# On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

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## Abstract

We study the bifurcation of periodic travelling waves of the capillary–gravity Whitham equation. This is a nonlinear pseudo-differential equation that combines the canonical shallow water nonlinearity with the exact (unidirectional) dispersion for finite-depth capillary–gravity waves. Starting from the line of zero solutions, we give a complete description of all small periodic solutions, unimodal as well bimodal, using simple and double bifurcation via Lyapunov–Schmidt reductions. Included in this study is the resonant case when one wavenumber divides another. Some bifurcation formulas are studied, enabling us, in almost all cases, to continue the unimodal bifurcation curves into global curves. By characterizing the range of the surface tension parameter for which the integral kernel corresponding to the linear dispersion operator is completely monotone (and, therefore, positive and convex; the threshold value for this to happen turns out to be \(T = \frac{4}{\pi ^2}\), not the critical Bond number \(\frac{1}{3}\)), we are able to say something about the nodal properties of solutions, even in the presence of surface tension. Finally, we present a few general results for the equation and discuss, in detail, the complete bifurcation diagram as far as it is known from analytical and numerical evidence. Interestingly, we find, analytically, secondary bifurcation curves connecting different branches of solutions and, numerically, that all supercritical waves preserve their basic nodal structure, converging asymptotically in \(L^2(\mathbb {S})\) (but not in \(L^\infty \)) towards one of the two constant solution curves.

## Keywords

Whitham-type equations Water waves Multi-dimensional bifurcation Nonlinear waves## Mathematics Subject Classification

Primary 35Q35 37K50 76N10## 1 Introduction

The overarching technique follows an approach similar to that used for the gravity Whitham equation in [18] and the Euler equations in [14], where a Lyapunov–Schmidt reduction is used to prove the existence of wave solutions through the application of the implicit function theorem. Here, however, the symbol of the linear dispersion has a different large-frequency behaviour: whereas it is \(\sim |\xi |^{-1/2}\) in the gravity case, it changes to \(\sim |\xi |^{1/2}\) in the presence of surface tension. Inspired by recent work on large waves for very weakly dispersive equations, we tackle the equation by inverting the linear operator, see (2.3), presenting us with a smoothing operator with good properties but that now acts nonlocally on a nonlinear term. Apart from the results presented in this paper, we see this as a first step toward handling large-amplitude theory for equations with mixed nonlocal and nonlinear terms. A study in that direction, but with a different order and global structure of the solutions, has been carried out in [6].

The organisation of the paper corresponds to the development of our theory:

We start, in Sect. 2, with a study of the *inverse* of the Fourier multiplier operator *M* in (1.2). This is a smoothing operator of order \(-\frac{1}{2}\) on any Fourier-based scale of functions spaces (such as the Sobolev and Zygmund spaces), that is realised as a convolution operator with a surface tension-dependent integral kernel \(K_T\). We characterise the kernel \(K_T\) in Theorem 2.3, expressing it as a sum of three terms that are, optimally, in the regularity classes \(\mathcal {C}^{-\frac{1}{2}}\), \(\mathcal {C}^{-\frac{3}{2}}\) and \(C^{\omega }\), respectively, where \(\mathcal {C}^s\) is the scale of Zygmund spaces, and \(C^\omega \) is the class of real-analytic functions. This is different from the regular Whitham symbol which, although of the same order, has only two terms when decomposed in the same manner [20]. Additionally, we estimate the decay rate of \(K_T\) and its compactness properties (in suitable spaces) which will play an important role in the global bifurcation analysis in Sect. 3. Finally, as in [20] we apply complex analysis techniques and the theory of Stieltjes functions to determine further properties of the convolution kernel, in particular the signs of its derivatives to infinite order. When the surface tension is big enough, \(T \geqslant \frac{4}{\pi ^2}\), we are able in Theorem 2.8 to show that the kernel is *completely monotone*, a delicate structural property shared by the kernel for the linear dispersion in the pure gravity case (not its inverse). Moreover, we can show that neither complete monotonicity nor monotonicity on a half-line is preserved if \(0< T < \frac{4}{\pi ^2}\), showing in effect that the critical Bond number \(\frac{1}{3}\) separating weak from strong surface tension is *not* the break-off value for the positivity of the kernel (or its stronger properties). How this affects solutions is further discussed and studied in Sect. 5.

In Sect. 3 we perform the one-dimensional bifurcation of periodic waves from simple eigenvalues along the line of zero solutions. After an initial discussion of the eigenvalues of the linearised operator, and a scaling to reduce the problem to a fixed period, we use Lyapunov–Schmidt reduction to prove the existence of small-amplitude solutions in a vicinity of the simple eigenvalues (expressed using the wavespeed) in Theorem 3.1. The constructed waves are all unimodal and bell shaped in a minimal period. They arise for both strong and weak surface tension; for strong surface tension, they are the only type of waves in a \(\mathcal {C}^s(\mathbb {S})\)-vicinity of the line of zero solutions, \(s > 0\). Although one could have carried out the simple bifurcation using the Crandall–Rabinwitz theorem [26], we choose to prove Theorem 3.1 using a Lyapunov–Schmidt reduction as a preparation for the two-dimensional case (which would otherwise be harder to understand). Under a simple condition that relates the wavenumber to the surface tension and period, we prove the continuation of the local solution curves to global ones in Theorem 3.6. This condition may be related to sub- and supercritical bifurcation, and we see in Remark 3.7 that both cases can appear. The modulational stability of these waves in the small-amplitude case has been studied in [23]

