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Water Waves

, Volume 1, Issue 1, pp 41–70 | Cite as

Babenko’s Equation for Periodic Gravity Waves on Water of Finite Depth: Derivation and Numerical Solution

  • Nikolay KuznetsovEmail author
  • Evgueni Dinvay
Original Article
  • 122 Downloads

Abstract

The nonlinear two-dimensional problem describing periodic steady gravity waves on water of finite depth is considered in the absence of surface tension. It is reduced to a single pseudo-differential operator equation (Babenko’s equation), which is investigated analytically and numerically. This equation has the same form as the equation for waves on infinitely deep water; the latter had been proposed by Babenko and studied in detail by Buffoni, Dancer and Toland. Instead of the \(2 \pi \)-periodic Hilbert transform \({\mathcal {C}}\) used in the equation for deep water, the equation obtained here contains a certain operator \({\mathcal {B}}_r\), which is the sum of \({\mathcal {C}}\) and a compact operator depending on a parameter related to the depth of water. Numerical computations are based on an equivalent form of Babenko’s equation derived by virtue of the spectral decomposition of the operator \({\mathcal {B}}_r \mathrm {d}/ \mathrm {d}t\). Bifurcation curves and wave profiles of the extreme form are obtained numerically.

Keywords

Steady water waves Babenko’s equation Froude number 

Mathematics Subject Classification

76B15 35Q35 

1 Introduction

Near the end of his remarkable career in both pure and applied mathematics (see [1] for highlights of major achievements), Konstantin Ivanovich Babenko (1919–1987) published four brief notes [4, 5, 6, 7] (the last two appeared posthumously) on a classical nonlinear problem in the mathematical theory of water waves, namely, the two-dimensional problem of steady periodic waves on infinitely deep water. In this paper dedicated to the centenary of Babenko’s birth, we extend the approach developed in [4] to the case of water of finite depth and deduce a single pseudo-differential operator equation (Babenko’s equation) equivalent to the corresponding free-boundary problem in some sense explained below (see Sect. 3.3). Moreover, using the spectral decomposition of a linear operator involved in the equation, we transform it to a form convenient for discretization and then apply a very robust numerical method that allows us to obtain results concerning global bifurcation branches, secondary bifurcations and free surface profiles including those of the extreme form.

It was Stokes [41] who initiated studies in this field. On the basis of approximations developed for waves with a single crest per wavelength (now referred to as Stokes waves), he made some conjectures about the behaviour of such waves on deep water. These conjectures were essential for the development of the mathematical approach to water waves in the twentieth century; see the paper [38] and references cited therein for an idea of how these conjectures were proved. In particular, the so-called Nekrasov equation was essential for this purpose. The first version of this nonlinear integral equation for waves on deep water was derived in [33] (see also [35], Part 1). Soon thereafter Nekrasov generalized his equation to the case of finite depth (see [34, 35], Part 2). Much later, Amick and Toland [2] proposed and investigated a more sophisticated version of the latter equation.

Levi–Civita [28] and Struik [42] considered (independently of Nekrasov) the problem of periodic waves on deep water and on water of finite depth, respectively. The hodograph transform allowed them to reduce the question of existence of waves to that of finding a pair of conjugate harmonic functions satisfying nonlinear Neumann boundary conditions. The existence proofs given in [28, 42] are based on a majorant method for demonstrating the convergence of power series that provide formal solutions. In his book [48], Chapter 71, Zeidler writes about these proofs that they are ‘very complicated’ in view of ‘voluminous computations’ involved. Today, both techniques (Nekrasov’s equations and the method of Levi–Civita and Struik) have been investigated in detail by means of analytic bifurcation theory. An account of this theory can be found in the books [14, 47], while many authors have studied its application to equations describing steady water waves; these results are summarised in [44] (deep water) and in [48], Chapter 71 (water of finite depth), where one also finds detailed historical remarks. It should be mentioned that Krasovskii [27] extended studies of water waves to the case of a periodic wavy bottom.

