that give rise to a bifurcating solution: *d*=2, *σ*=0, *h*=∞, *g*=1, 2π-periodicity, *κ*_{1}=1, *c*_{0}=1, for which we present results of a suitable FE implementation up to order *N*=40. In particular, all convolution products in *S*_{n} and *T*_{n} are performed using fast Fourier acceleration in vectors of length *N*_{x}=128 (which represent wavenumbers [−*N*_{x}/2, *N*_{x}/2−1]) greater than 2*N*=80 to prevent aliasing.

The first evidence we present concerns the accurate computation of the Fourier coefficients of the wave profile *η*. It is not difficult to see that, beginning with equation (2.18), for all *p*>0, while *d*_{n,n} and *a*_{n,n} will not be equal to zero and represent a ‘leading edge’ of non-zero Fourier coefficients at order *n*. In table 1 we report the results of the computation of |*d*_{n,n}|, via the FE recursion (2.17) with *α*=1/2 in equation (2.18), in both double and quadruple precision (*N*_{x}=128). In the final column we treat the quadruple precision calculations as ‘exact solutions’ and count the digits of accuracy in the double precision calculation. We point out the precipitous loss of accuracy in the coefficients |*d*_{n,n}| through all orders of *n*, which rapidly accelerates beyond *n*=17, resulting in approximations which contain *no accurate information* by *n*=27.

In the previous calculation, one could argue that for large *n* an accompanying factor of *ϵ*^{n} (where *ϵ* is typically much less than 1) in the approximation of *η*(*x*) might disguise the inaccurate computation of *d*_{n,n}. However, as the next calculation illustrates, such a hope is unfounded and the accurate computation of high-frequency information is *crucial* for a correct representation. To substantiate this claim, we next approximate a more physically relevant quantity, the *L*^{2}-norm of the wave form *η*(*x*). More specifically, if we denote(2.19)which represents the FE approximation to *η*(*x*), then in figure 1 we present the difference (measured in *L*^{2}) between double precision and sextuple precision approximations of , and a highly resolved calculation (sextuple precision with *N*_{x}=128 and *n*=40); sextuple precision was necessary as we found that quadruple precision calculations were inadequate beyond *n*=33. We note that at *ϵ*=0.3 the sextuple precision calculation is fully converged at *n*=40, indicating that *ϵ*=0.3 is *within* the disc of convergence of the Taylor series (2.10). We point out in this figure the explosive divergence of the double precision calculation as *n* is increased. We also note the roughly linear shape of the curve on the log-linear axes indicating the exponential growth of errors.

## 3. Transformed field expansions

As the calculations of the previous section indicate, the cancellations in equation (2.17) are present for all *n* and increase in severity with increasing *n*. As explained above, this has consequences for more than just numerical simulation. Indeed, as we mentioned, the cancellations preclude the use of the most natural approach to estimating the convergence of the series (2.10) based on the derivation of bounds, e.g. of the form , from the recurrence (2.17).

However, as we explain next and further demonstrate in §4 a direct estimation of the terms in the series (2.10) can be realized upon a change of independent variables in advance of the perturbation expansion, much as in the application of boundary perturbation methods to boundary value problems (Nicholls & Reitich 2004*a*,*b*). Indeed, as in these latter applications, the transformation has the effect of implicitly accounting for all significant cancellations so that the terms in the corresponding recurrence can be inductively estimated. To derive these transformed field expansions (TFE) we begin by considering the transformation(3.1)which maps the domain *S*_{1,η} to the strip *S*_{1,0}. The equations (2.2) become, upon dropping primes,(3.2a)(3.2b)(3.2c)(3.2d)where *c*_{0} will be defined as in equation (2.7), , and(3.3a)(3.3b)(3.3c)The functions *Q*(*x*) and *R*(*x*) are represented by similar formulas.

To solve equations (3.2) in the transformed variables we now propose the following expansions for *ϵ*∈**R**^{d−1} and multi-index *n*∈**N**^{d−1}(3.4)(the ‘TFE’) and find that we must solve the following problems(3.5a)(3.5b)(3.5c)(3.5d)Here, *δ*_{k,p} is the Kronecker delta, where *e*_{j} is non-zero at index *j*, and for multi-indices *m*, *n*∈**N**^{d−1}, *m*≤*n* if *m*_{j}≤*n*_{j} for all . Furthermore, , where(3.6a)(3.6b)(3.6c)The functions *Q*_{n} and *R*_{n} can be similarly derived. Note that (and *Q*_{n}, and *R*_{n}) depend only on {*η*_{j}}_{|j|<|n|}, {*φ*_{j}}_{|j|<|n|}, and {*c*_{j}}_{|j|<|n|−1}, and vanish for |*n*|=1.

