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Water Waves

, Volume 1, Issue 2, pp 275–313 | Cite as

On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

  • Mats EhrnströmEmail author
  • Mathew A. Johnson
  • Ola I. H. Maehlen
  • Filippo Remonato
Original Article

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

We consider periodic travelling wave solutions of the capillary–gravity Whitham equation
$$\begin{aligned} u_t+M_Tu_x+2uu_x=0, \end{aligned}$$
(1.1)
where \(M_T\) is a Fourier multiplier operator defined via its symbol \(m_T\) as
$$\begin{aligned} {\widehat{M_Tf}}(\xi )= m_T(\xi )\widehat{f}(\xi ) = \left( \frac{(1+T\xi ^2)\tanh (\xi )}{\xi }\right) ^{\frac{1}{2}}~\widehat{f}(\xi ), \end{aligned}$$
(1.2)
and the coefficient \(T>0\) denotes the strength of the surface tension. The symbol \(m_T\) arises as the linear dispersion relation for capillary–gravity water waves over a finite depth described by the Euler equations [29]. In the purely gravitational case, that is, when \(T=0\), the use of this symbol was proposed by Whitham as a way to generalise the KdV equation and remedy its strong dispersion [41]. Bifurcation in the gravitational setting has been investigated in [17, 18, 20]. We are here interested in completely characterising the local theory for travelling wave solutions of (1.1), and understanding their global extensions.

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]

A challenge and interesting feature of the capillary–gravity case is that weak surface tension allows for a non-monotone dispersion relation (see Fig.  1) and double eigenvalues of the corresponding linearised operator (in spaces of even functions). We handle this case in Sect. 4. To analytically capture the larger dimension of the space of solutions nearby the trivial ones, one requires an additional free parameter in addition to the wavespeed, used in the one-dimensional bifurcation. In line with [19] we choose to use the period as this extra parameter, while holding the surface tension fixed. The result, presented in Theorem 4.1, depends on the resonances between the two frequencies appearing in the nullspace: if one of the wavenumbers is a multiple of the other, one obtains a slit disk of solutions, excluding bifurcation straight in the direction of the higher wavenumber; if not, one obtains a full open disk of solutions, see Fig. 2. These results are in line with similar ones in [14, 33, 37] and include—when projecting the full disk onto a fixed period—a curve of bimodal rippled waves connecting waves of different wavenumbers (secondary bifurcation). This technique has later been used also in [2]. The existence of these interconnecting branches of waves has been corroborated numerically, showing persistence with respect to perturbations in the surface tension parameter [34]. The nonexistence of the pure higher mode in the resonant case of Theorem 4.1 (ii) has also been confirmed numerically in the same paper. More generally, Wilton ripples, as these kinds of waves are sometimes called, have earlier been found to exist for the Euler equations with surface tension [33, 37], and their spectral stability has been numerically investigated in [38]. They also exist in the presence of vorticity [30], even without capillarity [14, 19]. In that case, one may even construct arbitrary large kernels [1, 15], and corresponding multi-dimensional solution sets [28].
Fig. 1

Schematic drawings of the behaviour of the symbol \(l_T(\xi )\) for a weak surface tension \(0<T<1/3\) and for b strong surface tension \(T>1/3\). In both cases, the symbol is strictly positive and decays as \(|\xi |^{-1/2}\) as \(|\xi |\rightarrow \infty \)

Fig. 2

The local solution disks for the steady capillary–gravity Whitham equation (2.1) around a point where the bifurcation kernel is two-dimensional. The left-hand drawing depicts the situation in Theorem 4.1 (i), whereas the right-hand drawing refers to case (ii) of the same theorem. The blue and red colours represent the proximity of the solutions to the pure \(k_1\)- and \(k_2\)-modes, respectively. In particular, when \(k_1\) divides \(k_2\) we have not found any waves bifurcating in the direction of \(\cos (k_1 \cdot )\)

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.

Finally, we give in “Appendix A” some bifurcation formulas.
Fig. 3

A schematic drawing of the global bifurcation diagram in the case of strong surface tension \(T>\frac{1}{3}\) (partly \(T = \frac{4}{\pi ^2}\)). The diagram is discussed in detail in Section 5.2

2 Properties of the Convolution Kernel \(K_T\)

Travelling wave solutions of the form \(u(x-ct)\) satisfy the (profile) equation
$$\begin{aligned} -cu+M_Tu+u^2=0, \end{aligned}$$
(2.1)
where we have integrated once and used Galilean invariance to set the constant of integration to zero. Since \(m_T\) is strictly positive on \(\mathbb {R}\), the operator \(M_T\) is invertible (for example in any Fourier-based space) with inverse \(L_T\) defined via
$$\begin{aligned} {\widehat{L_Tf}}(\xi )=l_T(\xi )\widehat{f}(\xi ), \quad l_T(\xi )=(m_T(\xi ))^{-1}. \end{aligned}$$
(2.2)
In particular, the capillary–gravity Whitham equation (2.1) can be rewritten in the “smoothing” form
$$\begin{aligned} u-cL_T(u)+L_T(u^2)=0, \end{aligned}$$
(2.3)
where \(L_T = K_T*\) and \(K_T\) is the convolution kernel corresponding to the symbol \(l_T\). Note that the form (2.3) is resemblant of the Whitham equation itself, but with a nonlocal nonlinearity. By a solution of (2.1) [respectively (2.3)], we shall mean a real-valued, continuous and bounded functionuthat satisfies (2.1) [respectively (2.3)] everywhere.
In the rest of this work, we shall make heavy use of the properties of the convolution kernel \(K_T\) and its symbol. Our choice of Fourier transform is
$$\begin{aligned} \hat{f}(\xi )=\int _{\mathbb {R}}f(x)\mathrm{{e}}^{-ix\xi }\,\,{\mathrm {d}}x. \end{aligned}$$
To start, note that \(K_T=\mathcal {F}^{-1}l_T\) is smooth away from the origin with
$$\begin{aligned} \int _\mathbb {R}K_T(x) \,{\mathrm {d}}x = \lim _{\xi \rightarrow 0}l_T(\xi )=1, \end{aligned}$$
(2.4)
and
$$\begin{aligned} \lim _{x\rightarrow 0}K_T(x) = \frac{1}{2\pi }\int _{\mathbb {R}} l_T(\xi ) \,{\mathrm {d}}\xi = +\infty . \end{aligned}$$
Moreover, since \(l_T\) is analytic, \(K_T\) has rapid decay at \(\pm \infty \), whence \(K_T\in L^1(\mathbb {R})\) provided that the blow-up at \(x=0\) is not too fast. Later in this section, we will show that the singularity at the origin is of order \(|x |^{-\frac{1}{2}}\), with a second-leading term of somewhat smoother order, and that the convolution kernel is completely monotone for strong enough surface tension.

2.1 Analyticity of the Symbol

We start by studying the analytic extension of \(l_T\) to the complex plane; the results to come will be important to establish both the decay and the complete monotonicity of \(K_T\). Define the meromorphic function
$$\begin{aligned} \varrho _T(\zeta ) = \frac{\zeta }{(1+T\zeta ^2) \tanh (\zeta )}, \end{aligned}$$
(2.5)
with \(\zeta \) a complex number. We want to understand the complex extension \(\sqrt{\varrho _T}\) of \(l_T\), where \(\sqrt{\cdot }\) denotes the principal branch of the square root. Thus, we determine the pre-image \(\varrho _T^{-1}((-\infty ,0])\) and the set of singularities of \(\varrho _T\). As it turns out, the union of these (problematic) sets lie solely on the imaginary axis. To show this, we introduce the sets
$$\begin{aligned} Z_c&= \left\{ \pi \left( k - \textstyle \frac{1}{2}\right) :k \in \mathbb {Z}\right\} , \\ Z_s&= \left\{ \pi k :k \in \mathbb {Z}{\setminus }\{0\} \right\} , \\ Z_T&= \left\{ {\textstyle {-\frac{1}{\sqrt{T}}, \frac{1}{\sqrt{T}}}} \right\} , \end{aligned}$$
that is, the zeros of \(\cos (\zeta )\), \(\frac{\sin (\zeta )}{\zeta }\), and \(1-T\zeta ^2\), respectively. Finally, recall that the symmetric difference between two sets A and B is the set \(A \bigtriangleup B\) of elements either in A and not B, or contrariwise2
Fig. 4

A schematic drawing of the global bifurcation diagram in the case of weak surface tension \(T<\frac{1}{3}\). The diagram is discussed in detail in Sect. 5.3