Another method for periodic waves on deep water (with and without surface tension) was developed in detail by Buffoni, Dancer and Toland [12, 13]. In the absence of surface tension, it is based on the so-called Babenko equation
$$\begin{aligned} \mu \, {\mathcal {C}} (v') = v + v \, {\mathcal {C}} (v') + {\mathcal {C}} (v' v) , \quad t \in (-\pi , \pi ) . \end{aligned}$$
(1)
Here, \(\mu \) is a dimensionless bifurcation parameter (it is related to the speed of wave propagation), which must be found along with the \(2 \pi \)-periodic function v(t) that describes the wave profile parametrically in certain dimensionless coordinates; \('\) stands for differentiation with respect to t and \({{{\mathcal {C}}}}\) is the \(2 \pi \)-periodic Hilbert transform (the conjugation operator in the theory of Fourier series; see, for example, [51]). It is defined on \(L^2 (-\pi , \pi )\) by linearity from the following relations:
$$\begin{aligned} {{\mathcal {C}}} (\cos n t) = \sin n t \ \ \text{ for } \ n \ge 0 , \quad {{\mathcal {C}}} (\sin n t) = - \cos n t \ \ \text{ for } \ n \ge 1 . \end{aligned}$$
(2)
The original form of (1) was announced by Babenko [4] (see also [36], Section 3.7), where the equation is derived and expressed in terms of the self-adjoint operator \(J_0 = \mathcal {C} \, \mathrm {d}/ \mathrm {d}t\). In his second note [5], Babenko outlines the proof of a local existence theorem for his equation in a neighbourhood of the first bifurcation point—equal to unity. Setting \(\mu = 1 + \epsilon ^2\), he also changes the unknown function by applying the invertible operator \(I + J_0\) and seeks the new unknown in the form of an expansion in powers of \(\epsilon \). It is shown that the expansion converges for \(\epsilon \le 1 / 25\). Some numerical computations related to Babenko’s version of equation (1) are presented in [6, 7].
In [4], Babenko derived his equation by applying a conformal mapping of the infinitely deep water domain corresponding to a single wave onto the complement of the unit disc. In the case of finite depth, it is reasonable to follow the approach proposed by Nekrasov [35] and map the water domain corresponding to a single wave onto an annular domain whose radii are unity and \(r < 1\). Combining this mapping and the procedure used in [4] (see details in Sect. 3.1), we arrive at the equation
$$\begin{aligned} \mu \, \mathcal {B}_r (v') = v + v \, \mathcal {B}_r (v') + \mathcal {B}_r \, (v' v) , \quad t \in (-\pi , \pi ) . \end{aligned}$$
(3)
This is Babenko’s equation for water of finite depth, where \(\mu \) is the Froude number squared (the nondimensional parameter introduced in Sect. 2.2 and related to the velocity of wave propagation and wavelength), and the operator \(\mathcal {B}_r\) is defined by the relations
$$\begin{aligned} \mathcal {B}_r (\cos n t) = \frac{1 + r^{2 n}}{1 - r^{2 k}} \sin n t \ \ \text{ for } \ n \ge 0 , \quad \mathcal {B}_r (\sin n t) = - \frac{1 + r^{2 n}}{1 - r^{2 n}} \cos n t \ \ \text{ for } \ n \ge 1,\nonumber \\ \end{aligned}$$
(4)
which are modifications of formulae (2); here, \(t \in (-\pi , \pi )\).

The aim of this paper is to derive and analyse this equation and to provide numerical results based on its discretization. The most important of these are bifurcation diagrams that present families of solutions as curves plotted in the \((\mu , \Vert \eta \Vert _\infty )\)-plane; here, \(\Vert \eta \Vert _\infty \) is the nondimensional wave height. To the authors’ knowledge, numerical algorithms based on formulations other than Babenko’s equation are not efficient enough for computation of these diagrams, which demonstrate two interesting features of steady waves. The first of these is the presence of secondary bifurcation curves branching from the original ones. The so-called Tanaka phenomenon is another significant property taking place near the ends of bifurcation curves. Moreover, the efficiency of the numerical algorithm based on Babenko’s equation and its implementation allow us to compute wave profiles of extreme form.