Our motivation for this investigation has arisen from two different directions: one is the study of the (very) weakly dispersive equations with nonlocal nonlinearities, and especially their large-amplitude theories; the other is the mathematically qualitative analogues between the full water-wave problem and the family of fully dispersive Whitham-type equations. While numerical bifurcation of steady water waves with surface tension has been earlier carried out [8] and displays striking resemblances to our case, it is not known how to control the waves along the bifurcation curves when surface tension is present, and our results show that, at least for weak surface tension, the looping alternative in Theorem 3.6 is possible.^{1} Our initial hope was that, using methods as in [16, 20], one would be able to reach a conclusion for larger waves. In Sect. 5 we turn to this question, as well as discussing the general picture of bifurcation in the capillary–gravity Whitham equation. While we are indeed able to reach a partial result, preserving the nodal properties to \(\mathcal {O}(1)\)-height of the solutions in Proposition 5.4, the final evolution of solution curves is very challenging to handle analytically. While both our preliminary calculations and numerical simulations for this paper indicate that one can follow curves of supercritical bell-shaped solutions all the way to \(c \rightarrow \infty \), and that they converge, asymptotically in \(L^2(\mathbb {S})\), towards the curve of constant solutions \(u = c -1\), they do not converge in \(L^\infty \), and the analysis is complicated by that the equation lies exactly at the Sobolev-critical balance \(s = \frac{1}{2}\), \(p = 2\) and \(n = 1\). We discuss both our findings and conjectures in detail in Sect. 5. For a quick overview, we refer to Figs. 3 and 4.

## 2 Properties of the Convolution Kernel \(K_T\)

*By a solution of*(2.1) [

*respectively*(2.3)]

*, we shall mean a real-valued, continuous and bounded function*

*u*

*that satisfies*(2.1) [

*respectively*(2.3)]

*everywhere*.

### 2.1 Analyticity of the Symbol

*symmetric difference*between two sets

*A*and

*B*is the set \(A \bigtriangleup B\) of elements either in

*A*and not

*B*, or contrariwise

^{2}

### Lemma 2.1

Let \(\zeta = \xi + i \eta \). Then, \(\varrho _T(\zeta )\) takes a zero or infinite value exactly if \(\xi =0\) and \(\eta \in Z_s \cup ( Z_c \bigtriangleup Z_T )\). Further, \(\varrho _T(\zeta )\) is negative exactly when the following three conditions hold: \(\xi =0\), \(\eta \notin Z_s \cup ( Z_c \bigtriangleup Z_T )\), and the intersection \((0, |\eta |) \cap ( ( Z_c \cup Z_s) \bigtriangleup Z_T )\) contains an odd number of elements.

### Proof

In Sect. 2.2 we will use Paley–Wiener theory to establish the decay rate of \(K_T\); we will need to know the maximal vertical analytic extension of \(l_T\) into the complex plane. This is immediate from the previous result, and so we record the following corollary.

### Corollary 2.2

We shall also use decay of symbols on horizontal lines in \(\mathbb {R}\times i(-\delta ^*,\delta ^*)\). While \(l_T\) decays too slow (\(\sim |\xi |^{-\frac{1}{2}}\)) to be in \(L^2(\mathbb {R})\), its derivatives decay sufficiently fast (at least as \(|\xi |^{-\frac{3}{2}}\)). In particular, there is an increasing function \(\tau :[0,\delta ^*)\rightarrow \mathbb {R}^+\) such that \( |l'_T(\zeta )|\leqslant \tau (|\eta |) (1+|\xi |)^{-\frac{3}{2}}, \) which is readily seen by differentiating and exploiting that \(\coth '\) decays exponentially along fixed horizontal lines in the complex plane.

### 2.2 Regularity Properties and Decay

^{3}

*k*. We refer the reader to [36, Section 13.8] and [21, Section 1.4] for further details.

### Theorem 2.3

### Proof

We conclude this subsection by recording some mapping properties of the convolution operator \(L_T = K_T *\) that will be vital to the global bifurcation analysis in Sect. 3 and additionally employed in the analysis in Sect. 5. Let \(\mathbb {S}\) be the one-dimensional unit sphere of circumference \(2\pi \), and note that the Hölder and Zygmund spaces are straightforward to define on the compact manifold \(\mathbb {S}\) (these are the \(2\pi \)-periodic functions in the larger spaces \(C^s(\mathbb {R})\) and \(\mathcal {C}^s(\mathbb {R}))\).

### Lemma 2.4

For each \(T> 0\) and each \(s \geqslant 0\), \(L_T\) is a continuous linear mapping \(\mathcal {C}^s(\mathbb {R}) \rightarrow \mathcal {C}^{s+1/2}(\mathbb {R})\) and is hence compact on \(\mathcal {C}^s(\mathbb {S})\).

### Proof

### 2.3 Montonicity and Complete Monotonicity

*T*. This result will be employed in our analysis in Sect. 5. A function \(g : (0, \infty ) \rightarrow [0,\infty )\) is called

*completely monotone*if

*g*is infinitely differentiable with

*Stieltjes*. Our interest in such functions is motivated by the following two results, taken from [20, 35].

### Lemma 2.5

[20] Let \(f : \mathbb {R}\rightarrow \mathbb {R}\) and \(g : (0, \infty ) \rightarrow \mathbb {R}\) be two functions satisfying \(f(\xi ) = g(\xi ^2)\) for \(\xi \ne 0\). Then, *f* is the Fourier transform of an even, integrable, and completely monotone function if and only if *g* is Stieltjes with \(\lim _{\lambda \searrow 0} g(\lambda ) < \infty \) and \(\lim _{\lambda \rightarrow \infty } g(\lambda ) = 0\).

### Lemma 2.6

[35] Let *g* be a positive function on \((0, \infty )\). Then, *g* is Stieltjes if and only if \(\lim _{\lambda \searrow 0} g(\lambda )\) exists in \([0, \infty ]\) and *g* extends analytically to \(\mathbb {C}{\setminus } (-\infty , 0]\) such that \({\mathrm {Im}}(z)\cdot {\mathrm {Im}}(g(z)) \leqslant 0\).