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

By the infinite product formulas for \(\sinh \zeta \) and \(\cosh \zeta \) we obtain
$$\begin{aligned} \varrho _T(\zeta ) = \frac{1}{1+T\zeta ^2}\prod _{n=1}^\infty \frac{ 1 + \frac{\zeta ^2}{\pi ^2 \left( n - \frac{1}{2}\right) ^2} }{ 1 + \frac{\zeta ^2}{\pi ^2 n^2} }. \end{aligned}$$
(2.6)
The first part of the lemma now follows immediately, where the symmetric difference accounts for removable singularities should the term \((1+T\zeta ^2)\) coincide with a term of the form \(1 + \frac{\zeta ^2}{\pi ^2 (n - \frac{1}{2})^2}\). For the second part, we start by showing that \(\varrho _T\) is never negative away from the imaginary axis. As \(\varrho _T\) is symmetric about zero, we restrict our attention to \(\xi >0\). We have
$$\begin{aligned} {\mathrm {Re}}\Big [\cosh (\zeta )\overline{\sinh (\zeta )}\Big ]&= \frac{1}{2}\sinh (2\xi )>0, \\ {\mathrm {Re}}\Big [\zeta \,\overline{(1+T\zeta ^2)}\Big ]&= \xi + \xi T(\xi ^2+\eta ^2)>0, \end{aligned}$$
and consequently \(|\arg (\frac{\zeta }{1+T\zeta ^2})|,|\arg (\frac{1}{\tanh (\zeta )})|<\frac{\pi }{2}\). This in turn implies that \(|\arg (\varrho _T(\zeta ))|<\pi \), and so \(\varrho _T(\zeta )\) cannot be negative. Restricting our attention to the imaginary axis \((\zeta =i\eta )\) and away from zeroes and singularities, it is clear from (2.6) that \(\varrho _T(i\eta )\) is real valued and satisfies
$$\begin{aligned} \text {sgn}(\varrho _T(i\eta )) = \text {sgn}(1-T\eta ^2)\prod _{n=1}^\infty \text {sgn}\left( 1 - \frac{\eta ^2}{\pi ^2 \left( n - \frac{1}{2}\right) ^2}\right) \text {sgn}\left( 1 - \frac{\eta ^2}{\pi ^2 n^2}\right) . \end{aligned}$$
As \(\varrho _T(i\eta )\) is positive for \(\eta =0\), it is negative exactly when an odd number of factors in the expression above has swapped sign. This is equivalent to the last part of the lemma. \(\square \)

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

The symbol \(l_T\) extends analytically onto the strip \(\mathbb {R}\times i(-\delta ^*,\delta ^*)\), where
$$\begin{aligned} \delta ^* = {\left\{ \begin{array}{ll} \min \left\{ \frac{1}{\sqrt{T}}, \frac{\pi }{2} \right\} , \quad &{} T \ne 4/\pi ^2, \\ \pi \quad &{} T = 4/\pi ^2. \end{array}\right. } \end{aligned}$$

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

In this subsection we split \(K_T\) into three canonical parts and determine the precise regularity of these. We also record the rapid decay and smoothing properties of \(K_T\). Write
$$\begin{aligned} l_T = l_{-\frac{1}{2}} + l_{\frac{3}{2}} + l_\omega , \end{aligned}$$
with \(l_{-\frac{1}{2}}(\xi ) = \frac{1}{\sqrt{T |\xi |}}\), \(l_{\frac{3}{2}}(\xi ) = \sqrt{\frac{|\xi |}{1 + T \xi ^2}} - \frac{1}{\sqrt{T |\xi |}}\) and \(l_\omega (\xi ) = l_T(\xi ) - \sqrt{\frac{|\xi |}{1 + T \xi ^2}}\). The subscripts represent the regularity of each corresponding term of \(K_T\), as will be seen. The decay of \(l_{-\frac{1}{2}}(\xi ) \eqsim |\xi |^{-\frac{1}{2}}\) for \(|\xi |\gg 1\) is clear and, for any fixed \(T>0\), it is readily seen that
$$\begin{aligned} l_\frac{3}{2}(\xi ) \eqsim -|\xi |^{-\frac{5}{2}}, \end{aligned}$$
and
$$\begin{aligned} l_\omega (\xi ) = \sqrt{\frac{|\xi |}{1 + T \xi ^2}} \left( \sqrt{\coth (|\xi |)} - 1 \right) \eqsim |\xi |^{-\frac{1}{2}} \, \mathrm{{e}}^{-2|\xi |}, \end{aligned}$$
both for \(|\xi | \gg 1\).
To establish the regularity of the corresponding parts of \(K_T\) we shall use Zygmund spaces. Let \(\{\psi _j^2\}_{j = 0}^\infty \) be a partition of unity with \(\psi _0(\xi )\) supported in \(|\xi | \leqslant 1\), \(\psi _1(\xi )\) supported in \(\frac{1}{2} \leqslant |\xi | \leqslant 2\), and \(\psi _j(\xi ) = \psi _1(2^{1-j}\xi )\) for \(j \geqslant 2\). Then, the support of each \(\psi _j\) is concentrated around \(\xi \eqsim 2^j\). With \(D = -i\partial _x\), the Fourier multiplier operators \(\psi _j(D) :f \mapsto \mathcal {F}^{-1}(\psi _j \hat{f})\) characterise the Zygmund spaces: we say \(u \in \mathcal {C}^s(\mathbb {R})\) if
$$\begin{aligned} \Vert u \Vert _{\mathcal {C}^s(\mathbb {R})} = \sup _j \,2^{js}\, \Vert \psi _j^2(D) u \Vert _{L^\infty }, \end{aligned}$$
(2.7)
is finite. For non-integer values of \(s \geqslant 0\) the Zygmund spaces coincide with the standard (inhomogeneous) Hölder spaces,3
$$\begin{aligned} \mathcal {C}^s(\mathbb {R}) \cong C^s(\mathbb {R}), \quad s \in \mathbb {R}_+ {\setminus } \mathbb {N}_0, \end{aligned}$$
and one furthermore has the embedding \(C^k(\mathbb {R}) \hookrightarrow \mathcal {C}^k(\mathbb {R})\) for integer values of k. We refer the reader to [36, Section 13.8] and [21, Section 1.4] for further details.
Now, the symbols \(l_{-\frac{1}{2}}\), \(l_{\frac{3}{2}}\) and \(l_\omega \) all have well-defined Fourier transforms, and we let
$$\begin{aligned} K_{-\frac{1}{2}}(x)&= \mathcal {F}^{-1}(1/\sqrt{T |\cdot |})(x), \\ K_{\frac{3}{2}}(x)&= \mathcal {F}^{-1}(l_{\frac{3}{2}})(x), \\ K_\omega (x)&= \mathcal {F}^{-1}(l_\omega )(x), \end{aligned}$$
so that
$$\begin{aligned} K_T(x) = \mathcal {F}^{-1}(l_T)(x) = K_{-\frac{1}{2}}(x) + K_{\frac{3}{2}}(x) + K_\omega (x). \end{aligned}$$
From Fourier analysis, we know that \(\mathcal {F}^{-1}(1/\sqrt{ |\cdot |})(x)=1/\sqrt{ 2\pi |x |}\) and, additionally, that the exponential decay of \(l_\omega (\xi )\) for \(|\xi |\gg 1\) implies that \(K_\omega \) is real-analytic by Paley–Wiener’s first theorem [32]. The optimal regularity of \(K_{\frac{3}{2}}\) follows from the following theorem about the integral kernel \(K_T\).

Theorem 2.3

The integral kernel \(K_T\) may be written as
$$\begin{aligned} K_T(x) = \frac{1}{\sqrt{2\pi T|x|}} + K_{\frac{3}{2}}(x) + K_\omega (x), \end{aligned}$$
where the second term belongs to the optimal Hölder class \(C^\frac{3}{2}\) and the third is real-analytic. The singularity of \(K_T\) thus has the characterisation
$$\begin{aligned} \lim _{x \rightarrow 0} \sqrt{|x |} \, K_T(x) = \frac{1}{\sqrt{2\pi T}}. \end{aligned}$$
Moreover
$$\begin{aligned} |K_T(x)|\lesssim \mathrm{{e}}^{-\delta |x |} \quad \text { for } |x| > 1, \end{aligned}$$
with \(\delta < \delta ^*\) as given in Corollary 2.2. As a consequence, \(K_T\in L^1(\mathbb {R})\).

Proof

Most of the first claim was established in the preceding discussion, and only the regularity of \(K_{\frac{3}{2}}\) remains. Notice that \(l_{\frac{3}{2}}\) is always of negative sign, and thus so is the product \(\psi _j^2(\xi )l_{\frac{3}{2}}(\xi )\). This means
$$\begin{aligned} \left\| \psi _j^2(D) K_{\frac{3}{2}} \right\| _{L^\infty }=\left\| \psi _j^2(\xi ) l_{\frac{3}{2}} \right\| _{L^1}. \end{aligned}$$
Further, we exploit the decay of \(l_{\frac{3}{2}}\) and the compact support of \(\psi ^2_j\), to obtain
$$\begin{aligned} \left\| \psi _j^2(\xi ) l_{\frac{3}{2}} \right\| _{L^1}\eqsim \int _{2^{j-1}}^{2^{j+1}}|\xi |^{-\frac{5}{2}} \,{\mathrm {d}}\xi \eqsim 2^{-\frac{3}{2}j}. \end{aligned}$$
Combining these two equations, we conclude in view of (2.7) and the equivalence between Hölder and Zygmund norms for non-integer indices that \(K_{\frac{3}{2}}\) lies in the optimal Hölder class \(C^{\frac{3}{2}}(\mathbb {R})\). As for the decay rate of \(K_T\), we instead prove this estimate for the more regular expression \(x\mapsto xK_T(x)\), which again proves it for \(K_T\). The exponential decay of \(x\mapsto xK_T(x)\) is a direct consequence of Corollary 2.2 and the discussion thereafter combined with Paley–Wiener theory (see, for example, [32, Theorem IV]). One can obtain further asymptotic estimates as in [20, Prop. 2.1 and Cor. 2.26]. \(\square \)

