## Abstract

Crapper waves are a family of exact periodic travelling wave solutions of the free-surface irrotational incompressible Euler equations; these are pure capillary waves, meaning that surface tension is accounted for, but gravity is neglected. For certain parameter values, Crapper waves are known to have multi-valued height. Using the implicit function theorem, we prove that any of the Crapper waves can be perturbed by the effect of gravity, yielding the existence of gravity–capillary waves nearby to the Crapper waves. This result implies the existence of travelling gravity–capillary waves with multi-valued height. The solutions we prove to exist include waves with both positive and negative values of the gravity coefficient. We also compute these gravity perturbed Crapper waves by means of a quasi-Newton iterative scheme (again, using both positive and negative values of the gravity coefficient). A phase diagram is generated, which depicts the existence of single-valued and multi-valued travelling waves in the gravity–amplitude plane. A new largest water wave is computed, which is composed of a string of bubbles at the interface.

## 1. Introduction

We study the irrotational, incompressible Euler equations for a fluid bounded above by a free surface, with vacuum above the free surface. We consider a fluid region that is infinitely deep in the vertical direction and periodic in the horizontal direction. We seek travelling wave solutions, or solutions for which the free surface is of permanent form and steadily translating. We consider the effect of surface tension at the fluid boundary.

For this problem, in the absence of gravity, a family of exact solutions is known; these solutions are called Crapper waves, as they were discovered by Crapper [1]. There is an exposition of these waves in the book of Crapper [2], and also, in more detail, in the book of Kinsman [3]. A formula for Crapper waves is given in §2.

Kinnersley extended the Crapper waves to the case of finite depth [4]. As in the infinite depth case that Crapper studied [1], the travelling waves are found as exact solutions, in this case involving elliptic functions. Building upon the work by Tanveer for translating bubbles [5], Crowdy gives a different derivation of the Crapper waves, using conformal maps [6]. The formulation of Crowdy also allows for the identification of other, previously unknown, exact solutions of the free-surface Euler equations with surface tension. Crowdy further shows that exact steady free-surface Euler flows such as the Crapper waves yield, through a transformation, exact steady Hele–Shaw flows [7].

Uniqueness and stability of Crapper waves have also been studied. Assuming a certain positivity condition, Okamoto has proved uniqueness of the Crapper waves (that is, any travelling pure capillary water wave on infinite depth that satisfies the positivity condition is a Crapper wave) [8]; see also the discussion in the book of Okamoto & Shōji [9]. Tiron & Choi [10] studied stability of the Crapper waves. Stability was also studied by Hogan [11] and by Chen & Saffman [12].

Given an exact solution or family of exact solutions of a nonlinear partial differential equation, a natural question to ask is, can we perturb these solutions to find other, nearby solutions? Since the Crapper waves are irrotational, pure capillary water waves, there are then several natural directions in which to perturb them: through the inclusion of gravity, through the inclusion of vorticity in the bulk of the fluid and through the addition of an upper fluid to replace the vacuum above the waves. In this contribution, we prove that it is indeed possible to perturb any Crapper wave solution through the inclusion of gravity, either positive or negative. We do this by using a modified Liapunov–Schmidt analysis. We call the waves we find gravity perturbed Crapper waves. Perturbing the Crapper waves with vorticity or an additional fluid is expected to be the subject of future research.

In addition to proving they exist, we compute gravity perturbed Crapper waves and study the differences in features of the waves caused by the presence of gravity. The computational method used is the method introduced by the authors in [13]. This method uses a normalized arclength parametrization of the free surface, as developed for a numerical method for initial-value problems for vortex sheets in [14,15] and used analytically (again for the initial-value problem) in [16]. Since we describe waves using the arclength parametrization, there is no assumption that the interface has single-valued height, and thus the method works in exactly the same way whether or not the interface has overturned.

