## Abstract

We prove the existence of solutions to the irrotational water-wave problem in finite depth and derive an explicit upper bound on the amplitude of the nonlinear solutions in terms of the wavenumber, the total hydraulic head, the wave speed and the relative mass flux. Our approach relies upon a reformulation of the water-wave problem as a one-dimensional pseudo-differential equation and the Newton–Kantorovich iteration for Banach spaces.

This article is part of the theme issue ‘Nonlinear water waves’.

## 1. Introduction

The existence of small-amplitude, irrotational water waves under the influence of gravity, propagating over a flat bed with constant speed, is well established [1]. In recent existence proofs, which are derived for waves with constant, possibly non-zero, vorticity, the governing equations are reformulated as a one-dimensional pseudo-differential equation [2,3]. A similar formulation has been derived for Stokes waves in deep water [4,5]. Existence of small- and large-amplitude waves is then established through an analytical bifurcation theory on Banach spaces [6]. Based on topological degree theory, another global bifurcation approach yields the existence of waves of large amplitude [1,7]. While this approach captures, in principle, more solutions than the existence results in [3], the structure of the solution set is more complicated, being merely a continuum, rather than a curve with a local real-analytic parametrization.

These bifurcation methods, however, cannot give a qualitative description of the obtained solutions as they rely upon topological arguments. Only based on a thorough analysis of the underlying equations, *a priori* estimates and bounds on physical quantities have been derived in different settings of the water-wave problem, both with and without vorticity [8,9].

In this paper, we pursue a slightly different strategy from that in [2]. After carefully balancing the free parameters of the problem—the total hydraulic head, the wave number, the wave speed and the relative mass flux—we show that the Newton iterates of an appropriately chosen functional converge to a non-trivial solution of the full problem. Iterative procedures on function spaces have been successfully applied to the water-wave problem before [10]. In our case, the convergence is guaranteed by Kantorovich's theorem for the Newton iterates on Banach spaces [11,12].

As an initial guess, we choose a pure -frequency with small enough amplitude—in accordance with the leading-order term of the bifurcating solutions obtained in [2]. Moreover, the Newton–Kantorovich approach allows us to estimate how close the non-trivial solution lies to the initial guess in terms of the *H*^{1}-norm. Sobolev embedding [13] then implies an upper bound on the amplitude of the solution to the full problem. The accurate proportions of the numerical constants involved is, to some extent, reminiscent of the analysis carried out in [14].

## 2. Preliminaries

In this section, we give some basic definitions and prove some estimates.

### (a) Notation

Let denote the space of periodic and real-valued square-integrable functions defined on the interval [−*π*,*π*]. The space can be endowed with the inner product
2.1
for , which defines a Hilbert space structure on . The corresponding norm in is then given by
2.2
Any element can be expanded as a Fourier series (cf. [15])
2.3
where
2.4
denotes the *n*th Fourier coefficient of *u*. Specifically, we write
2.5
for the mean of *u*. By virtue of Parseval's theorem (cf. [15]) equation (2.1) can be written equivalently in terms of the Fourier series of *u* and *v* as
2.6
Here, *z** denotes the complex conjugate of the complex number *z*. For any , define
2.7
the Sobolev space of order *s*. The space can be endowed with the inner product
2.8
which makes it into a Hilbert space. Let
2.9
denote the linear subspace of mean-free functions and write
2.10
for the orthogonal projection onto . We then define
2.11
for . Furthermore, we define
2.12
and set
2.13
which is a Banach space with the inclusion . Again, we write
2.14
To ease later computations, we also introduce the *l*^{p}-spaces. For any sequence , *s*(*n*)=*s*_{n}, we define
2.15
and set
2.16
The *l*^{p}-spaces are Banach spaces (cf. [15]) and the space can be endowed with a Hilbert space structure by means of the inner product
2.17
for . Parseval's theorem can be stated equivalently as
2.18
for any function . Similar to the space , we define
2.19
and the corresponding orthogonal projection As for the projection *P*_{0}, we write
2.20
For any function , its *amplitude* is given by
2.21
which defines a norm that is equivalent to the -norm by means of the inequality
2.22
For *D*>0, we define the Fourier multiplier
2.23
which is, indeed, well defined because .

The operator defined in (2.23) is related to the Dirichlet-to-Neumann map for a strip of width *D*, as shown in [2]. For later computations, we point out that
2.24
because is monotonially decreasing for *x*>0.