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

Consider \(T>0\) fixed. We want to show that the inequality
$$\begin{aligned} \Vert \psi ^2_j(D)L_Tu\Vert _{L^\infty }\lesssim 2^{-\frac{j}{2}}\Vert \psi ^2_j(D)u\Vert _{L^\infty }, \end{aligned}$$
(2.8)
is valid for all \(j\in \mathbb {N}_0\). We prove this estimate for \(j\geqslant 1\); the case \(j=0\) must be done separately, but the calculation is similar to what follows and so we exclude it. Pick a smooth function \(\varphi \) supported in \(\frac{1}{3}\leqslant |\xi |\leqslant 3\) satisfying \(\varphi (\xi )=1\) whenever \(\frac{1}{2}\leqslant |\xi |\leqslant 2\). For \(j\geqslant 1\), we define \(\varphi _j(\xi )=\varphi (2^{1-j}\xi )\), and observe that \(\varphi _j\psi _j^2=\psi _j^2\). Exploiting this relationship, we deduce
$$\begin{aligned} \Vert \psi ^2_j(D)L_Tu\Vert _{L^\infty }&=\Vert \mathcal {F}(\psi ^2_j(\xi )l_T\hat{u})\,\Vert _{L^\infty }\\&=\Vert \mathcal {F}(\varphi _j\psi ^2_j(\xi )l_T\hat{u}) \Vert _{L^\infty }\\&=\Vert \mathcal {F}(l_T\varphi _j) *(\psi ^2_j(D)u)\Vert _{L^\infty }\\&\leqslant \Vert \mathcal {F} (l_T\varphi _j) \Vert _{L^1}\Vert \psi ^2_j(D)u\Vert _{L^\infty }, \end{aligned}$$
where we have used Young’s inequality for convolution. The proof will be complete if we can establish \(\Vert \mathcal {F}(l_T\varphi _j)\Vert _{L^1}\lesssim 2^{-\frac{j}{2}}\); we do this by splitting the integral, \(\Vert \cdot \Vert _{L^1}=\Vert \cdot \Vert _{L^1(|x|\leqslant 2^{-j})}+\Vert \cdot \Vert _{L^1(|x|>2^{-j})}\), and then prove the bound for each part separately. From the general fact \(\Vert f\Vert _{L^p}\leqslant |\text {supp}(f)|^{\frac{1}{p}}\Vert f\Vert _{L^\infty }\), we deduce two important inequalities for the calculations to come
$$\begin{aligned} \Vert l_T\varphi _j\Vert _{L^1}\lesssim 2^{\frac{j}{2}}, \quad \Vert (l_T\varphi _j)'\Vert _{L^2}\lesssim 2^{-j}. \end{aligned}$$
These follow from the bounds \(|l_T(\xi )|\lesssim |\xi |^{-\frac{1}{2}}\), \(|l_T'(\xi )|\lesssim |\xi |^{-\frac{3}{2}}\) and \((\varphi _j)'\eqsim 2^{-j}(\varphi ')_j\), and the observation that the support of \(\varphi _j\) (and \(\varphi _j'\)) is of size \(2^j\) and located about \(|\xi |\eqsim 2^j\). We now conclude the proof with the two calculations promised above; the first is straight forward
$$\begin{aligned} \Vert \mathcal {F} (l_T\varphi _j)\Vert _{L^1(|x|\leqslant 2^{-j})}\lesssim 2^{-j}\Vert \mathcal {F}(l_T\varphi _j)\Vert _{L^\infty } \leqslant 2^{-j}\Vert l_T\varphi _j\Vert _{L^1} \lesssim 2^{-\frac{j}{2}}. \end{aligned}$$
For the second, we use basic Fourier analysis, the Cauchy–Schwarz inequality, and the Plancherel theorem:
$$\begin{aligned} \Vert \mathcal {F}(l_T\varphi _j)\Vert _{L^1(|x|>2^{-j})}&= \Vert \tfrac{1}{x} \mathcal {F}((l_T\varphi _j)')\Vert _{L^1(|x|>2^{-j})}\\&\leqslant \Vert \tfrac{1}{x}\Vert _{L^2(|x|>2^{-j})}\Vert \mathcal {F}((l_T\varphi _j)')\Vert _{L^2(|x|>2^{-j})}\\&\lesssim 2^{\frac{j}{2}}\Vert (l_T\varphi _j)'\Vert _{L^2}\\&\lesssim 2^{-\frac{j}{2}}, \end{aligned}$$
and so we have established (2.8). It is immediate that \(L_T\) maps \(\mathcal {C}^s(\mathbb {R})\) to \(\mathcal {C}^{s+\frac{1}{2}}(\mathbb {R})\) continuously, and combining this with the compact embedding \(\mathcal {C}^{s+\frac{1}{2}}(\mathbb {S})\hookrightarrow \mathcal {C}^s(\mathbb {S})\) we get the full result. \(\square \)

2.3 Montonicity and Complete Monotonicity

We conclude this section by showing that \(K_T\) is completely monotone for sufficiently large 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
$$\begin{aligned} (-1)^n g^{(n)}(\lambda ) \geqslant 0 \end{aligned}$$
for \(n = 0,1,2,\ldots \) and all \(\lambda > 0\). If it can furthermore be written in the form
$$\begin{aligned} g(\lambda ) = \frac{a}{\lambda } + b + \int _{(0, \infty )} \frac{1}{\lambda + t} \,{\mathrm {d}}\sigma (t) \end{aligned}$$
for some constants \(a,b>0\), with \(\sigma \) a Borel measure satisfying \(\int _{(0, \infty )} \frac{1}{1+t} \,{\mathrm {d}}\sigma (t) < \infty \), then it is called 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

The following statements are true.
  1. (i)

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

     
  2. (ii)

    [Schur’s Theorem] A countable product of positive definite functions is positive definite.

     
  3. (iii)

    If \(f:\mathbb {R}\rightarrow \mathbb {C}\) is positive definite, then the global maximum of f occurs at \(x=0\).

     
  4. (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

We first prove that \(K_T\) is completely monotone for \(T\geqslant \frac{4}{\pi ^2}\). By Lemmas 2.5 and 2.6 and the discussion thereafter, we conclude that \(K_T\) is completely monotone exactly if \({\mathrm {Im}}(\zeta )\cdot {\mathrm {Im}}\sqrt{\varrho _T(\sqrt{\zeta })} \leqslant 0\) for \(\zeta \in \mathbb {C}{\setminus } (-\infty , 0]\). This property is satisfied for \(\sqrt{\varrho _T(\sqrt{\cdot })}\) if and only if it is satisfied for \(\varrho _T(\sqrt{\cdot })\), as the latter function maps \(\mathbb {C}{\setminus } (-\infty , 0]\) to itself (Lemma 2.1). Moving the first factor of \(\cosh \zeta \) out of the infinite product in (2.6), we obtain
$$\begin{aligned} \varrho _T(\xi )=\frac{1+\frac{4}{\pi ^2}\xi ^2}{1+T\xi ^2}\prod _{n=1}^\infty \frac{ 1 + \frac{\xi ^2}{\pi ^2 \left( n + \frac{1}{2}\right) ^2} }{ 1 + \frac{\xi ^2}{\pi ^2 n^2} }. \end{aligned}$$
(2.9)
Substituting \(\xi \mapsto \sqrt{\zeta }\) in (2.9), and taking the complex argument of both sides, we obtain
$$\begin{aligned} \begin{aligned} \arg \left( \varrho _T\left( \sqrt{\zeta }\right) \right) =&\left[ \arg \left( 1+\frac{4}{\pi ^2}\zeta \right) -\arg (1+T\zeta )\right] \\&+\sum _{n=1}^\infty \left[ \arg \left( 1 + \frac{\zeta }{\pi ^2 (n + \frac{1}{2})^2}\right) -\arg \left( 1 + \frac{\zeta }{\pi ^2 n^2}\right) \right] . \end{aligned} \end{aligned}$$
(2.10)
This equation is valid whenever the right hand side takes values in \((-\pi ,\pi )\), which in turn is always true when \(\zeta \in \mathbb {C}{\setminus }(-\infty ,0]\) as the RHS is continuous in \(\zeta \), zero for \(\zeta >0\) and prevented from taking the values \(\pm \pi \) as \(\varrho _T(\sqrt{\zeta })\) is non-negative (Lemma 2.1). Moreover, when Im\((\zeta ) > 0\), it is easily seen that \(t\mapsto \arg (1+t\zeta )\) is strictly increasing in \(t\in \mathbb {R}\), and so each square bracket in (2.10) is negative (the first non-positive), further implying \({\mathrm {Im}}(\zeta )\cdot {\mathrm {Im}}\sqrt{\varrho _T(\sqrt{\zeta })} < 0\). After a similar argument for Im\((\zeta )<0\), we obtain the desired conclusion.
For the second part of the theorem, we observe that by Bochner’s Theorem in Lemma 2.7(i), \(K_T\) is non-negative if and only if its Fourier transform \(l_T\) is a positive definite function; we now prove that the latter statement is false when \(0<T<\frac{4}{\pi ^2}\). Note first that for \(0<T<\frac{1}{3}\), this follows immediately from Lemma 2.7(iii) as \(l_T\) does not have a global maximum at \(\xi =0\) (see Fig. 1). Suppose instead that \(\frac{1}{3}\leqslant T<\frac{4}{\pi ^2}\). If \(l_T\) is positive definite, then Lemma 2.7(ii) implies the same would be true for its square \(\xi \mapsto \varrho _T(\xi )\). To this end, we write (2.9) as
$$\begin{aligned} \varrho _T(\xi )=\frac{1+\frac{4}{\pi ^2}\xi ^2}{1+T\xi ^2}~\varphi (\xi ), \end{aligned}$$
which, after introducing the positive constants \(\alpha =4/(T\pi ^2)\) and \(\beta =\alpha -1\), can be further rewritten as
$$\begin{aligned} \varrho _T(\xi )=\left( \alpha -\frac{\beta }{1+T\xi ^2}\right) \varphi (\xi )=:\alpha \varphi (\xi )-\beta \psi (\xi ). \end{aligned}$$
By Lemma 2.7, both \(\varphi \) and \(\psi \) are positive definite as they are (countable) products of positive definite functions, and thus \(\hat{\varphi },\hat{\psi }\geqslant 0\) by Bochner’s Theorem. Note that \(\varphi \) has a complex analytic extension to the strip \(\mathbb {R}\times i(-\pi ,\pi )\), while \(\psi \) cannot be extended to a larger strip than \(\mathbb {R}\times i(\frac{-1}{\sqrt{T}},\frac{1}{\sqrt{T}})\). Since \(\frac{1}{\sqrt{T}}\leqslant \sqrt{3}<\pi \), we can pick some \(\gamma \in (\frac{1}{\sqrt{T}},\pi )\) and use Paley–Wiener theory [32] and Cauchy–Schwarz to conclude that
$$\begin{aligned} 0<\int _{\mathbb {R}}\widehat{\varphi }(x)\mathrm{{e}}^{\gamma |x|}\,\,{\mathrm {d}}x<\infty \quad {\mathrm{and}}\quad \int _{\mathbb {R}}\widehat{\psi }(x)\mathrm{{e}}^{\gamma |x|}\,\,{\mathrm {d}}x=+\infty , \end{aligned}$$
which further implies
$$\begin{aligned} \int _{\mathbb {R}}{\widehat{\varrho _T}}(x)\mathrm{{e}}^{\gamma |x|}\,\,{\mathrm {d}}x=-\infty . \end{aligned}$$
By Bochner’s Theorem, \(\xi \mapsto \varrho _T(\xi )\) is not positive definite, and so neither is \(l_T\), which concludes the proof. \(\square \)