With \(f(\xi ) = l_T(\xi )\) and \(g(\xi ) = l_T(\sqrt{\xi })\) we want to employ the two above results to conclude that \(K_T=\mathcal {F}^{-1}(l_T(\xi ))\) is completely monotone for *T* sufficiently large. Since \(l_T\) has a unit limit at the origin and a vanishing limit at infinity, it only remains to prove that \(l_T(\sqrt{\cdot })\) is Stieltjes. In light of Lemma 2.6 it is useful to note that \(l_T(\sqrt{\cdot })\) indeed extends analytically to \(\mathbb {C}{\setminus } (-\infty , 0]\). Its extension is \(\zeta \mapsto \sqrt{\varrho _T(\sqrt{\zeta })}\), where \(\varrho _T\) is as in (2.5) and \(\sqrt{\cdot }\) is the principal branch of the square root. To see that this extension is well defined, note that \(\sqrt{\cdot }\) maps \(\mathbb {C}{\setminus } (-\infty , 0]\) into the right half-plane \(\mathbb {C}_{\xi >0}\), while Lemma 2.1 guarantees that \(\varrho _T\) maps \(\mathbb {C}_{\xi >0}\) into \(\mathbb {C}{\setminus } (-\infty , 0]\). Consequently, \(\varrho _T(\sqrt{\mathbb {C}{\setminus } (-\infty , 0]})\subseteq \mathbb {C}{\setminus } (-\infty , 0]\), and so it has principal branch square root. We are ready to prove Theorem 2.8, where we determine a critical value \(T_*=\frac{4}{\pi ^2}\) of the surface tension *T*, for which \(K_T\) is completely monotone whenever \(T\geqslant T_*\). Note that \(T_*\)*does not* correspond to the, likewise critical, Bond number \(T = \frac{1}{3}\) that separates strong from weak surface tension; in fact, \(T_*>\frac{1}{3}\). Further, this result is sharp since \(K_T\) is not monotone for \(T\in (0,T_*)\). As we shall see, the image of \(K_T\) in this regime contains negative values which rules out monotonicity as Theorem 2.3 guarantees that \(K_T\) is positive near zero and decays to zero at infinity.

In the calculations to come, we will make use of the class of so-called positive definite functions. A function \(f:\mathbb {R}\rightarrow \mathbb {C}\) is said to be *positive definite* if for every \(n\in \mathbb {N}\) and \(\varvec{\xi }\in \mathbb {R}^n\) the \(n\times n\) matrix \([f(\xi _i-\xi _j)]_{i,j=1}^n\) is positive semi-definite. We point out the following standard results [9].

### Lemma 2.7

- (i)
[Bochner’s Theorem] Any positive definite function is the Fourier transform of a non-negative, finite Borel measure.

- (ii)
[Schur’s Theorem] A countable product of positive definite functions is positive definite.

- (iii)
If \(f:\mathbb {R}\rightarrow \mathbb {C}\) is positive definite, then the global maximum of

*f*occurs at \(x=0\). - (iv)
The function \(f(x)=\frac{1+ax^2}{1+bx^2}\) is positive definite if and only if \(b\geqslant a\geqslant 0\).

With the above preliminaries, we now state the main result for this section.

### Theorem 2.8

For \(T\geqslant \frac{4}{\pi ^2}\), the kernel \(K_T\) is completely monotone on \((0,\infty )\). Further, for \(0<T<\frac{4}{\pi ^2}\), the image of \(K_T\) includes negative values. Consequently, \(K_T\) is not monotone on \((0,\infty )\).

### Proof

Before we end this section, we note that there is a range of values of strong surface tension \(T\in (\frac{1}{3},\frac{4}{\pi ^2})\) where the kernel \(K_T\) is not monotone. As we will see, this has implications when trying to establish monotonicity of solutions along the supercritical global solution branches described in Sect. 3.3 below; see Proposition 5.4 and the discussion in Sect. 5 in general.

## 3 One-Dimensional Bifurcation

*f*through a convolution of

*f*with a \(2\pi \)-periodic kernel \(K_p\) over a single period:

*F*correspond to the even and \(2\pi \)-periodic solutions of (2.1) with wavespeed

*c*. The choice \(s > \frac{1}{2}\) is by convenience, as functions of that regularity have absolutely convergent Fourier series [25].

^{4}

Note that for each fixed *k*, the function \(T_*(\cdot ;k)\) tends to zero as \(n\rightarrow \infty \), as does \(T_*(0;k)\) when \(k \rightarrow \infty \). It is also not hard to see that \(T_*(0;k)\) is a strictly decreasing function of *k*. Numerical plots indicate that also the function \(T_*(\cdot ;k)\) is strictly decreasing, but we will not use this monotonicity property in our proofs.

Throughout the remainder of this section, we turn our attention to the branches of solutions \(\{(u,c)\}\) bifurcating from the trivial line \(u=0\) at some wavespeed \(c_*\) for a fixed value of the surface tension \(T > 0\) and where \(\ker D_u F[0,c_*]\) is one-dimensional; two-dimensional bifurcation in the case \(0< T < \frac{1}{3}\) is dealt with in Sect. 4. Note that while one-dimensional kernels appear both for sub- and supercritical wave speeds, separated by \(c = 1\), two-dimensional kernels only appear for \(c \in (0,1]\): see Sect. 4 below.

### 3.1 The Parameters

*c*, the surface tension

*T*, and a scaling in the period of the waves, \(\kappa \). While the first two appear directly in the steady problem (2.1), the scaling \(\xi \mapsto \kappa \xi \) is realised by introducing the corresponding dependence in the convolution operator

*L*, so that

Since surface tension is a property of the medium, while the speed and wavenumber are properties of particular waves, it is physically more relevant to use the two latter as bifurcation parameters, while holding the surface tension fixed. This is what we will do in the following.