A number of other papers have computed overturning travelling waves in free-surface Euler flows. Baker *et al*. [17] formulated the travelling wave problem in such a way as to allow for overturning waves, but only computed waves with single-valued height in interfacial flows. Other studies that computed interfacial waves with multi-valued height are those of Saffman & Yuen [18], Meiron & Saffman [19], Turner & Vanden-Broeck [20] and Grimshaw & Pullin [21].

The self-intersection and extreme forms of gravity–capillary waves have been studied previously. In particular, a detailed study was made by Hogan [22]. The method of Hogan [22] is a boundary perturbation method, in the style of the expansions of Stokes [23] and Wilton [24,25]. Similar boundary perturbation methods have been applied to the water wave problem, with and without surface tension [26–31]. As noted in [30], these methods can be susceptible to floating point instabilities. Hogan observed this problem in his results, and reacted by working in quadruple precision and restricting the number of terms in his series expansions. The transformed field expansions method (of Nicholls and collaborators), a boundary perturbation algorithm, is stable to such errors [30,31]; however, it cannot compute overturned travelling waves.

A popular alternative to boundary perturbation is the combination of Fourier collocation and a quasi-Newton solver. Such an approach has been applied to compute gravity–capillary waves in a variety of settings, including finite and infinite depth, periodic and solitary waves [32]. For overturning waves, the majority of these computations are based on a conformal mapping applied to the fluid domain. The closest works in this latter class to the present are those of Debiane & Kharif [33] and Debiane *et al.* [34], in which the largest travelling waves are computed for a sampling of Bond numbers, that of Schwartz & Vanden-Broeck [35], where a host of travelling waves are computed at various amplitudes and Bond numbers, and Vanden-Broeck & Keller [36] computed the analogue of Crapper waves with larger bubbles (by varying the internal bubble pressure). Our numerical method also uses Fourier collocation and a quasi-Newton solver, but rather than being based on a conformal mapping, it instead parametrizes the interface by arclength [13]. This allows us to keep a uniform grid spacing along the interface, as opposed to having our grid points chosen by the mapping.

To complement our existence results, we numerically compute wave profiles and speeds, varying both the amplitude and the Bond number *σ*=*g*/*k*^{2}*τ* (these quantities will be specified in more detail in the sequel). The Bond number is the relevant non-dimensional quantity measuring the relative importance of gravity and surface tension. It is common to see the inverse of this quantity also called the Bond number; we choose this version of the Bond number as it is bounded near the Crapper wave at *g*=0. Because the existence proof relies on the implicit function theorem, it does not require that gravity be positive; therefore, in our computations, we perturb the Crapper wave with both positive and negative Bond number. We compute continuous branches of travelling waves connecting Crapper waves (*σ*=0) to gravity–capillary waves with *σ*=20, as well as to waves with negative Bond number −1<*σ*<0. Unlike methods based on amplitude expansions, our method need not be altered to compute the resonant Wilton ripples [37,38]. Our results support the conclusions of Schwartz & Vanden-Broeck [35]; we observe travelling waves at the Wilton ripple Bond numbers that are continuously embedded among the travelling waves computed elsewhere.

For the branches of waves bifurcating from the Crapper waves that we compute, as Bond number increases, we observe that the Crapper wave is continuously connected to *solitary* gravity–capillary waves. These computations serve as numerical verification of the argument of Longuet–Higgins, whereby asymptotics about the Crapper wave are used to approximate gravity–capillary solitary waves [39]. For the branches of waves with *σ*<0, we observe that the large-amplitude limit is typically a stationary (or standing) wave, rather than a travelling wave with a self-intersecting profile. We have generated a phase portrait of the existence of multi-valued and single-valued travelling waves in the gravity–amplitude plane. A new, globally largest water wave is computed, which is both standing and self-intersecting. This wave consists of a string of bubbles and droplets at the fluid interface.