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

Since \(K\in L^1(\mathbb {R})\), it may be periodised to an arbitrary period. In particular, given a \(2\pi \)-periodic \(f \in L^\infty (\mathbb {R})\) we can define the action of \(L_T=K_T*\) on f through a convolution of f with a \(2\pi \)-periodic kernel \(K_p\) over a single period:
$$\begin{aligned} L_Tf(x) = \int _\mathbb {R}K_T(x-y) f(y) \,{\mathrm {d}}y&= \int _{-\pi }^{\pi } \left( \sum _{k\in \mathbb {Z}} K_T(x-y+2k\pi ) \right) f(y) \,{\mathrm {d}}y \\&=: \int _{-\pi }^{\pi } K_p(x-y) f(y) \,{\mathrm {d}}y. \end{aligned}$$
Clearly \(K_p\) is even, strictly positive on \(\mathbb {R}\) and satisfies \(\Vert K_p\Vert _{L^1(-\pi ,\pi )}=1\). Further, by Theorem 2.3 we know that \(K_p\) is smooth on \(\mathbb {R}{\setminus } 2\pi \mathbb {Z}\), and that for \(T>\frac{4}{\pi ^2}\) it follows by Theorem 2.8 and [20, Proposition 3.2] that \(K_p\) is completely monotone function on the half period \((0,\pi )\). To find nontrivial solutions of the Eq. (2.1), or, equivalently, of (2.3), we fix \(s>1/2\) and define a map \(F :\mathcal {C}^s_{\mathrm {even}}(\mathbb {S})\times \mathbb {R}\rightarrow \mathcal {C}^s_{\mathrm {even}}(\mathbb {S})\) via
$$\begin{aligned} F(u,c)=u-cL_T(u)+L_T(u^2), \end{aligned}$$
(3.1)
where \(\mathcal {C}^s_{\mathrm {even}}(\mathbb {S})\) is the subspace of even functions in \(\mathcal {C}^s(\mathbb {S})\). Note this map is well defined since \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\) is a Banach algebra for any \(s>0\). Then, the roots of 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].
Now, we begin with the observation that \(F(0,c)=0\) for all \(c\in \mathbb {R}\) and that the linearised operator
$$\begin{aligned} D_u F[0,c]={\mathrm {Id}}-cL_T, \end{aligned}$$
has a nontrivial kernel in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\) if and only if \(c \, l_T(k) = 1\) for some \(k\in \mathbb {N}_0\) (we intentionally include the case \(k=0\) as it will play a role in the two-dimensional bifurcation to come). Consequently, for a fixed \(c\in \mathbb {R}\) we have
$$\begin{aligned} \ker D_u F[0,c] ={\mathrm {span}}\left\{ \cos (kx) :k\in \mathbb {N}_0 \text { such that } cl_T(k)=1 \right\} , \end{aligned}$$
(3.2)
and hence the multiplicity of the kernel depends sensitively on the graph of the function \(l_T(\xi )\). In particular, if \(T>1/3\) then \(l_T(\xi )\) is monotone decreasing on \(\mathbb {R}_+\) and hence the above kernel is simple: see Fig. 1. If \(0<T<1/3\), however, the function \(l_T\) has exactly one local extremum (a maximum) in the interior of \(\mathbb {R}_+\), whence opening the possibility of two different positive integers for which \(l_T(m) = l_T(k)\): again, see Fig. 1. A simple calculation shows that for a fixed \(k\in \mathbb {N}_0\), the kernel will be simple if and only if \(T\notin \{T_*(n;k)\}_{n\in \mathbb {N}_0}\), where4
$$\begin{aligned} T_*(n;k):=\frac{n\tanh (k)-k\tanh (n)}{kn\left( n\tanh (n)-k\tanh (k)\right) }, \end{aligned}$$
while it will have multiplicity exactly two when \(T = T_*(n;k)\) for some \(n \in \mathbb {N}_0\).

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

To investigate the bifurcations, we will make use in the following sections of three positive quantities—the wavespeed 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
$$\begin{aligned} {\widehat{L_{\kappa ,T}}}(\xi ) = l_{\kappa ,T}(\xi ) := l_T (\kappa \xi ). \end{aligned}$$
(3.3)
This operator agrees with the original one for \(\kappa = 1\). In particular, finding \(2\pi \)-periodic solutions of (2.1) with symbol \(L_{\kappa ,T}\) is equivalent to finding \(2\pi /\kappa \)-periodic solutions of (2.1) with symbol \(L_T=L_{1,T}\). This allows us to treat different wavelengths in the same equation by moving the wavelength parameter to \(L_{\kappa ,T}\). Of course, the family of operators \(L_{\kappa ,T}\) all enjoy the embedding properties of Lemma 2.4, as the proof is identical for an arbitrary, fixed, \(\kappa >0\). In what follows, we will thus modify (3.1) and seek non-trivial solutions of the map
$$\begin{aligned} F_\kappa (u,c)=u-cL_{\kappa ,T}(u)+L_{\kappa ,T}\left( u^2\right) \end{aligned}$$
(3.4)
in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\) for a fixed \(\kappa >0\).

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

Let \(k\in \mathbb {N}\) and set \(c_0 = l_{\kappa ,T}(k)^{-1}\). For any \(T, \kappa >0\) such that \({\mathrm {dim}} \ker D_u F_\kappa (0, c_0) = 1\) there exists a smooth curve
$$\begin{aligned} \{ \left( u(t), c(t) \right) :0 < |t|\ll 1 \} \end{aligned}$$
of small-amplitude, \(2\pi \)-periodic even solutions of the steady capillary–gravity Whitham equation (2.1) with symbol given by (3.3). These solutions satisfy
$$\begin{aligned} u(t)&= t\cos (kx) + \mathcal {O}(t^2) \\ c(t)&= c_0 + \mathcal {O}(t). \end{aligned}$$
in \(\mathcal {C}^s_{\mathrm {even}}(\mathbb {S}) \times \mathbb {R}\), and constitute all nontrivial solutions in a neighbourhood of \((0,c_0)\) in that space.

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

By considering the role of \(\kappa \) in the proof of Theorem 3.1 one can see that by varying \(\kappa \) one obtains a one-dimensional family of solution curves, the starting points of which depend smoothly on \(\kappa \). This may be seen also by applying the implicit function theorem directly to 3.1. For each \(k \in \mathbb {N}\) we thus obtain a two-dimensional sheet of solutions,
$$\begin{aligned} \mathcal {S}^k = \left\{ (u(t,\kappa ), c(t,\kappa ), \kappa ) :0 < |t| \ll 1, |\kappa - \kappa _0| \ll 1\right\} \end{aligned}$$
(3.5)
parameterised by \((t,\kappa )\) in a neighbourhood of a bifurcation point \((0,\kappa _0)\).