### 3.2 Local Bifurcation via Lyapunov–Schmidt

The following theorem establishes, for fixed wavelength and surface tension, the local bifurcation of small amplitude steady solutions the capillary–gravity Whitham equation (1.1). Although this is by now a standard Crandall–Rabinowitz type result [26], we prove the result using a direct Lyapunov–Schmidt reduction as to prepare for the two-dimensional bifurcation in Sect. 4. This is similar to the strategy in [14]. As \(\kappa \) and *T* will be fixed—assuming that we already have a one-dimensional kernel as described in the beginning of this section—we shall here suppress the dependence upon these parameters.

### Theorem 3.1

### Remark 3.2

There is an additional but qualitatively different bifurcation taking place at \(c =1\), where the straight curve of constant solutions \((u,c) = (c-1,c)\) crosses the trivial solution curve (0, *c*). These solutions must be taken into consideration when constructing non-constant waves at \(c=1\) when the kernel is two-dimensional, see Theorem 4.1.

### Remark 3.3

### Proof

*T*and \(\kappa \) throughout. According to the assumptions and the discussion after (3.2), on \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\) we have

*v*and

*r*such that for \(|t|\ll 1\) we have

*t*-dependence in

*v*and

*r*. Noting that

*v*is invertible on \(({\mathrm {Id}}-\Pi )\, \mathcal {C}^s_{\mathrm {even}}(\mathbb {S})\) (because \(D_uF(0,c_0)\) is). Therefore, by the implicit function theorem on Banach spaces, the second line of (3.9) has a unique solution \(v(t, r) \in ({\mathrm {Id}}-\Pi )\, \mathcal {C}^s_{\mathrm {even}}(\mathbb {S})\) defined in a neighbourhood of \((t,r) = (0, 0)\), and depending analytically on its arguments. By uniqueness, \(v(0,r)=0\) for all \(|r| \ll 1\). Moreover, differentiation with respect to

*t*at \((t,r) = (0,0)\) in (3.8) shows that \(\frac{\partial }{\partial t}v(0,r) = 0\), which implies that

*v*has no constant or linear terms in

*t*. As it is smooth in

*t*, it may be expanded in an (at least) quadratic series around \(t=0\).

*r*, with

*r*, which together with the symmetry of

*L*implies that we can write

*R*is analytic with \(R(0,0) = \partial _r R(0,0) = 0\), again due to the properties of

*v*(here, \(l = l_{T, \kappa }\)). An application of the implicit function theorem to the equation \(r \, l(k) \pi + R(t,r) = 0\) at \((t,r) = (0,0)\) then yields the existence of a locally unique smooth function \(r :t \mapsto r(t)\) with \(r(0) = 0\) such that

### 3.3 Global Bifurcation (Analytic)

*N*are solutions of the steady capillary–gravity Whitham equation (2.1), and conversely. Let

*N*). Note that Lemma 2.4 implies that \(S \subset \mathcal {C}^\infty _{\mathrm{even}} \times \mathbb {R}\), so that all solutions are smooth: for details, see Proposition 5.1 below. By combining this with a diagonal argument one obtains the following compactness result.

### Lemma 3.4

Bounded and closed sets in *S* are compact in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\).

### Proof

*L*is compact on \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\) thanks to Lemma 2.4, it follows that after passing to a further subsequence \((u_l,c_l)_l\subset K\) that \((N(u_l,c_l))_l\) converges in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\) to some function

*u*. Since \(u_l=N(u_l,c_l)\) by definition, passing to limits implies the sequence \((u_l,c_l)_l\) converges in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\) with limit \((u,c)\in S\). As

*K*is closed it follows that \((u,c)\in K\), establishing that

*K*is compact. \(\square \)

### Corollary 3.5

The Fréchet derivative \(D_u F(u,c)\) is a Fredholm operator of index 0 at any point \((u,c) \in C^s_{\mathrm {even}}(\mathbb {S})\times \mathbb {R}\).

### Proof

Before embarking on to the next theorem, we recall the shorthand \(l(\cdot )\) for \(l_{\kappa ,T}(\cdot )=l_T(\kappa \cdot )\).

### Theorem 3.6

- (i)
\(\Vert (u(t), c(t)) \Vert _{\mathcal {C}^s(\mathbb {S}) \times \mathbb {R}} \rightarrow \infty \) as \(t \rightarrow \infty \).

- (ii)
\(t \mapsto \left( u(t), c(t) \right) \) is

*P*-periodic for some finite*P*, so that the curve forms a loop.

### Remark 3.7

*k*, and a subcritical pitchfork bifurcation for large

*k*. Note numerically, we observe there exists a unique \(k_*=k_*(T)>0\) such that \(\ddot{c}(0)>0\) for \(0<k<k_*\) and \(\ddot{c}(0)<0\) for \(k>k_*\). For \(0<T<1/3\), both the numerator and denominator of (3.10) change signs. Note that one may be able to do global bifurcation when \(\ddot{c}(0)=0\) but inspecting \(c^{(4)}(0)\): see, for example, [20, Theorem 6.1]. We do not pursue this here.

### Proof

This theorem is a version of the global analytic bifurcation theorem in [10], and—apart from the bifurcation formulas—the proof goes as in the purely gravitation case in [18, 20]. The assumptions are fulfilled from Lemma 3.4 and Corollary 3.5 if one can just show that some derivative \(c^{(k)}(0)\) is non-vanishing. We give the calculations for \(\dot{c}(0)\) and \(\ddot{c}(0)\) in the Appendix; the first is 0, and the second is given by (3.10). Note that a third alternative in the theorem in [10] does not happen here, as the set \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S}) \times \mathbb {R}\) lacks a boundary. \(\square \)

There are a few more things one can say about the global bifurcation curves, both numerically and analytically, and we discuss the global bifurcation diagram in detail in Sect. 5. In particular, the cases of strong and weak surface tension are summarised in Figs. 3 and 4, respectively.