For a possibly unbounded operator *T*:*X*→*X*, defined on some Banach space *X*, with domain , we denote its spectrum by *σ*(*T*). The operator norm of an operator *S*:*X*→*Y* is defined as
2.25

### (b) Two estimates

In this section, we give the proofs of two elementary estimates that are used in the proof of our main theorem. We remark that the numerical value of the constants appearing in the following estimates are crucial in order to make the Newton–Kantorovich procedure converge to a non-trivial solution.

### Lemma 2.1

*Let* *and* . *Then the inequality*
2.26
*holds true.*

### Proof.

Using the Cauchy–Schwartz inequality for the inner product of , we readily compute 2.27 where, in the last step, we have used the partial fraction decomposition of the function: 2.28 for any (cf. [16]). ▪

### Lemma 2.2

*Let* . *Then the inequality*
2.29
*holds true.*

### Proof.

First, we note that
2.30
for |*k*|,|*n*|>1. Using inequality (2.30) and the subadditivity of the square-root function, we find that
2.31
Using equation (2.31) and Young's inequality for discrete convolution in *l*^{p}-norm (cf. [15]), we obtain
2.32
As in the proof of lemma 2.1, we note that
2.33
and conclude with
2.34
□

## 3. The governing equations and the main theorem

Consider a two-dimensional, *L*-periodic wave train propagating with a fixed wave speed *c* over a flat bed of constant (physical) mean depth *d*. We denote the flat bed as
3.1
The free surface of the wave, denoted by , is parametrized by a differentiable curve
3.2
and the fluid domain enclosed by the free surface and the flat bed is denoted by *Ω* (cf. figure 1). The physical mean depth can be recovered from the parametrization of the free surface,
3.3
The parameter *d* fixes the position of the wave profile in the (*X*,*Y*)-plane. For a flat surface, (*ξ*,*η*)=(0,*d*).

The governing equations for water waves in finite depth can be formulated as the free-boundary problem
3.4a
3.4b
3.4c
3.4d
for an unknown, *L*-periodic *stream function* *ψ* and an unknown surface . Here, *g* denotes the gravity constant, *Q* the total hydraulic head and *m* the relative mass flux of the fluid flow. The fluid velocity field (*u*,*v*) can then be reconstructed from the stream function via the relation ∇*ψ*=(−*v*,*u*), while the wave speed can be reconstructed via
3.5
The above integral is, indeed, independent upon *X* and *Y* by equation (3.4a), as long as the segment joining (*X*,*Y*) and (*X*+2*L*,*Y*) lies entirely inside *Ω*.

To ease notation in the subsequent computations, we define the *mean depth* (cf. [2]) of system (3.4) as the ratio between the mass flux and the wave speed,
3.6
and the *wavenumber* as
3.7
for the wave period *L*.

Equation (3.4a) is equivalent to the incompressible Euler equations for an irrotational fluid, while equation (3.4b) expresses that the velocity field at the free surface is purely tangential (kinematic boundary condition). Equation (3.4c) is equivalent to the constant pressure constraint at the free surface (dynamic boundary condition), while equation (3.4d) reflects the fact that the velocity field is purely tangential at the flat bed.

In [2], it has been proved that system (3.4) is equivalent to the following one-dimensional pseudo-differential equation:
3.8
for an unknown function , as well as an unknown constant *h*. Here, is the Fourier multiplier defined in (2.23), *h* is the mean depth (3.6), *k* is the wavenumber (3.7) and
3.9
is a parameter measuring the total energy of the wave. We also define the parameter
3.10
The wave profile can be reconstructed from the solution (*w*,*h*) as
3.11
which uniquely determines the solution to problem (3.4) (cf. [2]). We are now ready to formulate our main theorem. To ease notation, we define the numerical constants
3.12

and 3.13 which will frequently appear in the later computations.

### Theorem 3.1

*Assume that the mean depth h is given and that the parameters μ and k are such that*
3.14
*Then, there exists a non-trivial (w≠0) solution w** *to equation (3.8), which satisfies the amplitude estimate*
3.15

## 4. Proof of the main theorem

In this section, we will prove the existence and uniqueness of solutions to equation (3.8) based on the following abstract theorem for iterative procedures in Banach spaces.

### Theorem 4.1 Newton–KantorovichNewton--Kantorovich

*Let (X,∥.∥*_{X}*) and (Y,∥.∥*_{Y}*) be Banach spaces, let F:X→Y be a continuously differentiable map and let DF(x) denote its derivative at x∈X. Assume that there exists an x*_{0}*∈X such that DF(x*_{0}*) is an invertible operator. Assume further that there exist constants α,β and γ such that*

(1)

*∥DF(x)−DF(y)∥≤α∥x−y∥*_{X}*for all x,y∈X,*(2)

*∥DF(x*_{0})^{−1}*∥*_{op}≤*β*,(3)

*∥F(x*_{0}*)∥*_{Y}*≤γ.*

*If*
(4.1)
*then there there exists a solution x***∈X to the equation F(x)=0, i.e. F(x***)=0. Moreover, setting*
(4.2)
*the solution x** *is contained in the open ball*
(4.3)

For a proof of theorem 4.1, we refer the reader to [11,12].