The plan of the paper is as follows. We begin with the dimensional statement of the problem describing steady periodic waves on water of finite depth and reduce it to the equivalent nondimensional one in Sect. 2. Babenko’s equation is derived from the nondimensional formulation in Sect. 3.1, and the existence of local bifurcation branches for this equation is proved on the basis of the Crandall–Rabinowitz theorem in Sect. 3.3. In Sect. 3.2, we outline how to obtain a solution of the nondimensional problem from a given solution of Babenko’s equation. A numerical procedure for solving Babenko’s equation is presented in Sect. 4 along with various bifurcation curves and wave profiles obtained with its help. Section 5 contains concluding remarks. In two appendices, we compare our version of Babenko’s equation with those used in [17, 18]; it is also demonstrated that our numerical method is at least as accurate as that applied in [17].

1.1 Further Background

Longuet–Higgins [29] derived an infinite system of algebraic equations equivalent to (1) (see also [15, 36], Section 3.6). He used this system for numerical computations of Stokes-wave bifurcations (see [30] and also [36], Sections 3.2 and 3.8). It is worth mentioning that this system naturally appears from the Lagrangian formalism developed in [9]. In the paper [31], a similar system of quadratic relations between the Fourier coefficients of the wave profile was obtained in the case of water having finite depth.

Interesting results concerning the secondary or subharmonic bifurcations from branches describing Stokes-wave solutions of (1) are proved in the comprehensive articles [12, 13], and it is essential that Babenko’s equation is used for this purpose. It is shown in [12] that this type of bifurcation does not occur near the points where Stokes waves bifurcate from a trivial solution. On the other hand, it is demonstrated in [13] that there exist secondary bifurcations from symmetric Stokes waves. This confirms rigorously another line of investigation (appearing in the 1980s), providing significant numerical evidence about the existence of steady periodic waves that differ from classical Stokes waves. These new waves have more than one crest per period and bifurcate from Stokes waves. Branches of subharmonic bifurcations for deep water were first computed by Chen and Saffman [16], whereas Vanden-Broeck [45] obtained a similar result for water of finite depth by solving numerically an integrodifferential system arising after the hodograph transform; this system was proposed in [46]. Craig and Nicholls [19] obtained numerical results generalising those of Vanden-Broeck. For this purpose, they used the numerical technique introduced in [20] and based on the Taylor expansion of a Dirichlet–Neumann operator in homogeneous powers of the surface elevation \(\eta \). Moreover, it was shown that asymmetric periodic waves exist on water of finite depth, for which purpose a weakly nonlinear Hamiltonian model was applied in [49].

References to other works containing numerical results on subharmonic bifurcations can be found in [8, 36]. The latter paper gives some theoretical insights concerning these bifurcations. Moreover, it was concluded in [13] ‘that the subharmonic bifurcations [...] are an inevitable consequence of the formation of Stokes highest wave’. A characteristic property of the latter wave is the angle \(2\pi /3\) formed at the crest by two smooth symmetric curves. Concerning this wave see [38] and references cited therein. It is also worth mentioning that a generalization of Babenko’s equation was studied by Shargorodsky and Toland [40].

Apart from Nekrasov’s equation, the approach of Struik and numerical methods mentioned above, there arose a quite different direction of studies for waves on water of finite depth. Namely, a pseudodifferential equation in terms of variables arising after the hodograph transform was derived in [26]. This equation describes all waves for which the rate of flow per unit span and the Bernoulli constant are prescribed, and so it is suitable for justifying the Benjamin–Lighthill conjecture for near-critical waves. However, it deals only with waves of small amplitude and is hardly applicable for studying subharmonic bifurcations. Meanwhile, the results obtained for equation (1) in [12, 13] show that Babenko’s equation serves well for this purpose and this is confirmed numerically in this paper (see Fig. 7 below).

Finally, we turn to two recent papers that are of paramount importance for us since they provide an alternative approach to Babenko’s equation. In the first article [18], Constantin, Strauss and Vărvărucă obtained a version of Babenko’s equation for irrotational waves on water of finite depth that differs from (3). It should be emphasised that Babenko’s equation is a subsidiary result in the comprehensive work [18] aimed at investigating waves on flows with constant vorticity. It is shown that to study this more difficult problem one needs a system of quasilinear pseudo-differential equations instead of a single equation. Nevertheless, in the irrotational case (that is, for zero vorticity) and for some particular value of a parameter involved, the system reduces to a single equation that has the same form as (1) and (3), but the operator involved in this equation (the so-called periodic Hilbert transform) differs from both \({\mathcal {C}}\) and \({{\mathcal {B}}}_r\). We compare this equation with (3) in Appendix A.