## 4 Two-Dimensional Local Bifurcation

*c*, while the surface tension

*T*is assumed to be fixed. We shall, therefore, study for \(\kappa >0\) the operator

*T*, as it will not be used apart from in this assumption). A two-dimensional kernel can arise only for \(c_0 \in (0,1]\). Let now \(1 \leqslant k_1 \leqslant k_2\). With \(\mathcal {S}^k\) being the sheet of \(2\pi /k\)-periodic solutions defined in (3.5) we shall show that in addition to the solutions in \(\mathcal {S}^{k_1}\) and \(\mathcal {S}^{k_2}\), we may obtain solutions in a set called \(\mathcal {S}^{{mixed}}\) consisting of perturbations of functions in the span of \(\cos (k_1 \cdot )\) and \(\cos (k_2 \cdot )\). Assuming that \(k_1 \leqslant k_2\), the resonant case when \(k_2\) is an integer multiple of \(k_1\) (sometimes referred to as Wilton ripples) is more difficult than the generic case, but we follow here the procedure in [14, 19] to construct a slit disk of solutions also in that case. Numerical calculations indicate that this set is optimal [34].

When one of the wavenumbers is zero (meaning \(c_0 = 1\)), we instead call that one \(k_2\), and we will automatically have the resonant case, as then \(k_1 \mid k_2\). That case is included in the below theorem. Hence, at \(c=1\) there is a nontrivial bifurcation, but the arising waves always have a non-zero component in the constant direction.

### Theorem 4.1

- (i)When \(k_1\) does not divide \(k_2\) there is a full, smooth, sheetof solutions in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\times \mathbb {R}_+\) of the form$$\begin{aligned} \mathcal {S}^{{mixed}} = \{ \left( u(t_1, t_2), c(t_1, t_2), \kappa (t_1, t_2)\right) :0 < |(t_1, t_2)|\ll 1 \} \end{aligned}$$to the steady capillary–gravity Whitham equation (2.1). The set \(\mathcal {S}^{k_1} \cup \mathcal {S}^{k_2} \cup \mathcal {S}^{{mixed}}\) contains all nontrivial solutions in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\times \mathbb {R}_+\) of this equation in a neighbourhood of \((0, c_0, \kappa _0)\).$$\begin{aligned} u(t_1, t_2)&= t_1 \cos (k_1 x) + t_2 \cos (k_2 x) + \mathcal {O}(|(t_1, t_2)|^2), \\ c(t_1, t_2)&= c_0 + \mathcal {O}((t_1, t_2)), \\ \kappa (t_1, t_2)&= \kappa _0 + \mathcal {O}((t_1, t_2)), \end{aligned}$$
- (ii)When \(k_1\) divides \(k_2\) there exists for any \(\delta > 0\) a small but positive \(\varepsilon _\delta \) and a slit, smooth, sheetof solutions in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\times \mathbb {R}_+\) of the form$$\begin{aligned} \mathcal {S}^{{mixed}}_\delta = \{ \left( u(\varrho , \vartheta ), c(\varrho , \vartheta ), \kappa (\varrho , \vartheta )\right) :0<\varrho<\varepsilon _\delta ,\, \delta< |\vartheta | < \pi - \delta \} \end{aligned}$$to the steady capillary–gravity Whitham equation (2.1). In a neighbourhood of \((0, c_0, \kappa _0)\), the set \(\mathcal {S} = \mathcal {S}^{k_2} \cup \mathcal {S}^{{mixed}}_\delta \) contains all nontrivial solutions in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\times \mathbb {R}_+\) of (2.1) such that \(\delta< |\vartheta | < \pi -\delta \).$$\begin{aligned} u(\varrho , \vartheta )&= \varrho \cos (\vartheta ) \cos (k_1 x) + \varrho \sin (\vartheta ) \cos (k_2 x) + \mathcal {O}(\varrho ^2), \\ c(\varrho , \vartheta )&= c_0 + \mathcal {O}(\varrho ), \\ \kappa (\varrho , \vartheta )&= \kappa _0 + \mathcal {O}(\varrho ). \end{aligned}$$

### Remark 4.2

The order of vanishing of the functions \(c-c_0\) and \(\kappa -\kappa _0\) in Theorem 4.1 is analysed in Sect. A.2 of Appendix A.

### Remark 4.3

The bifurcation Theorem 4.1 shows that near a two-dimensional bifurcation point in the case where \(k_2/k_1 \notin \mathbb {N}_0\) there exists a full disk of solutions (for fixed \(\kappa \)), while if \(k_2/k_1 \in \mathbb {N}_0\) the disk is slit with one axis removed. This situation is summarised in Fig. 2. In particular, this means that it is possible to find curves connecting solutions with different wavenumbers, consistent with the recent numerical findings in [34].

### Proof

*v*,

*r*and

*p*parameterised by \((t_1, t_2)\) such that

*v*is orthogonal to \(\ker (D_u F_{\kappa _0}(0,c_0))\) the second equation in (4.5) reads \(v = D_u F_{\kappa _0}(0,c_0)^{-1}({\mathrm {Id}}-\Pi )g(t_1, t_2, r, p, v)\). It is clear that

*v*is \(D_u F_{\kappa _0}(0,c_0)\), which is invertible on \(({\mathrm {Id}}-\Pi )C_{\mathrm {even}}^s(\mathbb {S})\). The implicit function theorem then ensures the existence of a solution \(v = v(t_1, t_2, r, p) \in ({\mathrm {Id}}-\Pi )C_{\mathrm {even}}^s(\mathbb {S})\). By uniqueness, we have that \(v(0,0,r,p) = 0\) for all small enough values of

*r*and

*p*. Moreover, note that \(\frac{\partial }{\partial t_1}v(0,0,0,0) = 0\) and \(\frac{\partial }{\partial t_2}v(0,0,0,0) = 0\). This follows by differentiating (4.4) with respect to \(t_1\) or \(t_2\), and evaluating at \((t_1, t_2, r, p) = (0, 0, 0, 0)\) recalling that \(D_u F_{\kappa _0}(0,c_0)\) is invertible on its range. As a consequence,

*v*depends at least quadratically on \(t_1\) and \(t_2\).