The paper is organized as follows: in §2, we describe Crapper waves. In §3, we give our proof, using the implicit function theorem, to show the existence of gravity perturbed Crapper waves. In §4, we describe our numerical method and give our numerical results.

## 2. Crapper waves

In §1.1 of the book of Okamoto & Shōji [9], they give the following equation whose solutions give travelling spatially periodic gravity–capillary waves in a two-dimensional fluid of infinite depth:
2.1Here, the independent variable *a* is in **T**:=[−*π*,*π*], *θ* is the tangent angle to the surface and satisfies periodic boundary conditions, and *H* is the periodic Hilbert transform,
For background on the periodic Hilbert transform, the interested reader could consult, for instance, [40]. The non-dimensional constants *p* and *q* are given by
where *L* is the spatial period of the wave, *c* is the travelling wave speed, *g* is the acceleration due to gravity and *τ* is the surface tension constant. We remark that in the sequel, we will always take *q* to be positive. This formulation of the travelling water wave problem follows from complex variable methods initiated by Stokes [23], with important contributions from Levi-Civita [41].

An important feature here is that the free surface is not parametrized by arclength; reconstruction of the surface from *θ* requires some more information. Specifically, if *θ*(*a*) is a solution of (2.1), the parametrization of the free surface (*x*(*a*),*y*(*a*)) is given by
2.2

If one sets *p*=0, then one can find exact formulae for solutions *θ* of (2.1), which are called *Crapper waves*. We briefly explain these solutions now. When *p*=0, (2.1) becomes
The left-hand side is a perfect derivative,
Integrating gives
2.3In [9], the following is shown.

### Lemma 2.1

For any *θ*∈*L*^{2}(** T**),
2.4

With this, one can show that the constant in (2.3) is 1/2. Equation (2.3) can then be rewritten as
2.5Take *q*≥1 and let *A* be either of the solutions of
Note that *A*∈(−1,1). Let
and
Then, we have the following.

### Theorem 2.2 (Crapper [1])

*Note that θ*_{q} *(a) is an odd function of a. The function θ*_{q} *gives the Crapper wave.*

Linearizing (2.5) about *θ*_{q} gives the operator
(Here and below, the prime indicates differentiation with respect to *a*.) The following lemma, which is equivalent to [9], lemma 2.1, is used there to prove that the Crapper waves are locally unique.

### Lemma 2.3 (Okamoto & Shōji [9])

For *q*>1, zero is an eigenvalue of *Γ*: *X*^{1}→*L*^{2}(**T**) with geometric multiplicity one and algebraic multiplicity two. The eigenfunction is d*θ*_{q}/d*a* and the generalized eigenfunction is ∂*θ*_{q}/∂*q*.

In the above, and

In this article, we show that Crapper waves perturb to solutions of (2.1) when *p*∼0. In particular, we prove the following theorem.

### Theorem 2.4

*For all q>1, there exist P=P(q)>0 and a* *function
**such that F(Θ*_{q}*(p);p,q)=0 for |p|<P and Θ*_{q}*(0)=θ*_{q}*. Moreover, Θ*_{q}*(p) is an odd function of a, and Θ*_{q} *is smooth with respect to q.*

### Remark 2.5

Note that this theorem gives the existence of travelling gravity–capillary waves nearby to the Crapper waves, for small values of *p*, with either *p*>0 or *p*<0, i.e. with either positive or negative values of the acceleration due to gravity. There are some theorems in the literature on the non-existence of travelling water waves with negative gravity, in the case of pure gravity waves [42,43].

The method of proof is, roughly speaking, a Liapunov–Schmidt analysis, and follows relatively quickly from theorem 2.2 and lemma 2.3. Despite the brevity of its proof, we point out a particularly important and novel feature of theorem 2.4: it implies the existence of periodic travelling capillary–gravity waves that overhang. It is well known that for *q* large enough, the profile of the Crapper wave is not given by a function over the horizontal Eulerian coordinate. As our theorem applies for any *q*>1 and *Θ*_{q} is continuous with respect to *p*, we have the same feature for the gravity perturbed Crapper wave.