The second recent paper [17] has a very different aim; its authors, Clamond and Dutykh, describe an algorithm for computing steady gravity waves on the free surface of water in irrotational motion. The proposed algorithm is based on the same Babenko equation as in [18], but rewritten into a form suitable for applying the Petviashvili method [37] originally proposed for computing solitary waves. For example, equation (4.10) in [17] is equivalent to the Babenko equation obtained in [18], and to show this an appropriate transformation of the operator and parameters must be carried out. In Appendix B, we compare our numerical method with that used in [17] as well as the results of computations.

2 Statements of the Problem

In its simplest form, the problem of steady surface waves concerns the two-dimensional irrotational motion of an inviscid, incompressible heavy fluid, say water, bounded above by a free surface and below by a rigid horizontal bottom. (For example, this kind of motion occurs in water occupying an infinitely long channel with rectangular cross-section and having uniform width.) In an appropriate frame of reference, the velocity field of steady motion is time independent, as is the free-surface profile, and they can be described in two equivalent ways that differ by prescribed parameters.

2.1 The Benjamin–Lighthill Statement for Steady Waves

The classical formulation proposed by Benjamin and Lighthill is convenient for justification of their conjecture (see [10, 26], where it was justified for Stokes waves and all near-critical waves, respectively). In this formulation, Q—the constant rate of flow per unit span—is prescribed along with the total head R, also referred to as the Bernoulli constant. Let Cartesian coordinates (XY) be chosen so that the bottom coincides with the X-axis and gravity acts in the negative Y-direction, while the wave profile has a crest on the Y-axis (this does not restrict generality). Thus, the profile is given by the graph of an unknown positive function \(\xi \) (that is, \(Y = \xi (X)\), \(X \in \mathbb {R}\)), attaining its maximum at \(X=0\). Moreover, we suppose that \(\xi \) is continuously differentiable and even. In the longitudinal section of the water domain \({{\mathcal {D}}} = \{ X \in \mathbb {R},\ 0< Y < \xi (X) \}\), the velocity field is described by the stream function \(\Psi (X, Y)\), that is, the projections of the velocity vector at (XY) on the X- and Y-axes are \(\Psi _Y\) and \(-\Psi _X\), respectively. It is assumed that \(\Psi \) belongs to \(C^2 ({{\mathcal {D}}}) \cap C^1 (\bar{{\mathcal {D}}})\) and is an even function of X (hence, it is bounded on \(\bar{{\mathcal {D}}}\)).

If surface tension is neglected, then \(\Psi \) and \(\xi \) must satisfy the following free-boundary problem:
$$\begin{aligned}&\Psi _{XX} + \Psi _{YY} = 0, \quad (X,Y) \in {{\mathcal {D}}}; \end{aligned}$$
(5)
$$\begin{aligned}&\Psi (X, 0) = - Q, \quad X \in \mathbb {R}; \end{aligned}$$
(6)
$$\begin{aligned}&\Psi (X, \xi (X)) = 0, \quad X \in \mathbb {R}; \end{aligned}$$
(7)
$$\begin{aligned}&\frac{1}{2} |\nabla \Psi (X, \xi (X))|^2 + g \, \xi (X) = R , \quad X \in \mathbb {R}. \end{aligned}$$
(8)
In the left-hand side of the last relation, usually referred to as Bernoulli’s equation, \(g > 0\) is the constant acceleration due to gravity. It is known that nontrivial solutions of problem (5)–(8) exist only when \(Q \ne 0\) and \(R > R_c = \frac{3}{2} (Q g)^{2/3}\). For a proof of the first relation, see Proposition 1.1 in [25], whereas the last inequality is proved in [24] under weaker assumptions than those listed above. In what follows, these restrictions on Q and R hold; moreover, we suppose (without loss of generality) that \(Q > 0\).