*The non-resonant case*. Assume that \(k_2/k_1 \notin \mathbb {N}_0\). Using the properties of

*v*and \(\Pi _1\), a direct calculation shows that

*p*and

*r*in a neighbourhood of \((t_1,t_2,r,p)=(0,0,0,0)\). There are clearly four cases: \(t_1 = t_2 = 0\) represents the trivial solutions. When \(\Psi _1 = 0\) and \(t_2=0\) we can apply Theorem 3.1 concerning one-dimensional bifurcations along with the remark following it to obtain the solutions in \(\mathcal {S}^{k_1}\). Similarly, when \(t_1=0\) and \(\Psi _2=0\) we instead retrieve the solutions in \(\mathcal {S}^{k_2}\). To obtain the mixed-period solutions we apply the implicit function theorem to solve \(\Psi _1 = \Psi _2 = 0\) near the origin. Indeed, note that \(\Psi _1(0,0,0,0)=\Psi _2(0,0,0,0)=0\) and that the Jacobian of the map

*p*and

*r*both vanishing to at least second order at \((t_1,t_2)=(0,0)\), as claimed.

*The resonant case*. Assume now that \(k_2/k_1 \in \mathbb {N}_0\). In this case, we are not guaranteed that \(Q_2(t_1,0,r,p)=0\) for all \(|t_1|\ll 1\) due to a possible resonance in the inner product term in (4.11). Nevertheless, we do know that \(Q_2(0,0,r,p)=0\). Using polar coordinates to introduce the function

*r*and

*p*in a neighbourhood of \((\varrho ,\vartheta ,r,p)=(0,0,0,0)\). The case \(\varrho =0\) clearly corresponds to trivial solutions, while the case \(\cos (\vartheta )=0, {\widetilde{\Psi }}_2 = 0\) corresponds to solutions in \(\mathcal {S}^{k_2}\) via the application of Theorem 3.1. For the case that \({\widetilde{\Psi }}_1=0, {\widetilde{\Psi }}_2=0\) we again apply the implicit function theorem near the origin. Indeed, note that both \({\widetilde{\Psi }}_1\) and \({\widetilde{\Psi }}_2\) both vanish at the origin and that the Jacobian of the map

## 5 Global Bifurcation Diagram

In this section, we give some additional properties of solutions of (2.1), that is, of continuous and finitely periodic solutions. Our goal is to communicate the global bifurcation picture, as gathered from both analytic and numerical evidence, as well as to relate this to some comparable studies. We first present and prove the additional analytic results, after which we discuss the bifurcation diagram of the periodic capillary–gravity Whitham with the help of Figs. 3 and 4.

### Proposition 5.1

Any \(L^\infty (\mathbb {R})\)-solution of the steady capillary–gravity Whitham equation (2.1) is smooth.

### Proof

This is immediate from writing the equation in the form (2.3). For any \(T > 0\), the operator \(L_T\) is a smoothing Fourier multiplier operator of order \(-\frac{1}{2}\). This applies in particular to the scale of Zygmund spaces \(\mathcal {C}^s(\mathbb {R})\), \(s \geqslant 0\), see Lemma 2.4. As \(L^\infty (\mathbb {R})\) is an algebra embedded in \(\mathcal {C}^{0}(\mathbb {R})\) [36, Section 13.8], and the spaces \(\mathcal {C}^s(\mathbb {R})\) are Banach algebras for \(s > 0\), the result follows by bootstrapping. \(\square \)

### Proposition 5.2

- (i)There are no periodic solutions of (2.1) in the region$$\begin{aligned} \max u < \min \{0, c-1\}.\\ \end{aligned}$$
- (ii)
Except for the bifurcation points when \(c = \frac{1}{l_T(k)} > 0\) there are no small periodic solutions in a vicinity of any point along the curve of trivial solutions \((u,c) = (0,c)\), \(c \in \mathbb {R}\). Similarly, there are no periodic solutions that are small perturbations of the constant solutions \((u,c) = (c-1,c)\), \(c \in \mathbb {R}\), except for the bifurcation points that appear along this line for \(c < 2\).

- (iii)
The solution \(u=0\) is the only periodic solution for \(c=1\).

- (iv)For \(T\geqslant \frac{4}{\pi ^2}\), all periodic solutions satisfywith equality if and only if$$\begin{aligned} \textstyle \max u \leqslant \frac{c^2}{4}, \end{aligned}$$
*u*is a constant solution and either \(c = 0\) or \(c=2\).

### Remark 5.3

The qualifier ’periodic’ is here used only to guarantee that solutions, which we have defined to be continuous, are integrable over their period.

### Proof

For the second statement, consider first \(c < 1\). As the symbol \(l_T\) is positive, and the operator \(L_T\) is a linear isomorphism \(\mathcal {C}^s(\mathbb {S}) \rightarrow \mathcal {C}^{s+\frac{1}{2}}(\mathbb {S})\) unless \(c l_T(k) = 1\) (cf. 3.2), the implicit function theorem implies that there are no small solutions in a vicinity except for the bifurcation points found in Theorems 3.1 and 4.1 when \(c < 1\). In particular, there are no such solutions for \(c < 1\) in the case of strong surface tension \(T \geqslant \frac{1}{3}\), and none for \(c < 0\) in the case of weak surface tension \(0< T < \frac{1}{3}\). By Galilean invariance, the corresponding result applies to the line \(u = c - 1\) for \(c \geqslant 1\).

The proposition (iii) is immediate from (5.1).

*L*is then everywhere positive. This proves that \(\max u \leqslant \frac{c^2}{4}\), with equality if and only if \((u,c) = (1,2)\) or \((u,c) = (0,0)\), as these are the only constant solutions along the line \(\max u = \frac{c}{2}\). \(\square \)

### Proposition 5.4

- (i)
is bounded in wavespeed but with \(\min u\) unbounded; or

- (ii)
eventually leaves every set \(\{\max u \leqslant \lambda c\}\) for \(\lambda < \frac{1}{2}\).