We will be employing the implicit function theorem several times in our proof and for completeness, we state the version of this theorem we use here.

### Theorem 2.6 (The implicit function theorem)

*X, Y and Z are Banach spaces and ζ:X×Y →Z is C*^{k}*, k≥1. If ζ(x***,y***)=0 and D*_{x}*ζ(x***,y***) is a bijection from X to Z, then there exists ϵ>0 and a unique C*^{k} *map χ:Y →X such that χ(y***)=x** *and ζ(χ(y),y)=0 when ∥y−y***∥*_{Y}*<ϵ.*

Statements of implicit function theorems can be found in many standard texts; for example, a similar statement to theorem 2.6 can be found in [44], ch. 13 .

## 3. Existence of gravity perturbed Crapper waves

Our first task is to rewrite the problem (2.1) as a perturbation of (2.5). Note that we can write (2.5) as
Integrating this from 0 to *a*, we get
3.1where *γ* is a constant. Note that it is possible to determine the value of *γ* in advance for a solution, as is done in the derivation of (2.5) above. However, it is to our advantage in the proof to leave this constant undetermined at this time, as it gives us an extra parameter at our disposal. Recalling the constant of integration when *p*=0 was 1/2, we set
where *κ*∈**R**. Also, we define
We then recast (3.1) as
A quick rearrangement of terms converts this to
3.2

Observe that theorem 2.2 shows that *Φ*(*θ*_{q};0,0)=0. Moreover, if we linearize *Φ* at *θ*=*θ*_{q} and *p*=*κ*=0, we have
Lemma 2.3 tells us that *Γ* is not invertible. As such, we cannot directly employ the implicit function theorem to find solutions of *Φ*=0.

Nevertheless, we can eliminate the kernel of *Γ* by restricting attention to odd functions. Let
and
Of course is a subspace of *X*^{1} and is a subspace of *L*^{2}(**T**).

### Proposition 3.1

*Φ*(*θ*;*p*,*κ*) *is a* *map from* *into*

We omit the proof of this. The key thing is that d/d*a*, *H* and *I* all map odd functions to even ones.

Recall that *θ*_{q} is odd, and thus *θ*′_{q} is even. Therefore, the kernel of *Γ* as a map on is trivial. That is, we have the following corollary of lemma 2.3.

### Corollary 3.2

*When Γ is viewed as a map from* *to*

It might at this time appear that we could apply the implicit function theorem to find solutions of (3.2). However, this is not the case. While *Γ* is injective as a map from to , it is not surjective, as the following proposition demonstrates.

### Proposition 3.3

*Let* . *Then, there exists* *with*
*if and only if*

### Proof.

Lemma 2.1 implies, for any function *w*∈*L*^{2}(** T**),
Taking the real part of this gives
Also, as , we have . So, for any function

*w*∈

*H*

^{1}(

**), 3.3**

*T*Fix *u*∈*H*^{1}(** T**) and let

*w*=

*θ*

_{q}+

*ϵu*, where

*ϵ*∈

**R**. Inserting this into (3.3) gives, for all

*ϵ*, Differentiating this with respect to

*ϵ*, and then setting

*ϵ*=0 gives The first line vanishes by theorem 2.2. The term in square brackets in the second line is

*Γu*. Thus, we have, for all

*u*∈

*H*

^{1}(

**), 3.4**

*T*If and satisfy *Γu*=*f*, then (3.4) tells us
And so we have shown the ‘only if’ part of the proposition.