2.2 A Nondimensional Statement for Periodic Waves

Let us assume that \(\xi \) is a \(2 \ell \)-periodic function (\(\ell > 0\)), while \(\Psi (X, Y)\) is \(2 \ell \)-periodic in X, but the constant R is to be found along with these functions from problem (5)–(8). To reduce the reformulated problem to a nondimensional form, we average Bernoulli’s equation over \((-\ell , \ell )\). Since \(\Psi \) is constant on the free surface, we obtain that \(c^2 = 2 (R - g H)\), where
$$\begin{aligned} H = \frac{1}{2 \ell } \int _{-\ell }^\ell \xi (X) \, \mathrm {d}X \quad \text{ and } \quad c^2 = \frac{1}{2 \ell } \int _{-\ell }^\ell \left| \frac{ \partial \Psi }{\partial n} (X, \xi (X)) \right| ^2 \mathrm {d}X . \end{aligned}$$
(9)
Here, n is the unit normal to \(Y = \xi (X)\) directed out of \(\mathcal{D}\). One can show that the last equality (9) is true with the same constant \(c^2\) when this curve is changed to \(Y = \tilde{\xi }(X)\)—an arbitrary streamline—and n is changed to \({{\tilde{n}}}\)—the unit normal to this streamline. Therefore, \(c > 0\) is the unknown mean velocity of flow in the positive direction of the X-axis.
It is convenient to introduce the following nondimensional quantities: \(h = \pi H / \ell \) (the mean depth of flow) and \(Q_0 = Q / \sqrt{ g (\ell / \pi )^3}\) (the rate of flow). Now, we scale the dimensional variables and shift the vertical variables downwards as follows:
$$\begin{aligned} x = \frac{\pi }{\ell } X ,\ y = \frac{\pi }{\ell } Y - h ; \quad \eta (x) = \frac{\pi }{\ell } \, \xi (X) - h ; \quad \psi (x,y) = \frac{Q_0}{Q} \Psi (X,Y).\quad \end{aligned}$$
(10)
This is advantageous because the new unknown \(\eta \) is a \(2 \pi \)-periodic and even function of x satisfying the following condition:
$$\begin{aligned} \int _{-\pi }^\pi \eta (x) \, \mathrm {d}x = 0 . \end{aligned}$$
(11)
Furthermore, the function \(\psi \) has the same properties on \({\bar{D}}\) as \(\Psi \) has on \(\bar{{\mathcal {D}}}\), namely,
$$\begin{aligned} \psi \in C^1 (\bar{D}) \cap C^2 (D), \quad \text{ where }\quad \ D = \{ x \in \mathbb {R}, -h< y < \eta (x) \}, \end{aligned}$$
and is a \(2 \pi \)-periodic and even function of x. Moreover, the change of variables (10) reduces relations (5)–(8) to the following:
$$\begin{aligned}&\psi _{xx} + \psi _{yy} = 0, \quad (x,y) \in D; \end{aligned}$$
(12)
$$\begin{aligned}&\psi (x, -h) = -Q_0 , \quad x \in \mathbb {R}; \end{aligned}$$
(13)
$$\begin{aligned}&\psi (x, \eta (x)) = 0, \quad x \in \mathbb {R}; \end{aligned}$$
(14)
$$\begin{aligned}&|\nabla \psi (x, \eta (x))|^2 + 2 \eta (x) = \mu , \quad x \in \mathbb {R}. \end{aligned}$$
(15)
In the nondimensional Bernoulli equation, the parameter \(\mu = \pi c^2 / (g \ell )\) is the Froude number squared, which must be found along with \(\eta \) and \(\psi \). Also, \(\mu / 2\) serves as an upper bound for \(\eta \) which is independent of h, where equality is achieved only for the wave of extreme form with Lipschitz crest; see [19]. Thus, the nondimensional statement of the problem is as follows.

Definition 1

Let \(Q_0\) and h be given positive numbers. Then, problem P\((Q_0,h)\) is to find a triple \((\mu , \eta , \psi )\) satisfying relations (12)–(15) such that \(\mu \) is positive, \(\eta \in C^1 (\mathbb {R})\) is a \(2 \pi \)-periodic, even function satisfying (11), and \(\psi \in C^1 (\bar{D}) \cap C^2 (D)\) is a \(2 \pi \)-periodic, even function of x.

On the other hand, if we have a solution of problem P\((Q_0,h)\), formulae (10) yield a \(2 \ell \)-periodic solution of problem (5)–(8) for any \(\ell > 0\), for which purpose one has to set \(Q = Q_0 \sqrt{ g (\ell / \pi )^3}\) and to determine R from the first relation (9) with \(c^2 = \mu g \ell / \pi \) and \(H = h \ell / \pi \).