### Proof

*u*one may as in [16, 20] use (2.1) to write

*u*is decreasing on \((0,\pi )\) with \(u \leqslant \frac{c}{2}\), this implies that

*u*is strictly decreasing on the same interval (unless

*u*is a constant), and a standard argument [16, Lemma 5.5] yields that looping as in alternative (ii) is ruled out.

Let us therefore, for a contradiction, assume that the bifurcation curve remains within the set \(\{\max u < \frac{c}{2}\}\). Recalling that Theorem 2.8 and [20, Proposition 3.2] together imply that \(K_p\) is completely monotone on \((0,\pi )\) when \(T \geqslant \frac{4}{\pi ^2}\), it follows that alternative (i) in Theorem 3.6 has to hold. As solutions are smooth, this is equivalent to a sequence of solutions \((u_n, c_n) = (u(t_n),c(t_n))\) satisfying \(|u_n|_\infty + |c_n| \rightarrow \infty \) as \(n \rightarrow \infty \).

Assume first that \(\{c_n\}_n\) is bounded. Then, \(\{u_n\}_n\) is unbounded in \(L^\infty (\mathbb {R})\) and, therefore, \(\min u_n \rightarrow - \infty \) as \(n \rightarrow \infty \) is the only possibility, by Proposition 5.2 (iv).

If, on the other hand, \(\{c_n\}_n\) is unbounded, pick a subsequence with \(\lim _{n \rightarrow \infty } |c_n| = \infty \). Note that \(c_n\) cannot pass \(c=1\), as Proposition 5.2 (iii) shows that it would have to pass via \((u,c) = (0,1)\), but near that point there are only small constant solutions (see Remark 3.2 and Theorem 4.1). Hence, the solution curve would first have to connect to either the curve \(u = c-1\) or \(u = 0\). But, as described in Proposition 5.2 (ii), the first of these has no bifurcation points for strong surface tension and \(c > 1\), and connection back to the bifurcation points of the second is excluded by the argument used in [16, Lemma 5.5] (no looping). Hence, \(\lim _{n \rightarrow \infty } c_n = \infty \).

*u*is even, and strictly increasing on the half-period \((-\pi ,0)\), in view of the positivity of the integrand in (5.2). According to our assumptions, there exists \(\delta > 0\) such that \(\frac{c_n}{2} - \max u_n \geqslant \delta c_n\), pick \(x_n \in (0,\pi )\) such that

*n*. \(\square \)

### 5.1 Discussion and Summary of Results

Analytically, we have determined almost completely^{5} the solution set near the lines of constant solutions \(u = 0\) and \(u = c -1\). The result depends crucially on the strength of surface tension *T*, and, apart from the easily seen change in the dispersion relation at \(T = \frac{1}{3}\), we have seen in Sect. 2 that there is a more subtle change at \(T = \frac{4}{\pi ^2}\), at which the integral kernel of the dispersive operator *L* loses its positivity and monotonicity; that has made it possible to prove some additional, but not complete, results for the case of (very) strong surface tension \(T \geqslant \frac{4}{\pi ^2}\). To complete the picture where our analytical methods have so far proved insufficient, we have additionally run a spectral bifurcation code similar to the one used in [34]: a Fourier-collocation scheme is employed to discretise and solve the equation, while a pseudo-arclength strategy allows us to follow the branch of solutions in the presence of turning points and other complex behaviours. In these computations, the wavelength \(2\pi \) has been used, that is, \(\kappa = 1\). We will present the main result of these calculations as well, but only in overview form.

### 5.2 The Case of Strong Surface Tension

The initial direction of the curves is calculated in Remark 3.7: analytically, sub-critical bifurcation is established for small enough values of *k*, and super-critical bifurcation as \(k \rightarrow \infty \); numerically, this shift happens at exactly one value, and we have illustrated this with the last visible (third) curve bending leftwards from the bifurcation point, while the two first bend rightwards (the direction after the Galilean shift is opposite).

The result of the global bifurcation theory as carried out in Theorem 3.6 is that each curve, when considered in a space of \(2\pi /k\)-periodic functions, is either unbounded in \(\mathcal {C}^s \times \mathbb {R}\), or returns (loops) back to \((u,c) = (0,c_k)\) in a finite period of the bifurcation parameter. The standard tool for ruling out looping is by preserving the unimodal nodal pattern along the main bifurcation branch, an argument for which one relies on maximum principles/positivity of the underlying operators. As we prove in Theorem 2.8 that this property is present when the surface tension coefficient satisfies \(T \geqslant \frac{4}{\pi ^2}\) (and only then),^{6} the complete monotonicity of the kernel *K* established in Theorem 2.8 for that case provides hope for stronger results. Note that, regardless of the exact value of \(T > 0\), it follows from Lemma 2.4 that all solutions of (2.1) are smooth, so that alternative (i) in Theorem 3.6 is equivalent, by bootstrapping in (2.3), to a sequence of solutions satisying \(|u|_\infty + |c| \rightarrow \infty \) along the bifurcation curve.

While we cannot rule out alternative (ii) in Theorem 3.1 completely, see Proposition 5.4, we can at least show that looping would require leaving every set of the form \(\max u < \lambda c\) for \(\lambda < \frac{1}{2}\) (that is the consequence of Proposition 5.4, as an unbounded continuous bifurcation curve cannot be finitely periodic). Although alternative (i) in Proposition 5.4 is very unlikely, and never appears in our numerical calculations, we have been unable to rule it out (the reason for this might be that the balance between *Mu* and \(u^2\) is exactly at the critical threshold for Gagliardo–Nirenberg, so that control of a higher Sobolev norm of *u* in terms of a lower seems to require using precise properties of the integral kernel.) We have illustrated this with long-dashed lines in Fig. 3, showing the curves (probably) leaving the cone \(\max u \leqslant \frac{c}{2}\).