Proof

As stated above, we suppress the dependence on the fixed parameters T and \(\kappa \) throughout. According to the assumptions and the discussion after (3.2), on \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\) we have
$$\begin{aligned} {{\,\mathrm{ker}\,}}D_u F(0,c_0) = \ker ({\mathrm {Id}} - c_0 L) = {{\,\mathrm{span}\,}}\{\cos (k \cdot )\}. \end{aligned}$$
We first write
$$\begin{aligned} u(t)&= t \cos (k x) + v(t), \\ c(t)&= c_0 + r(t), \end{aligned}$$
with \(v(t) \in \mathcal {C}^s_{\mathrm {even}}(\mathbb {S})\) such that \(\int _{-\pi }^\pi \cos (k x) v \,{\mathrm {d}}x = 0\) and \(r(t) \in \mathbb {R}\), and proceed to show the existence of v and r such that for \(|t|\ll 1\) we have
$$\begin{aligned} F(t \cos (k x) + v(t), c_0 + r(t)) = 0. \end{aligned}$$
(3.6)
As a subspace of \(L^2(\mathbb {S})\), we equip \(\mathcal {C}^s_{\mathrm {even}}(\mathbb {S})\) with the \(L^2\) inner product \(\langle f, g \rangle = \frac{1}{\pi } \int _{-\pi }^{\pi } fg \, \,{\mathrm {d}}x\) and let \(\Pi :\mathcal {C}^s_{\mathrm {even}}(\mathbb {S}) \rightarrow {{\,\mathrm{ker}\,}}D_u F(0,c_0)\) be the projection onto \({{\,\mathrm{span}\,}}\{\cos (k\cdot )\}\) parallel to \({{\,\mathrm{ran}\,}}(D_u F(0,c_0))\). Since \(D_u F(0,c_0)\) is a symmetric Fredholm operator with index 0 by Corollary 3.5 below, it follows that \(\mathcal {C}^s_{\mathrm {even}}(\mathbb {S})\) may be decomposed as a direct sum between its kernel and range. In particular, (3.6) is equivalent to the system of equations
$$\begin{aligned} \begin{aligned} \Pi F(t \cos (k x) + v, c_0 + r)&= 0, \\ (I-\Pi ) F(t \cos (k x) + v, c_0 + r)&= 0, \end{aligned} \end{aligned}$$
(3.7)
where we have suppressed the t-dependence in v and r. Noting that
$$\begin{aligned}&F(t \cos (k x) + v, c_0 + r) \\&\quad = t \cos (k x) + v - (c_0+r)L (t \cos (k x)+v) + L (t \cos (k x) + v)^2 \\&\quad = D_u F(0,c_0) (v + t\cos (k x))\\&\qquad -\, r L (t \cos (k x) + v) + L (t \cos (k x) + v)^2, \end{aligned}$$
and that \(\cos (k\cdot )\) is in the kernel of \(D_u F(0,c_0)\), the Eq. (3.6) may be rewritten as
$$\begin{aligned} D_uF(0,c_0) v = r L (t \cos (k x) + v) - L (t \cos (k x) + v)^2 =: g(t, r, v) \end{aligned}$$
(3.8)
and hence, recalling that \(v\in (1-\Pi )\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\), (3.7) is equivalent to the system
$$\begin{aligned} \begin{aligned} 0&= \Pi g(t, r, v) \\ D_uF(0,c_0) v&= ({\mathrm {Id}} - \Pi )g(t, r, v). \end{aligned} \end{aligned}$$
(3.9)
Finally, observe that since \(D_uF(0,c_0)\) is invertible on \((I-\Pi ) \mathcal {C}^s_{\mathrm {even}}(\mathbb {S})\), the second equation in (3.9) can be rewritten as
$$\begin{aligned} v = [D_uF(0,c_0)]^{-1} ({\mathrm {Id}} -\Pi ) g(t, r, v). \end{aligned}$$
Concerning this latter equation, note that at \((t,r) = (0,0)\) we have both that \(v=0\) is a solution and that the Fréchet derivative with respect to 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 \(t=0\).
We now need to solve the equation
$$\begin{aligned} \Pi g(t, r, v(t, r)) = Q(r,t) \cos (kx) = 0 \end{aligned}$$
for r, with
$$\begin{aligned} Q(t,r) := \langle g(t, r, v(t, r)) , \cos (k \cdot ) \rangle . \end{aligned}$$
Notice that \(Q(0,r)=0\) since \(v(0,r)=0\) for all r, which together with the symmetry of L implies that we can write
$$\begin{aligned} Q(t, r) = t \left[ r \, l(k) + R(t,r)\right] , \end{aligned}$$
where 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
$$\begin{aligned} Q(t, r(t))) = t ( r(t)\, l(k) + \tilde{R}(t, r(t))) = 0 \end{aligned}$$
for all \(|t|\ll 1\). This concludes the proof. \(\square \)

3.3 Global Bifurcation (Analytic)

We now extend the local bifurcation curves from Sect. 3.2 to global ones by the means of the analytic bifurcation theory pioneered by Dancer [12, 13] and then developed further by Buffoni and Toland [10]. For fixed \(s>1/2\), we define \(N :\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\rightarrow \mathcal {C}^{s+1/2}_{\mathrm{even}}(\mathbb {S})\) by
$$\begin{aligned} N(u,c) = L(cu-u^2). \end{aligned}$$
Fixed points of N are solutions of the steady capillary–gravity Whitham equation (2.1), and conversely. Let
$$\begin{aligned} S =\left\{ (u,c)\in \mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}:F(u,c)=0\right\} \end{aligned}$$
be the set of solutions (fixed points of 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

Let \(K\subset S\subset \mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\) be closed and bounded, and pick a sequence \((u_j, c_j)_j \subset K\). Since \(\{c \in \mathbb {R}:(u,c)\in K\}\) is a closed and bounded subset of \(\mathbb {R}\), it is compact. This means that \((c_j)_j\) has a convergent subsequence, name it \((c_k)_k\). As the map
$$\begin{aligned} \mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\ni (u,c)\mapsto cu-u^2\in \mathcal {C}_{\mathrm{even}}^s(\mathbb {S}) \end{aligned}$$
is continuous for \(s>1/2\), and since the map 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

This follows immediately from Lemma 3.4 as then
$$\begin{aligned} D_u F(u,c) = {\mathrm {Id}}-L(c - 2 u) \end{aligned}$$
is a compact perturbation of the identity. \(\square \)

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

Whenever
$$\begin{aligned} \ddot{c}(0) = \frac{3c_0 l(2k) - l(2k) - 2}{(c_0-1) (c_0 l(2k) - 1)} \end{aligned}$$
(3.10)
is finite and non-vanishing the local bifurcation curve \(t \mapsto \left( u(t), c(t) \right) \), \(|t| \ll 1\), from Theorem 3.1 extends to a continuous and locally analytically re-parameterisable curve in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\) defined for all \(t \in [0, \infty )\). One of the following alternatives holds:
  1. (i)

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

     
  2. (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

We note that
$$\begin{aligned} \ddot{c}(0;k)=\left\{ \begin{aligned}&\frac{10}{(3T-1)(\kappa k)^2}+\mathcal {O}(1)\qquad \quad \qquad \qquad {\mathrm{for}}~|k|\ll 1\\&-(\sqrt{2}-1)(T\kappa k)^{-1/2}+\mathcal {O}\left( k^{-1}\right) \quad {\mathrm{for}} ~k\gg 1. \end{aligned}\right. \end{aligned}$$
For \(T>1/3\) it follows that \((0,c_0)\) undergoes a supercritical pitchfork bifurcation for small 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

We now focus our attention on the case of a two-dimensional bifurcation kernel in \(\mathcal {C}^s_{\text {even}}(\mathbb {S})\). To enable the necessary two degrees of freedom, we shall make use of the wavelength \(\kappa \) in addition to the wavespeed c, while the surface tension T is assumed to be fixed. We shall, therefore, study for \(\kappa >0\) the operator
$$\begin{aligned} F_{\kappa }(u,c) = u + L_{\kappa }(u^2 - c u) \end{aligned}$$
on \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\), along with its linearisation
$$\begin{aligned} \mathcal {L}= D_u F_{\kappa _0}(0, c_0) = {\mathrm {Id}} - c_0 L_{\kappa _0}, \end{aligned}$$
assuming that \(T, \kappa _0, c_0 > 0\) are constants such that
$$\begin{aligned} \ker (\mathcal {L}) = {\mathrm {span}}\{\cos (k_1 \cdot ), \cos (k_2 \cdot )\}, \end{aligned}$$
(4.1)
which happens when \(\kappa _0, c_0 > 0\) and \(k_1, k_2 \in \mathbb {N}_0\), \(k_1 \ne k_2\), are such that
$$\begin{aligned} c_0 = l_{\kappa _0}(k_1)^{-1} = l_{\kappa _0}(k_2)^{-1}, \end{aligned}$$
as described at the start of Sect. 3 (we suppress the dependence on 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

Let \(T>0\) be fixed and assume that (4.1) holds for some distinct \(k_1,k_2\in \mathbb {N}_0\).
  1. (i)
    When \(k_1\) does not divide \(k_2\) there is a full, smooth, sheet
    $$\begin{aligned} \mathcal {S}^{{mixed}} = \{ \left( u(t_1, t_2), c(t_1, t_2), \kappa (t_1, t_2)\right) :0 < |(t_1, t_2)|\ll 1 \} \end{aligned}$$
    of solutions in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\times \mathbb {R}_+\) of the form
    $$\begin{aligned} u(t_1, t_2)&= t_1 \cos (k_1 x) + t_2 \cos (k_2 x) + \mathcal {O}(|(t_1, t_2)|^2), \\ c(t_1, t_2)&= c_0 + \mathcal {O}((t_1, t_2)), \\ \kappa (t_1, t_2)&= \kappa _0 + \mathcal {O}((t_1, t_2)), \end{aligned}$$
    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 \((0, c_0, \kappa _0)\).
     
  2. (ii)
    When \(k_1\) divides \(k_2\) there exists for any \(\delta > 0\) a small but positive \(\varepsilon _\delta \) and a slit, smooth, sheet
    $$\begin{aligned} \mathcal {S}^{{mixed}}_\delta = \{ \left( u(\varrho , \vartheta ), c(\varrho , \vartheta ), \kappa (\varrho , \vartheta )\right) :0<\varrho<\varepsilon _\delta ,\, \delta< |\vartheta | < \pi - \delta \} \end{aligned}$$
    of solutions in \(\mathcal {C}^s_{\mathrm{even}}(\mathbb {S})\times \mathbb {R}\times \mathbb {R}_+\) of the form
    $$\begin{aligned} u(\varrho , \vartheta )&= \varrho \cos (\vartheta ) \cos (k_1 x) + \varrho \sin (\vartheta ) \cos (k_2 x) + \mathcal {O}(\varrho ^2), \\ c(\varrho , \vartheta )&= c_0 + \mathcal {O}(\varrho ), \\ \kappa (\varrho , \vartheta )&= \kappa _0 + \mathcal {O}(\varrho ). \end{aligned}$$
    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 \(\delta< |\vartheta | < \pi -\delta \).
     