To finish the proof, we need to show that if and then there exists *u*∈*X*^{1}_{odd} with *Γu*=*f*. First, note that
defines a bounded map from to . It is also bounded from *L*^{2}(**T**) to itself. Suppose that is a bounded sequence. As (owing to the Rellich–Kondrachov theorem), we know {*u*_{n}} contains a subsequence that converges in *L*^{2}(**T**). We abuse notation and call this subsequence {*u*_{n}}. Since the functions *u*_{n} are odd, so is the limit. Also, *K* is bounded on *L*^{2}(**T**), and thus {*Ku*_{n}} is a convergent sequence of even functions in *L*^{2}(**T**). The limit is even. Thus, *K* is a compact operator from to .

Now, *Γu*=*qu*′+*K*, which means that *Γ* is a compact perturbation of
We claim that *Γ*_{0} is a Fredholm operator from to with index −1. This is straightforward. We recall that *q* is non-zero and that the kernel of d/d*a* is the set of constant functions. The only odd constant function is 0. Therefore, the kernel of *Γ*_{0} in is trivial. Likewise, *Γ*_{0}*u*=*f* has a periodic solution *u* if and only if . This implies that the cokernel of *Γ*_{0} is one dimension. Thus, the index of *Γ*_{0} is −1.

As compact perturbations do not change the index of an operator [44], we know that *Γ* is also a Fredholm operator from to with index −1. As we know the kernel of *Γ* is trivial from corollary 3.2, we must therefore have that the dimension of the cokernel of *Γ* is equal to 1.

As , the work above tells us that is not in the range of *Γ*. Thus, the equivalence class of this function, which we call [*c*], spans the cokernel of *Γ*. Now take with . Let [*f*] be the equivalence class of *f* in the cokernel. There must be a constant *β*∈**R** such that [*f*]=*β*[*c*]. This means that there exists with . From the first part of this proof, we know that this implies , which implies *β*=0. This in turn implies that *Γu*=*f*, and we are done.

So *Γ* is not surjective. Our method is a modification of the Liapunov–Schmidt strategy commonly used in perturbation theory when the linearization of the problem fails to be injective to the case when the problem fails to be surjective. Define
where . A quick calculation shows *Π*^{2}=*Π*, and so *Π* is a projection. Moreover, (3.4) tells us that
3.5

Let and *M*:=*R*^{⊥}. Proposition 3.3 tells us that . Let
Note that *ΠΦ*_{red}=(*Π*−*Π*^{2})*Φ*=0, and so *Φ*_{red} is a map from into *R*.

From theorem 2.2, we know
Also,
As *ΠΓ*=0, we see that (1−*Π*)*Γ* is injective because *Γ* is. Moreover, by construction, (1−*Π*)*Γ* is surjective onto *R*. And thus we can, at last, apply the implicit function theorem: there exists a unique map
so that
for all (*p*,*κ*) sufficiently small and

Now set
This is a map from **R**^{2} to itself. That is to say, *g*(*p*,*κ*) satisfies . If we find *g*(*p**,*κ**)=0, then note that *Ξ*(*p**,*κ**) has
That is to say, it solves (3.2).

Clearly, *g*(0,0)=0. We claim that
3.6If so, we can call again on the implicit function theorem. This tells us that there exists a unique map, *ξ*(*p*), smooth, with *ξ*(0)=0 and defined for *p* sufficiently small, for which
(Note that this function *ξ*(*p*) selects *κ* and thus *γ*.) Let
This is the map whose existence we were hoping to establish. There are two things left to check. The first is to establish (3.6) and the second is to show that *Θ*_{q}(*p*)∈*X*^{2}, not just . This latter is important in that a solution of (3.2) in *X*^{2} will also be a solution of (2.1), which is our ultimate goal. We mention that the smoothness of *Θ*_{q} with respect to *q* follows from our repeated use of the implicit function theorem.

For (3.6), note that
As *ΠΓ*=0, we have
Then, we see from (3.2) that *D*_{κ}*Φ*(*θ*_{q};0,0)=−*e*^{−Hθq}, and therefore
Using lemma 2.1, we have *D*_{κ}*g*(0,0)=−2*π*, as claimed.