## 4. Analyticity of solutions

Clearly, the nature of equations (3.5) is quite similar to that of equations (2.16); however, there are some important differences. Most notably, in contrast to equations (2.16), the right-hand sides of equation (3.5) contain only derivatives of order one or two which act either on the field, *φ*_{l}, or the surface shape, *η _{l}*. In what follows, we show that this, in fact, allows for the inductive establishment of bounds:in appropriate function spaces

*X*and

*Y*, and for some constants

*C*,

*B*≥0. Furthermore, we will show that

*all*spatial derivatives of

*φ*

_{n}and

*η*

_{n}can be similarly bounded, implying that all quantities are

*jointly*analytic with respect to all arguments.

To set notation we recall that any *L*^{2} function *f* periodic on a (*d*−1)-dimensional lattice *Γ*⊂**R**^{d−1} can be represented as(4.1)where *Γ*′ is the conjugate lattice to *Γ*. Additionally, if an *L*^{2} function *u*(*x*, *y*) is periodic in *x* with respect to *Γ* and square integrable in the *y* variable on [−1, 0] then(4.2)Using this representation we can define the *L*^{2} based Sobolev spaces(4.3)for *s*∈**Z**^{+} and where(4.4)and . Note that if *f*=*f*(*x*) depends on *x* alone then the space *H*^{s} for *s*∈**R** can be defined by the norm(4.5)For future reference we note the following algebra property for *H*^{s} (Adams, 1975).

*If s*>*d*/*2 then H*^{s} *is an algebra*, *i.e. for u*, *v*∈*H*^{s}(4.6)*for a constant M=M*(*d*, *s*) *depending only on d and s*.

In the case of non-zero surface tension (*σ*>0), our main result is as follows.

*Given an integer s*>*d*/2, *if* |*n*|≥1 *the solutions φ*_{n}(*x*, *y*), *η*_{n}(*x*), *and* *of equation* (3.5) (*σ*>0) *satisfy*(4.7a)(4.7b)*for all j=*1,…, *d*−1, *p*>*d*−1, *and some constants C*_{1}, *B*, *D*, *A*>*0*.

We will prove theorem 4.2 by induction on *l*; thus our first objective is to establish the result for *l*=0. This is done in the following lemma, for all *k*, with an induction on the order |*n*|.

*Given an integer s*>*d*/2, *if* |*n*|≥1 *the solutions φ*_{n}(*x*, *y*), *η*_{n}(*x*), *and* *of equation* *(3.5)* (*σ*>0) *satisfy**for all j*=1, …, *d*−1, *p*>*d*−1, *and some constants C*_{1}, *B*, *A*>0.

To prove Lemma 4.3 we need two lemmas: the following which estimates the right hand sides of equation (3.5) of the inhomogeneous problems, and the sequel which provides estimates on solutions of these problems.

*Given an integer s*>*d*/2, *suppose that**for all* 1≤*n*<*N (i.e. n*_{j}<*N*_{j}*)*, *p*>*d*−1, *and for some constants C*_{1}, *B*, *A*>0. *Then there exists a constant C*_{2} *such that the functions* , , *and R*_{N} *in equation* *(3.5)* *satisfy*(4.8a)(4.8b)(4.8c)

*of* *lemma 4.4*. For the sake of brevity consider the first term of :every other term in *F*_{N}, *Q*_{N}, and *R*_{N} can be similarly estimated. We beginSincewe can deduce thatContinuing,andwherewhich is bounded uniformly in |*m*| for *p*>*d*−1. The proof is complete provided . ▪

The second lemma necessary to prove lemma 4.3 is now presented; the proof, based on classical elliptic estimates, is given in appendix A.

*Consider any integer s*≥0. *Given linearly independent wavenumbers* *there exists a unique speed* *satisfying equation* *(2.8)* *such that R*_{j}>0. *Given this c*, *if for all multi-indices* , *p*_{n}∈*H*^{s}, *q*_{n}∈*H*^{s+1/2}, *and r*_{n}∈*H*^{s+1/2} *then there exist for all* *real solutions w*_{n}∈*H*^{s+2}, *v*_{n}∈*H*^{s+5/2}, *and* *of*(4.9a)(4.9b)(4.9c)(4.9d)*where*(4.10a)(4.10b)*If*, *in addition, we require that*(4.11)*then this solution is unique. Furthermore there exists a constant C*_{e} *such that the solutions satisfy*(4.12a)(4.12b)(4.12c)

We can now complete the proof of lemma 4.3.