Define the functional
4.4
and note that equation (3.8) is equivalent to . In the following paragraphs, we will carry out a step-by-step check of the assumptions of theorem 4.1 for the functional (4.4). To ease notation, we set
4.5
the *dispersion parameter*, which is, indeed, positive by assumption (3.14). As a first approximation to the nonlinear solution, we choose the initial function
4.6
where the approximated amplitude satisfies
4.7
We can always choose such an *A* owing to assumption (3.14). For later computations, we note that (4.7) together with (3.14) implies that
4.8
Indeed, (2*ε*_{1}λ−1)*μ*<0<2*ε*_{1}/*k*, because λ<2 and *ε*_{1}≈0.0519. Observe that our initial guess is consistent with the leading-order term of the full, nonlinear solution obtained in [2].

### (a) The functional is well defined

First, we observe that for [*w*]=0. Using the estimate (2.24) together with lemma 2.1, we compute the norm of ,
4.9
where, in the second inequality, we have again used (2.24) as well as lemma 2.2.

### (b) A Lipschitz bound on

For , the derivative of is given by
4.10
Similar to the computations carried out in (4.9), we can estimate
4.11
for , which implies that we can choose
4.12
where *ϵ*_{0} is as in (3.12).

### (c) Estimating

For the following calculations, we write . This defines an unbounded, self-adjoint operator with domain . Indeed, using the symmetry of the operator with respect to *x*, we obtain, for ,
In particular, . Here, we have used that [*u*]=[*v*]=0 in the integral expressions.

### Lemma 4.2

*Assume that the initial amplitude satisfies (4.7). Then*
4.13

### Proof.

Expanding in a Fourier series and abbreviating , we find that
4.14
Using that *ϕ* is real-valued, i.e. , and that *c*_{n}=*c*_{−n}, we obtain
4.15
which is equivalent to
4.16
We recall that, for *n*≥1,
4.17
and estimate from above
4.18
Indeed, as λ<2, the inequality 1/*k*−*μn*+2*Aλn*+*Aλ*<1/*k*+λ(2*A*−*μ*) for all *n*≥2 is equivalent to *Aλ*<((*n*−λ)/(2*n*−1))*μ*≤*μ*/2, for all *n*≥2. The last inequality, however, is guaranteed by observation (4.8). This proves the claim. ▪

To estimate the norm of , we observe that the resolvent of is normal and hence 4.19 We may, therefore, set 4.20

### (d) Estimating

We find that because the function is monotonically decreasing and by assumption (4.7). We, therefore, set 4.21

Now that we have set the constants *α*,*β* and *γ* for our problem, let us check condition (4.1). Inserting the expressions (4.12), (4.20) and (4.21) into inequality (4.1), we find
4.22
which is equivalent to
4.23
This inequality is satisfied, *a fortiori*, by assumption (4.7) and (3.13). Theorem 4.1 now implies that there exists a solution *w** to equation (3.8) such that
4.24
To exclude the trivial solution, *w*≡0, from our analysis, we have to assume that
4.25
which is equivalent to
4.26
The above expression, in turn, simplifies to
4.27

The last inequality, again, follows from assumption (4.7) because *ε*_{1}≈0.0519<(*ε*_{0}+2)^{−1}≈0.1270. Taking a square and rewriting (4.27) as , we can solve for *A*,
4.28
This is assumption (4.7) and, owing to the sign condition in (4.27), we can reverse the inequalities to obtain (4.25). This proves the claim.

We conclude with the proof of the amplitude estimate (3.15). As in the proof of lemma 2.1, we observe that inequality (4.24) implies that
4.29
It follows from the inverse triangle inequality together with the observation that *t**<1/*αβ*=(*δ*−2*λA*)/*ε*_{0}λ,
4.30
which, due to (4.7), implies that
4.31
Equation (2.22) gives the desired estimate.

## Data accessibility

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## Competing interests

I declare I have no competing interests.

## Funding

No funding has been received for this article.

## Acknowledgements

F.K. would like to thank Eugen Varvaruca and Adrian Constantin for several useful comments. F.K. is very grateful to one of the anonymous referees for several constructive suggestions and the detailed report.

## Footnotes

One contribution of 19 to a theme issue ‘Nonlinear water waves’.

- Accepted August 1, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.