Finally, let *θ*:=*Θ*_{q}(*p*). We know that and that *θ* solves (3.2). Since *θ*∈*H*^{1}(**T**), we know *Hθ*∈*H*^{1}(**T**) as well. This in turn implies that and *e*^{−Hθ} are in *H*^{1}(**T**). Likewise for *I*(*θ*). Thus,
It is straightforward to see that as *θ* depends smoothly on *p*, so must *ρ*. As (3.2) states that *θ*′=−(1/*q*)*ρ*, we have *θ*′∈*H*^{1}(**T**), and thus *θ*∈*H*^{2}(**T**). Therefore, we can differentiate (3.2) to see that *θ* solves (2.1) and depends smoothly on *p*. (We remark that this argument could be repeated to conclude higher regularity of *θ*.) This completes the proof of theorem 2.4.

## 4. Numerical results

In this section, we augment the preceding existence proof with numerical computations of branches of travelling waves bifurcating from the Crapper waves. The branches are computed in the arclength parametrized vortex sheet formulation of the water wave problem [13]. Travelling waves are computed at different values of total displacement and Bond number *σ*=*g*/*k*^{2}*τ* (here, *τ* is the surface tension coefficient, *k* is a representative wavenumber and *g* is the acceleration due to gravity). The branches are continuous in both parameters. We observe that the amplitude along a branch is either limited topologically, by a profile that self-intersects or by a turning point, in which case, the largest profile is a stationary (i.e. standing) wave. As our preceding proof does not require positive values of gravity, we compute perturbations of the Crapper wave with both signs of *σ*. We allow large departures from *σ*=0, computing wave profiles with *σ*∈(−1,20], in which interval we observe continuous dependence on Bond number, including at the Wilton ripple configurations [25,45,46].

The numerical method used to compute these waves is based on the method of [13]. In this method, we find functions *θ* and *γ* that solve the equations
this is the travelling wave ansatz, as developed in [13]. Here, *U* is the normal velocity of the fluid interface and is equal to the normal component of the Birkhoff–Rott integral (which will be introduced in the following paragraph). A Fourier-collocation method is used to discretize spatial derivatives. We seek symmetric, even profiles in terms of tangent angle *θ* and vortex sheet strength *γ*. The frame of the wave is defined so that the mean of *γ* and *θ* are zero. When the system is discretized with *N* spatial points, we must then solve for *N* Fourier modes and the wave speed *c*. The projection of the partial differential equation into Fourier space gives *N* equations, to which we append an equation fixing the amplitude to close the system. The resulting nonlinear system of algebraic equations is then solved via Broyden's method, and continuation in amplitude or gravity, similar to [47,48].

The collocation method is standard, with the possible exception of the implementation of the Birkhoff–Rott integral, **W**=(*W*_{1},*W*_{2}), given by
which is computed using the method used in [13,49], wherein the integral is split into singular and regular parts. The singular portion is 1/2*H*(*γ*/*z*_{α}), where *H* is the Hilbert transform and is computed using its definition in Fourier space, . The remainder integral, *W*_{1}−*iW*_{2}−1/2*H*(*γ*/*z*_{α}), is non-singular and is computed using the trapezoidal rule at alternating grid points. The numerical results presented here use *n*=512 spatial points, resulting in an arclength step size of Δ*α*≈0.01. At this resolution, the largest computed waves, which have overturned crests and are near pinch-off, have Fourier modes that decay to 10^{−8} by wavenumber *k*=256.

We observe that travelling waves depend continuously on Bond number. The waves are also continuous in amplitude, and we use this fact to continue in both parameters to find the largest amplitude waves. As our numerical method is based on a vortex sheet formulation, we cannot compute waves that self-intersect, like those in [36,39] (that is, for self-intersecting waves, we would be unable to compute the Birkhoff–Rott integral). We are able to compute waves very close to the first self-intersecting configuration, which we refer to as pinch-off. The Crapper wave first self-intersects when the scaled displacement [1,10]. We have computed overturned profiles with scaled displacements up to *h*/2*π*=0.723, or 99% of the Crapper pinch-off displacement. Computed Crapper wave profiles at a sampling of amplitudes are superimposed in figure 1*a*.