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

We start by writing
$$\begin{aligned} u(t_1, t_2)&= t_1 \cos (k_1 x) + t_2 \cos (k_2 x) + v, \\ c(t_1, t_2)&= c_0 + r, \\ \kappa (t_1, t_2)&= \kappa _0 + p, \end{aligned}$$
where, generically, we want to find v, r and p parameterised by \((t_1, t_2)\) such that
$$\begin{aligned} F_{\kappa _0 + p}(t_1 \cos (k_1 x) + t_2 \cos (k_2 x) + v, c_0 + r) = 0, \end{aligned}$$
(4.2)
for sufficiently small values of \((t_1, t_2)\). As in the proof of Theorem 3.1, we let \(\Pi :C_{\mathrm {even}}^s(\mathbb {S}) \rightarrow \ker (D_u F_{\kappa _0}(0,c_0))\) be the projection onto \(\ker (D_u F_{\kappa _0}(0,c_0))\) parallel to \({{\,\mathrm{ran}\,}}(D_u F_{\kappa _0}(0,c_0))\), where we have equipped \(C_{\mathrm {even}}^s(\mathbb {S})\) with the \(L^2\) inner product \(\langle f, g \rangle = \frac{1}{\pi } \int _{-\pi }^{\pi } fg \, \,{\mathrm {d}}x\). According to Corollary 3.5, Eq. (4.2) is then equivalent to
$$\begin{aligned} {\left\{ \begin{array}{ll} \Pi F_{\kappa (t_1, t_2)}\left( u(t_1, t_2), c(t_1, t_2)\right) = 0 \\ ({\mathrm {Id}}-\Pi )F_{\kappa (t_1, t_2)}\left( u(t_1, t_2), c(t_1, t_2)\right) = 0. \end{array}\right. } \end{aligned}$$
(4.3)
Note that under the above ansatz, where it is assumed that \(\Pi v = 0\),
$$\begin{aligned} F_\kappa \left( u, c\right) =&t_1\cos (k_1 x) + t_2\cos (k_2 x) + v\\&+ L_{\kappa _0+p} \left[ (t_1\cos (k_1 x) + t_2\cos (k_2 x) + v)^2 \right. \\&\left. - (c_0 + r)\,(t_1\cos (k_1 x) + t_2\cos (k_2 x) + v) \right] \\ =&(v - c_0 L_{\kappa _0+p}v) + t_1 \left( \cos (k_1 x) - c_0 L_{\kappa _0+p} \cos (k_1 x) \right) \\&+ t_2 \left( \cos (k_2 x) - c_0 L_{\kappa _0+p} \cos (k_2 x) \right) \\&- r L_{\kappa _0+p} \left( t_1 \cos (k_1 x) + t_2 \cos (k_2 x) + v \right) \\&+ L_{\kappa _0+p} \left( t_1 \cos (k_1 x) + t_2 \cos (k_2 x) + v \right) ^2, \end{aligned}$$
and writing \(L_{\kappa _0+p} = L_{\kappa _0} + (L_{\kappa _0+p} - L_{\kappa _0})\) we have
$$\begin{aligned} F_\kappa \left( u, c\right) =&D_u F_{\kappa _0}(0,c_0) v - c_0 (L_{\kappa _0+p} - L_{\kappa _0}) v\\&- t_1 c_0 (L_{\kappa _0+p} - L_{\kappa _0}) \cos (k_1 x) - t_2 c_0 (L_{\kappa _0+p} - L_{\kappa _0}) \cos (k_2 x)\\&- r L_{\kappa _0+p} \left( t_1 \cos (k_1 x) + t_2 \cos (k_2 x) + v \right) \\&+ L_{\kappa _0+p} \left( t_1 \cos (k_1 x) + t_2 \cos (k_2 x) + v \right) ^2\\ =:&D_u F_{\kappa _0}(0,c_0) v - g(t_1, t_2, r, p, v). \end{aligned}$$
Therefore, (4.2) is equivalent to
$$\begin{aligned} D_u F_{\kappa _0}(0,c_0) v = g(t_1, t_2, r, p, v), \end{aligned}$$
(4.4)
and we can rewrite (4.3) as
$$\begin{aligned} {\left\{ \begin{array}{ll} 0 = \Pi g(t_1, t_2, r, p, v) \\ D_u F_{\kappa _0}(0,c_0) v = ({\mathrm {Id}}-\Pi ) g(t_1, t_2, r, p, v). \end{array}\right. } \end{aligned}$$
(4.5)
Note that since 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
$$\begin{aligned} D_u F_{\kappa _0}(0,c_0) v - ({\mathrm {Id}}-\Pi ) g(t_1, t_2, r, p, v) = 0 \end{aligned}$$
has the solution \((t_1, t_2, r, p, v) = (0, 0, 0, 0, 0)\) and at that point the Fréchet derivative with respect to 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 \(t_2\).
We are now left with solving the finite-dimensional problem given by the first equation in (4.5). To this end, we decompose the projection \(\Pi \) as \(\Pi = \Pi _1 + \Pi _2\), where \(\Pi _1\) is the projection onto \(\cos (k_1 \cdot )\), and \(\Pi _2\) is the projection onto \(\cos (k_2 \cdot )\). Then
$$\begin{aligned} \Pi g = \Pi _1 g + \Pi _2 g = Q_1 \cos (k_1 x) + Q_2 \cos (k_2 x), \end{aligned}$$
with \(Q_j = \langle g, \cos (k_j \cdot ) \rangle \), and the first line of (4.5) is equivalent to showing that
$$\begin{aligned} Q_1 = Q_2 = 0. \end{aligned}$$
(4.6)
To solve (4.6) we consider two cases.
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
$$\begin{aligned} \begin{aligned} Q_1 =&t_1 \left[ c_0 \big (l((\kappa _0+p)k_1) - l(\kappa _0 k_1)\big ) + r \, l((\kappa _0+p)k_1) \right] \\&- l((\kappa _0+p)k_1) \left\langle \cos (k_1 \cdot ) , \left( t_1 \cos (k_1 \cdot ) + t_2 \cos (k_2 \cdot ) + v(t_1, t_2, r, p) \right) ^2 \right\rangle . \end{aligned} \end{aligned}$$
(4.7)
As \(v(0,t_2,r,p)\) is \(2\pi /k_2\)-periodic and \(k_2 \ne k_1\), the above inner term product vanishes for \(t_1 = 0\). Therefore, we may write
$$\begin{aligned} Q_1(t_1, t_2, r, p) = t_1 \, \Psi _1 (t_1, t_2, r, p) \end{aligned}$$
(4.8)
with
$$\begin{aligned} \Psi _1(t_1,t_2,r,p) = \int _0^1 \frac{\partial Q_1}{\partial t_1}(z t_1, t_2, r, p) \;{\mathrm {d}}z, \end{aligned}$$
(4.9)
and note (4.7) implies
$$\begin{aligned} \Psi _1(0, 0, r, p) = c_0 \left[ l((\kappa _0+p)k_1) - l(\kappa _0 k_1)\right] + r\, l((\kappa _0+p)k_1). \end{aligned}$$
(4.10)
Similarly, we have
$$\begin{aligned} \begin{aligned} Q_2 =&t_2 \left[ c_0 \big (l((\kappa _0+p)k_2) - l(\kappa _0 k_2)\big ) + r \, l((\kappa _0+p)k_2) \right] \\&- l((\kappa _0+p)k_2) \left\langle \cos (k_2 \cdot ) , \left( t_1 \cos (k_1 \cdot ) + t_2 \cos (k_2 \cdot ) + v(t_1, t_2, r, p) \right) ^2 \right\rangle \end{aligned} \end{aligned}$$
(4.11)
with the inner product term vanishing at \(t_2=0\) since we assumed \(k_2/k_1 \notin \mathbb {N}_0\). We can thus write
$$\begin{aligned} Q_2(t_1, t_2, r, p) = t_2 \, \Psi _2(t_1, t_2, r, p) \end{aligned}$$
(4.12)
with
$$\begin{aligned} \Psi _2(t_1,t_2,r,p) = \int _0^1 \frac{\partial Q_2}{\partial t_2}(t_1,z t_2, r, p) \;{\mathrm {d}}z \end{aligned}$$
(4.13)
so that
$$\begin{aligned} \Psi _2(0, 0, r, p) = c_0 \left[ l((\kappa _0+p)k_2) - l(\kappa _0 k_2)\right] + r\, l((\kappa _0+p)k_2). \end{aligned}$$
(4.14)
Hence, condition (4.6) is equivalent solving the system
$$\begin{aligned} {\left\{ \begin{array}{ll} t_1 \Psi _1(t_1, t_2, r, p) = 0 \\ t_2 \Psi _2(t_1, t_2, r, p) = 0 \end{array}\right. } \end{aligned}$$
for 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
$$\begin{aligned} (r,p)\mapsto (\Psi _1(0,0,r,p),\Psi _2(0,0,r,p)) \end{aligned}$$
at \((r,p)=(0,0)\) is given by
$$\begin{aligned}&\det \left. \begin{bmatrix} D_r \Psi _1(0,0,r,p)&D_p \Psi _1(0,0,r,p) \\ D_r \Psi _2(0,0,r,p)&D_p \Psi _2(0,0,r,p) \end{bmatrix}\right| _{(r,p)=(0,0)} \nonumber \\&\quad = c_0 \, l_{\kappa _0}( k_1) \left[ l'_{\kappa _0}( k_2)\, k_2 - l'_{\kappa _0}( k_1)\, k_1 \right] , \end{aligned}$$
(4.15)
which is always different from 0 since \(l_T\) has only one positive stationary point, \(l_{\kappa _0}( k_1) \ne 0\), and that the terms \(l'_{\kappa _0}( k_1)\) and \( l'_{\kappa _0}( k_2)\) necessarily have opposite signs. Applying the Implicit Function Theorem gives the solutions in \(\mathcal {S}^{{mixed}}\). Note in each of the above four cases, we find \(r = r(t_1, t_2)\) and \(p = p(t_1, t_2)\) with 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
$$\begin{aligned} \widetilde{Q}_2 (\varrho , \vartheta , r, p)=Q_2(\varrho \cos (\vartheta ),\varrho \sin (\vartheta ),r,p), \end{aligned}$$
defined for \(0\leqslant \varrho \ll 1\) and \(|(\vartheta ,r,p)|\ll 1\), we find from (4.11) that
$$\begin{aligned} \widetilde{Q}_2 (\varrho , \vartheta , r, p)=&\varrho \sin (\vartheta ) c_0 \big (l((\kappa _0+p)k_2) - l(\kappa _0 k_2)\big ) \nonumber \\&+ \varrho \sin (\vartheta ) r \, l((\kappa _0+p)k_2) \nonumber \\&- l((\kappa _0+p)k_2) \frac{1}{\pi }\int _{-\pi }^\pi \cos (k_2 x) \Big [ \varrho \cos (\vartheta ) \cos (k_1 x) \nonumber \\&+ \varrho \sin (\vartheta ) \cos (k_2 x) + v(\varrho \cos (\vartheta ), \varrho \sin (\vartheta ), r, p) \Big ] ^2 \;{\mathrm {d}} x . \end{aligned}$$
(4.16)
Since \(\widetilde{Q}_2(0,\vartheta ,r,p)=0\), we may as before write
$$\begin{aligned} \widetilde{Q}_2(\varrho , \vartheta , r, p) = \varrho \, \widetilde{\Psi }_2(\varrho , \vartheta , r, p) \end{aligned}$$
(4.17)
with
$$\begin{aligned} \widetilde{\Psi }_2(\varrho ,\vartheta ,r,p) = \int _0^1 \frac{\partial {\widetilde{Q}}_2}{\partial \varrho }(z \varrho ,\vartheta , r, p) \;{\mathrm {d}}z \end{aligned}$$
(4.18)
so that
$$\begin{aligned} \begin{aligned} \widetilde{\Psi }_2(0,\vartheta ,r,p) =&\sin (\vartheta )\, c_0 \left[ l((\kappa _0+p)k_2) - l(\kappa _0 k_2)\right] \\&+\, r \sin (\vartheta )\, l((\kappa _0+p)k_2). \end{aligned} \end{aligned}$$
(4.19)
For \(Q_1\), instead, all the previous calculations remain true and hence, similarly defining the function
$$\begin{aligned} \widetilde{\Psi }_1 (\varrho , \vartheta , r, p):=\Psi _1(\varrho \cos (\vartheta ),\varrho \sin (\vartheta )), \end{aligned}$$
(4.20)
it follows in this resonant case that (4.6) is equivalent to solving the system
$$\begin{aligned} {\left\{ \begin{array}{ll} \varrho \cos (\vartheta ) {\widetilde{\Psi }}_1(\varrho , \vartheta , r, p) = 0 \\ \varrho \, {\widetilde{\Psi }}_2(\varrho , \vartheta , r, p) = 0. \end{array}\right. } \end{aligned}$$
for 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
$$\begin{aligned} (r,p)\mapsto ({\widetilde{\Psi }}_1(0,0,r,p),{\widetilde{\Psi }}_2(0,0,r,p)) \end{aligned}$$
at \((r,p)=(0,0)\) is given by
$$\begin{aligned}&\det \left. \begin{bmatrix} D_r {\widetilde{\Psi }}_1(0,\vartheta ,r,p)&D_p {\widetilde{\Psi }}_1(0,\vartheta ,r,p) \\ D_r {\widetilde{\Psi }}_2(0,\vartheta ,r,p)&D_p {\widetilde{\Psi }}_2(0,\vartheta ,r,p) \end{bmatrix}\right| _{(r,p)=(0,0)} \nonumber \\&\quad = \sin (\vartheta ) \, c_0 \, l(\kappa _0 k_1) \left[ l'(\kappa _0 k_2)\, k_2 - l'(\kappa _0 k_1)\, k_1 \right] , \end{aligned}$$
(4.21)
which, by the same considerations we applied to (4.15), is non-zero so long as \(\sin (\vartheta ) \ne 0\) Therefore, for any fixed \(\delta >0\), restricting to \(\delta< |\vartheta | < \pi -\delta \) gives the solutions in \(\mathcal {S}_\delta ^{{mixed}}\), as desired \(\square \)