*of* *lemma 4.3*. The proof proceeds via induction on |*n*|; since *φ*_{n}, *η*_{n}, *c*_{n} satisfy equation (3.5), , , and *c*_{n−1} satisfy(4.13a)(4.13b)(4.13c)(4.13d)In the case |*n*|=1 we have the *explicit* formulas(4.14a)(4.14b)and, for *c*_{0}∈**R**^{d−1}, the set of (*d*−1) equations . From these formulas the estimates for |*n*|=1 follow easily. Next, consider order |*N*| and note that *N*=ν+*e*_{j} for some multi-index *ν*. Suppose thatfor all *k* and for all 1≤|*n*|<|*N*|, *n*<*N*. Using lemma 4.5 we haveFinally, using lemma 4.4 we obtainThe proof is complete, provided that *B*>6*C*_{e}*C*_{2} and *p*>*d*−1. ▪

Lemma 4.3 proves equation (4.7) for *l*=0. In order to complete the induction for *l*>0 we need the following two results.

*Given an integer s*>0 *the following estimate holds*(4.15)*for some constants C*_{1}, *D*, *A*>0.

*of* *lemma 4.6*. The proof comes immediately from the explicit formula (4.14) for . ▪

*Given an integer s*>*d*/2, *suppose that*(4.16a)(4.16b)*for all indices k*≥0, |*n*|≥1, *when l*<*L*; *for all k*≥0, |*n*|<|*N*| *when l*=*L*; *and p*>*d*−1. *Then there exists a constant C*_{3} *such that F*_{N}, *Q*_{N}, and *R*_{N} *in equations* *(3.6)* *satisfy*(4.17a)(4.17b)(4.17c)

*of* *lemma 4.7*. Estimates (4.17*b*) and (4.17*c*) are true by lemma 4.4. For brevity, in regard to equation (4.17*a*) let us consider only the third portion of , namelyThenSinceandwe can continueUsing equation (4.16) we obtainandprovided that *C*_{3} is chosen appropriately. ▪

We are now in a position to prove theorem 4.2.

*of* *theorem 4.2*. We begin by noting that lemma 4.3 has already established equation (4.7*b*) so we need only focus on equation (4.7*a*). For this estimate we work by induction on *l*. For *l*=0 and any *k*≥0, |*n*|≥1 we use lemma 4.3. Now we assumefor all *l*<*L* and any *k*≥0 and |*n*|≥1, and seek to provefor any *k*≥0 and |*n*|≥1. We accomplish this via a second induction on |*n*|. Lemma 4.6 establishes equation (4.7*a*) in the case |*n*|=1. We now assume thatfor all |*n*|<|*N*| and seek to proveWe make the estimatewhere *C* is a generic function of dimension alone and we have used the fact that *φ*_{N} solves equation (3.5). Finally, using lemma 4.7 we havewhich completes the proof provided that *D* >(1+*C*(*d*)*A*) and *B* >*C*_{3}. ▪

We remark here that the same existence and analyticity proofs given above for the case of capillary gravity waves (*σ*>0) can also be given for pure gravity waves (*σ*=0) provided that *d*=2. The key differences lie in the different nature of *Λ*_{σ} when *σ*=0, and the elliptic estimate, lemma 4.5.

## Disclaimer

*Disclaimer*. Effort sponsored by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under grant number F49620-02-1-0052, and by AHPCRC under the auspices of the Department of the Army, Army Research Laboratory cooperative agreement number DAAD19-01-2-0014. The US Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation thereon. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research, the Army Research Laboratory or the US Government.

D.P.N. gratefully acknowledges support from NSF through grants No. DMS-0196452 and DMS-0139822. F.R. gratefully acknowledges support from NSF through grant No. DMS-0311763, from AFOSR through contract No. F49620-02-1-0052 and from the Army High Performance Computing Research Center (AHPCRC) under Army Research Laboratory cooperative agreement number DAAD19-01-2-0014.

## Footnotes

As this paper exceeds the maximum length normally permitted, the authors have agreed to contribute to production costs.

- Received April 23, 2004.
- Accepted October 4, 2004.

- © 2005 The Royal Society