An advantage of quasi-Newton-based methods over amplitude expansions, like Hogan [22] or Haupt & Boyd [37], is that no modification is necessary to compute travelling waves at the Wilton ripple resonant configurations. We observe Wilton ripples that are continuously embedded in the branches of travelling waves. In figure 1*b*, the profiles of travelling waves are presented for at a sampling of Bond numbers *σ*∈[−0.95,20].

We have employed continuation schemes to follow continuous branches of travelling waves in both amplitude and Bond number. Beginning with the Crapper wave, we observe that waves become more localized with increased Bond number. As *σ* increases, profiles become more solitary than periodic, gaining oscillatory tails similar to those computed in [47,50]. These changes occur at all amplitudes, see figure 1 for the case of , however, the size of oscillations in the tail decreases with amplitude. These numerical results support the argument of Longuet-Higgins [39], that the large-amplitude Crapper wave is connected continuously to gravity–capillary solitary waves. It is natural that the waves become solitary, as increasing the Bond number has the dual interpretation of decreasing *k* and increasing the wavelength.

In figure 2, we report both the globally largest wave and a phase portrait of the existence of travelling waves in the gravity–displacement plane. We observe that branches of travelling waves culminate in one of two ways: either in a standing wave or a self-intersecting profile. We consider self-intersecting interfaces to be the terminus of a branch of travelling waves. It is possible that branches of travelling waves continue to exist after self-intersection, however, such profiles need to be computed with an alternative method, for example, that of [36]. In figure 2*b*, we report a phase portrait of the space of travelling waves. In the grey region, travelling waves are single-valued functions of the *x*-coordinate. In the black region, the waves have overturned crests. The boundary between the black and white region is an estimate of the displacement of the largest wave. This boundary corresponds to self-intersection of the wave profile to the right of the star and a standing wave (*c*=0) to the left of the star. The globally largest wave is marked by a star, and is both stationary and self-intersecting.

That travelling waves exist with negative Bond numbers (owing to negative gravity coefficients) is not surprising when one examines the linear situation. The speed of a linear wave is One might expect travelling waves whenever the phase speed is real—thus linear theory predicts travelling waves for *σ*>−1. When *σ*=−1, the speed of the wave is zero, and we are left with stationary, standing waves. This is exactly what we observe numerically. Moreover, we observe that for the most negative Bond numbers, the branches of nonlinear travelling waves have amplitudes limited by standing waves. These standing waves form the boundary of the white region in figure 2*b*, from *σ*=−1 to *σ*≈−0.0432 marked by the star. Profiles of these standing waves are shown in figure 3*a*. The water wave problem has a symmetry between waves moving to the left and right. Given a wave of a particular amplitude and speed, there is a corresponding wave with the same amplitude and negative speed. At stationary waves, these two branches of travelling waves intersect to form a closed loop. Whenever branches of travelling waves are thus connected, the standing wave will be the profile of limiting amplitude.

## Funding statement

D.M.A. gratefully acknowledges support from the National Science Foundation through grant nos DMS-1008387 and DMS-1016267. J.D.W. gratefully acknowledges support from the National Science Foundation through grant nos DMS-0908299 and DMS-1105635.

## Acknowledgements

Although stability has not been discussed here, there is preliminary evidence that gravity–capillary solitary waves with bubbles may be stable; the authors thank Zhan Wang for access to his numerical results of the time evolution of such waves (Z. Wang 2013, personal communication). This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process or service by trade name, trademark, manufacturer or otherwise does not necessarily constitute or imply its endorsement, recommendation or favouring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

- Received August 5, 2013.
- Accepted October 18, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.