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

  1. (i)
    There are no periodic solutions of (2.1) in the region
    $$\begin{aligned} \max u < \min \{0, c-1\}.\\ \end{aligned}$$
     
  2. (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 \(c < 2\).

     
  3. (iii)

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

     
  4. (iv)
    For \(T\geqslant \frac{4}{\pi ^2}\), all periodic solutions satisfy
    $$\begin{aligned} \textstyle \max u \leqslant \frac{c^2}{4}, \end{aligned}$$
    with equality if and only if 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

As all steady solutions are smooth, and the symbol of \(L_T\) satisfies \(l_T(0) = 1\), one may as in [20] integrate over any finite period to obtain
$$\begin{aligned} (c-1) \int _{-\pi }^{\pi } u\, \,{\mathrm {d}}x = \int _{-\pi }^{\pi } u^2\, \,{\mathrm {d}}x. \end{aligned}$$
(5.1)
(The same argument works for other periods as well.) This is an immediate contradiction for \(u < \min \{0, c - 1\}\).

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).

For (iv), note that
$$\begin{aligned} u(x) = L (cu - u^2) = \frac{c^2}{4} - L \left( \frac{c}{2} -u \right) ^2 \leqslant \frac{c^2}{4}, \end{aligned}$$
when \(T \geqslant \frac{4}{\pi ^2}\), as the integral kernel of 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

If the surface tension satisfies \(T \geqslant \frac{4}{\pi ^2}\), then the bifurcation curve found in Theorem 3.6 for \(k=1\) can be constructed such that it contains a subsequence of solutions that are all single-crested (bell shaped) in each minimal period and that either:
  1. (i)

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

     
  2. (ii)

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

     

Proof

For even and periodic solutions u one may as in [16, 20] use (2.1) to write
$$\begin{aligned} u'(x) = 2\int _0^\pi \left( K_p(x-y)-K_p(x+y)\right) \left( \frac{c}{2}-u(y)\right) u'(y) \,{\mathrm {d}}y. \end{aligned}$$
(5.2)
When \(K_p\) is completely monotone, and 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, \(\lim _{n \rightarrow \infty } c_n = \infty \).

We now show that this is impossible when \(\max u_n \leqslant \lambda c_n\), \(\lambda < \frac{1}{2}\). Recall that we are following a branch of the curve for which 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
$$\begin{aligned} -u_n'(x_n)&= \min _{y \in [\delta ,\pi -\delta ]} (-u_n'(y)). \\ -u_n'(x_n)&= 2\int _0^\pi \left( K_p(x_n-y)-K_p(x_n+y)\right) \left( \frac{c_n}{2}-u_n(y)\right) (-u'(y)) \,{\mathrm {d}}y\\&\geqslant 2 \delta c_n \int _\delta ^{\pi -\delta } \left( K_p(x_n-y)-K_p(x_n+y)\right) (-u_n'(y)) \,{\mathrm {d}}y\\&\geqslant -2 \delta c_n u_n'(x_n) \int _\delta ^{\pi -\delta } \left( K_p(x_n-y)-K_p(x_n+y)\right) \,{\mathrm {d}}y. \end{aligned}$$
On the interval of consideration, \(K_p(x_n-y)-K_p(x_n+y)\) is bounded from below by a positive constant (it is zero only for \(y = k\pi \), \(k \in \mathbb {Z}\)). Although it has a singularity at \(x_n = y\), it tends to \(\infty \) there, so we may estimate it from below, uniformly in \(x_n\), by
$$\begin{aligned} \min \left\{ \left( K_p(x_n-y)-K_p(x_n+y)\right) :(x,y) \in [\delta , \pi -\delta ] \times [\delta , \pi -\delta ]\right\} \gtrsim 1. \end{aligned}$$
Consequently,
$$\begin{aligned} -u_n'(x_n) \gtrsim -c_n u_n'(x_n), \end{aligned}$$
which is not possible, as \(c_n \rightarrow \infty \) and \(-u_n'(x_n) > 0\) for all n. \(\square \)

5.1 Discussion and Summary of Results

Analytically, we have determined almost completely5 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.