*u*crosses \(\frac{c}{2}\). Interestingly enough, the same pattern seems to persist even when the kernel is not everywhere positive and monotone, that is, for \(T < \frac{4}{\pi ^2}\).

As a comparison, for the Euler equations—in the presence of interfacial waves or waves with surface tension—analytically all alternatives along a global bifurcation curve are open: waves could be steepening, looping, speeding, lengthening or develop surface or vorticity singularities [4]; for interfacial waves without surface tension, unboundedness in speed, slope or in the form of a surge is necessary [5]. There is an indirect proof, however, of connection between the trivial state and waves with infinite slopes/overhanging profiles [3] and even self-touching surface (so-called splash singularities) [11], in that the former are perturbations of Crapper waves, the Crapper family being a continuum from undisturbed water [31]. Numerical investigations have further shown that waves with infinite slopes can re-appear higher up along bifurcation branches [39]. As the model we are dealing with cannot capture multi-valued profiles, the increased steepening visible in the numerical calculations is probably the closest one can come. Interestingly, in [4], an alternative is that two different flat states connect in a way very much resemblant to our curves approaching the line \(u = c -1\).

Finally, for surface tension \(T \geqslant \frac{4}{\pi ^2}\), Proposition 5.2 shows that no solutions pass the line \(c=0\) with \(\max u \geqslant 0\), indicated by red in Fig. 3.

### 5.3 The Case of Weak Surface Tension

When the surface tension is weak, \(T < \frac{1}{3}\), several things are very different. First of all, the first single bifurcation points \(c_k\) might, depending on the period, appear in the interval \(0< c < 1\), although for large enough values of the wavenumber *k* the waves will all be supercritical. Just as in the case of strong surface tension, Proposition 5.2 guarantees that solutions do not cross the lines marked with red in Fig. 4 (although these now do not include the positive vertical axis \(\max u > 0\)), and there are no solutions in the grey area. Similarly, there are no small, non-constant, solutions in a neighbourhood of any point along the constant solution axes \(u = 0\) and \(u = c-1\), except at the countable bifurcation points.

A peculiarity in the case of weak surface tension is the appearance of multimodal waves connecting different curves of *k*-modal waves. Analytically, we find a full disk of solutions by two-dimensional bifurcation in Theorem 4.1 (i), by varying the wavelength. Fixing the fundamental period, however, this yields a one-dimensional subset of this disk, where we continuously transform via only a curve between two main modes of waves. Numerically, this effect persists even for values slightly off the exact points of two-dimensional bifurcation: as the numerical investigation [34] shows, the looping alternative (i) in the global one-dimensional Theorem 3.6 happens in the form of one bifurcation curve of *k*-modal waves transforming into one of *n*-modal waves and thereby connecting back to the line of zero states. The same kind of connections has been found for the Euler equations, analytically for small waves [37], and numerically for small and large waves [8, Figures 4 and 5] (see also [22, 40] for perturbation theory and numerical calculations showing the rippling and non-uniqueness of small waves). These branch-to-branch connections are illustrated in Fig. 4 by a curve of small bimodal waves connecting two curves of unimodal waves bifurcating off the 0-axis for \(c \in (0,1)\). (In numerical calculations for this manuscript, there have even been instances of curves of waves bridging, consecutively, three different unimodal bifurcation curves, that is, a nontrivial path that connects three separate bifurcation points, but that is not indicated in the graphics.)

The curves of subcritical waves can be followed, again numerically, past zero wave speed, going left-ward without any indication to stop. In \(L^2(\mathbb {S})\), they seem to flatten out to 0, but not in \(L^\infty \). This feature reappears again and again in both numerics and our calculations: while \(L^\infty \)-bounds easily yield bounds on higher norms, and one has control of solutions in \(L^2\) with respect to the wave speed, it is extremely difficult to relate the \(L^\infty \)-norm of solutions to their \(L^2(\mathbb {S})\)-norm, even when the wave speed is bounded. Generally, all curves of solutions appear to asymptotically approach one of the curve of constant solutions (\(u = 0\) or \(u = c-1\)) in \(L^2(\mathbb {S})\), while an actual connection in a space of higher regularity is impossible for almost all wavespeeds because of the invertibility of the linear operator \(D_u F\) (note that it is not obvious how to make sense of the nonlinear mapping *F* in \(L^2(\mathbb {S})\)).

Finally, in the case of supercritical bifurcation, we find only single-crested (bell-shaped) waves even though the surface tension is weak. When these waves are small, it is a result of Theorem 3.1. These curves may be continued globally (Theorem 3.6), but the information about them is purely numerical. Just as in the case of strong surface tension, these supercritical waves show no ripples, and they asymptotically approach \(u = c -1\) in \(L^2(\mathbb {S})\), but not in \(L^\infty \). Any proof of preservation of the nodal properties in the case of supercritical bifurcation when the surface tension is weak is for the moment entirely out of our reach, even though it would be very interesting to obtain.

## Footnotes

- 1.
See also the discussion in Sect. 5 concerning related results for the Euler equations.

- 2.
That is, \((A \bigtriangleup B)=\left( A\cap B^c\right) \cup \left( B\cap A^c\right) \).

- 3.
Throughout, we use the notation that \(\mathbb {N}_0:=\mathbb {N}\cup \{0\}\).

- 4.
Note that the function \(T_*(\cdot ;\cdot )\) can be extended to the cases \(n=0\) and \(k=0\) through continuity.

- 5.
We lack a proof of non-existence of the \(k_2\)-modal waves in the resonant case of Theorem 4.1, but these waves do not seem to exist numerically.

- 6.
It is possible that the periodised kernel is positive even when the original kernel is not, depending on the period, but we have not investigated that here.

## Notes

### Acknowledgements

The authors would like to acknowledge valuable input from the referees. Their comments helped improve both the exposition and the mathematical precision of the paper.

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