To start our discussion, focus first on one of the Figs. 3 or 4. Just as the regular Whitham equation, the capillary–gravity Whitham equation (2.1) admits two lines of constant solutions, namely \(u = 0\) and \(u = c-1\). These cross at \(c = 1\), the point of a transcritical bifurcation (see Remark 3.2), and also a bifurcation point for solitary [7] and generalised solitary [24] waves ; additionally, \(c=1\) is the symmetry line for the Galilean invariance
$$\begin{aligned} c \mapsto 2 - c, \qquad u \mapsto u + 1 - c, \end{aligned}$$
that leaves (2.1) invariant, and is shared by the regular Whitham equation [20]. The two constants 0 and \(c-1\) correspond to the two natural depths that appear for steady flows in the water wave problem, see for example [27]. In addition to these two lines, there is a third, mathematical, constant arising from the structure of (2.1) when completing the square, namely \(\frac{c}{2}\). While this constant is of physical and absolute importance in the regular Whitham equation—being the height above surface of a highest wave—and while it appears as a technical difficulty when trying to expand the result of Proposition 5.4, numerical evidence indicates that this construct is probably only artificial in the presence of capillarity. Still, we have indicated it in Fig. 3 using the line \(\max u = \frac{c}{2}\) (but not in Fig. 4, as it did not prove any help in communicating our results). Additionally, in both Figs. 3 and 4 the greyed-out area illustrates Proposition 5.2, that there are no solutions in the region where
$$\begin{aligned} \max u < \min \{0, c-1\}. \end{aligned}$$
A final common feature of the strong and weak surface tension case is that solutions cannot pass \(c=1\), except via the transcritical bifurcation point \((u,c) = (0,1)\), where, locally, the only solutions are given by the constant functions \(u = 0\) and \(u = c-1\). This fact may be induced from Proposition 5.2 (iii) and Remark 3.2, and is indicated in the figures with a solid red line (no solutions pass). Note that both figures are for a fixed and finite period.

5.2 The Case of Strong Surface Tension

Now, let us focus on the strong surface tension case and especially the case \(T \geqslant \frac{4}{\pi ^2}\), which is depicted in Fig. 3. As described in Theorem 3.1, we have small waves of the approximate linear form \(\cos (k \cdot )\) bifurcating at
$$\begin{aligned} c_k = \frac{1}{l_T(k)} > 1. \end{aligned}$$
The bifurcation curves of these waves are indicated by solid blue lines, with a zoom-in on a small wave along the main bifurcation branch \(k=1\). The red line \(\{u = 0, 1 < c \ne c_k\}\) shows the result of Proposition 5.2 (ii), that there are no other supercritical solutions in a \(\mathcal {C}^s\)-vicinity of the line of vanishing solutions. By Galilean invariance, each of these curves (and non-existence results) has an exact counterpart for \(c < 1\) along the line \(u = c-1\), and we do not comment more on that in 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 \(\max u \leqslant \frac{c}{2}\).

After that point, our calculations are purely numerical, showing the solution curves asymptotically approaching the second curve of constant solutions \(u = c -1\). Indeed, if the quotient
$$\begin{aligned} \frac{u}{c-1} \end{aligned}$$
should converge to any constant along the bifurcation curve, it is immediate from (5.1) that the limit is either 0 or 1. The numerics indicate that the quotient \(\frac{\max (u)}{c-1}\) increases along the bifurcation curve to cover all of the interval (0, 1), with wave profiles that are monotone on a minimal half-period even though, by far, we have passed \(u = \frac{c}{2}\). Such a result, we believe, would be new in the setting of capillary–gravity water waves, but it is so far out of reach for us when 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. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 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. 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.

References

  1. 1.
    Aasen, A., Varholm, K.: Traveling gravity water waves with critical layers. J. Math. Fluid Mech. 20, 161–187 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Akers, B.F., Ambrose, D.M., Sulon, D.W.: Periodic travelling interfacial hydroelastic waves with or without mass II: Multiple bifurcations and ripples. Eur. J. Appl. Math. 30, 756–790 (2019)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Akers, B.F., Ambrose, D.M., Wright, J.D.: Gravity perturbed Crapper waves. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci 470, 20130526 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ambrose, D.M., Strauss, W.A., Wright, J.D.: Global bifurcation theory for periodic traveling interfacial gravity–capillary waves. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 1081–1101 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Amick, C.J., Turner, R.E.L.: A global theory of internal solitary waves in two-fluid systems. Trans. Am. Math. Soc. 298, 431–484 (1986)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Arnesen, M.N.: A non-local approach to waves of maximal height for the Degasperis-Procesi equation. arXiv:1808.08057
  7. 7.
    Arnesen, M.N.: Existence of solitary-wave solutions to nonlocal equations. Discret. Contin. Dyn. Syst. 36, 3483–3510 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Aston, P.: Local and global aspects of the (1, n) mode interaction for capillary–gravity waves. Phys. D 52, 415–428 (1991)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bhatia, R.: Positive Definite Matrices. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2007)Google Scholar
  10. 10.
    Buffoni, B., Toland, J.: Analytic Theory of Global Bifurcation—An Introduction, Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2003)CrossRefGoogle Scholar
  11. 11.
    Córdoba, D., Enciso, A., Grubic, N.: On the existence of stationary splash singularities for the Euler equations. Adv. Math. 288, 922–941 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dancer, E.: Bifurcation theory for analytic operators. Proc. Lond. Math. Soc. XXVI, 359–384 (1973)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dancer, E.: Global structure of the solutions set of non-linear real-analytic eigenvalue problems. Proc. Lond. Math. Soc. XXVI, 747–765 (1973)CrossRefGoogle Scholar
  14. 14.
    Ehrnström, M., Escher, J., Wahlén, E.: Steady water waves with multiple critical layers. SIAM J. Math. Anal. 43, 1436–1456 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ehrnström, M., Holden, H., Raynaud, X.: Symmetric waves are traveling waves. Int. Math. Res. Not. 24, 4578–4596 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ehrnström, M., Johnson, M.A., Claassen, K.M.: Existence of a highest wave in a fully dispersive two-way shallow water model. Arch. Ration. Mech. Anal. (2018).  https://doi.org/10.1007/s00205-018-1306-5 CrossRefzbMATHGoogle Scholar
  17. 17.
    Ehrnström, M., Kalisch, H.: Traveling waves for the Whitham equation. Differ. Integral Equ. 22, 1193–1210 (2009)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ehrnström, M., Kalisch, H.: Global bifurcation for the Whitham equation. Math. Model. Nat. Phenom. 8, 13–30 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ehrnström, M., Wahlèn, E.: Trimodal steady water waves. Arch. Rational Mech. Anal. 216, 449–471 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ehrnström, M., Wahlèn, E.: On Whitham’s conjecture of a highest cusped wave for a nonlocal dispersive equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 36, 1603–1637 (2019)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics, 3rd edn. Springer, New York (2014)Google Scholar
  22. 22.
    Hunter, J.K., Vanden-Broeck, J.-M.A.: Solitary and periodic gravity–capillary waves of finite amplitude. J. Fluid Mech. 134, 205–219 (1983)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hur, V.M., Johnson, M.A.: Modulational instability in the Whitham equation with surface tension and vorticity. Nonlinear Anal. 129, 104–118 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Johnson, M.A., Wright, J.D.: Generalized solitary waves in the gravity-capillary Whitham equation. arXiv:1807.11469
  25. 25.
    Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge Mathematical Library, 3rd edn. Cambridge Univerisity Press, Cambridge (2004)CrossRefGoogle Scholar
  26. 26.
    Kielhöfer, H.: Bifurcation Theory. Applied Mathematical Sciences, vol. 156. Springer, New York (2004)CrossRefGoogle Scholar
  27. 27.
    Kozlov, V., Kuznetsov, N.: Steady free-surface vortical flows parallel to the horizontal bottom. Q. J. Mech. Appl. Math. 64, 371–399 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kozlov, V., Lokharu, E.: N-Modal steady water waves with vorticity. J. Math. Fluid Mech. 2017, 1–15 (2017)zbMATHGoogle Scholar
  29. 29.
    Lannes, D.: The Water Waves Problem, Vol. 188 of Mathematical Surveys and Monographs. Mathematical Analysis and Asymptotics. American Mathematical Society, Providence (2013)Google Scholar
  30. 30.
    Martin, C., Matioc, B.-V.: Existence of Wilton ripples for water waves with constant vorticity and capillary effects. SIAM J. Appl. Math. 73, 1582–1595 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Okamoto, H., Shōji, M.: The Mathematical Theory of Permanent Progressive Water-Waves. Advanced Series in Nonlinear Dynamics, vol. 20. World Scientific Publishing Co., Inc., River Edge (2001)CrossRefGoogle Scholar
  32. 32.
    Paley, R.E.A.C., Wiener, N.: Fourier Transforms in the Complex Domain, Vol. 19 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (1987). (Reprint of the 1934 original)Google Scholar
  33. 33.
    Reeder, J., Shinbrot, M.: On Wilton ripples, II: Rigorous results. Arch. Ration. Mech. Anal. 77, 321–347 (1981)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Remonato, F., Kalisch, H.: Numerical Bifurcation for the Capillary Whitham Equation. Phys. D 343, 51–62 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Schilling, R.L., Song, R., Vondraĉek, Z.: Bernstain Functions. de Gruyter Studies in Mathematics, vol. 37, 2nd edn. Walter de Gruyter & Co., Berlin (2012)Google Scholar
  36. 36.
    Taylor, M.E.: Partial Differential Equations III. Nonlinear Equations, Applied Mathematical Sciences, 2nd edn. Springer, Berlin (2011)CrossRefGoogle Scholar
  37. 37.
    Toland, J.F., Jones, M.C.W.: The bifurcation and secondary bifurcation of capillary–gravity waves. Proc. R. Soc. Lond. Ser. A 399, 391–417 (1985)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Trichtchenko, O., Deconinck, B., Wilkening, J.: The instability of Wilton ripples. Wave Motion 66, 147–155 (2016)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Turner, R.E.L., Vanden-Broeck, J.-M.: The limiting configuration of interfacial gravity waves. Phys. Fluids 29, 372–375 (1986)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Vanden-Broeck, J.-M.: Nonlinear gravity–capillary standing waves in water of arbitrary uniform depth. J. Fluid Mech. 139, 97–104 (1984)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Whitham, G.: Variational methods and applications to water waves. Proc. R. Soc. Lond. A 299, 6–25 (1967)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA
  3. 3.Department of MathematicsUniversity of PaviaPaviaItaly

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