3 Babenko’s Equation

In this section, we derive the nonlinear pseudodifferential equation (3) starting from problem \(\hbox {P}(Q_0,h)\), demonstrate the existence of small-amplitude solutions, and finally show how solutions of this equation define periodic waves.

3.1 Derivation of Babenko’s Equation

Firstly, we follow the considerations in Section 8 of [35]. (Nekrasov derives his equation for water of finite depth in this section; his considerations can also be found in the rather recent paper [11], which is mainly concerned with a method for numerical solution of this Nekrasov equation and results of its implementation.) By \(w (z) = \varphi + \mathrm {i}\psi \,(z = x + \mathrm {i}y)\), we denote the complex potential restricted to the one-wave domain
$$\begin{aligned} D_{2 \pi } = \{ -\pi< x< \pi , -h< y < \eta (x) \} \end{aligned}$$
of some periodic wave on water of a certain depth h. Here, \(\varphi (x, y)\) is the harmonic conjugate to \(\psi \), odd in x, for which purpose the additive constant must be chosen properly. For some \(r \in (0,1)\), we consider a conformal mapping, say u(z), from the z-plane to the auxiliary u-plane; it maps \(D_{2 \pi }\) onto
$$\begin{aligned} A_r = \{ r< |u| < 1 ; \ {\text {Re}}u \notin (-1 , -r) \ \text{ when } \,\, {\text {Im}}u =0 \} . \end{aligned}$$
(16)
This annular domain has a cut that makes it simply connected, and the map is such that the images of the upper and lower parts of \(\partial D_{2 \pi }\) are
$$\begin{aligned} \{ |u|=1 ; {\text {Re}}u \ne -1 \} \quad \text{ and } \quad \{ |u|=r ; {\text {Re}}u \ne -r \}, \end{aligned}$$
respectively, whereas the left (right) side of \(\partial D_{2 \pi }\) is mapped onto the upper (lower, respectively) side of the cut \(\{ {\text {Re}}u \in (-1 , -r) \ \text{ when } \ {\text {Im}}u =0 \}\). Setting
$$\begin{aligned} u = \mathrm {e}^{-\mathrm {i}w} \quad \text{ and } \quad \frac{\mathrm {d}z}{\mathrm {d}u} = \mathrm {i}\left[ u^{-1} + f (u) \right] , \end{aligned}$$
(17)
where f(u) is a certain Laurent series, one obtains that
$$\begin{aligned} - \frac{\mathrm {d}w}{\mathrm {d}z} = \left[ 1 + u f (u) \right] ^{-1} . \end{aligned}$$
(18)
In [35], Section 8, this formula serves as the basis for deriving Nekrasov’s equation in the case of finite depth. An equivalent form of this equation is given in [11]; see equation (1.1) there.
According to the second equality (17), the general form of the inverse conformal mapping \(A_r \ni u \mapsto z \in D_{2 \pi }\) is as follows:
$$\begin{aligned} z (u) = \mathrm {i}\Big [ \log u - a_0 + \sum _{k=1}^\infty a_k \left( u^k - r^{2 k} u^{-k} \right) \Big ] , \quad \text{ where } \ a_k \in \mathbb {R}, \ k = 0,1,2,\dots . \end{aligned}$$
(19)
Here, the minus sign in front of \(a_0\) is chosen for convenience in the following. The fact that all coefficients \(a_k\) are real is a consequence of equality (18) because \(\psi \) is equal to a real constant on the bottom part of \(\partial D_{2 \pi }\), which corresponds to \(\{ |u|=r ; {\text {Re}}u \ne -r \}\)—the circumference cut on the negative real axis.
Let us derive some relations for the coefficients from (19). Firstly, for \(u=r\) we obtain
$$\begin{aligned} a_0 = h + \log r . \end{aligned}$$
(20)
Substituting \(u = \mathrm {e}^{\mathrm {i}t}\), \(t \in (- \pi , \pi )\), into (19) and separating the real and imaginary parts, we arrive at the following parametric representation of the free surface profile:
$$\begin{aligned} x (t)= & {} - t - \sum _{k=1}^\infty a_k \left( 1 + r^{2 k} \right) \sin kt ,\nonumber \\ v (t)= & {} \eta (x (t)) = - a_0 + \sum _{k=1}^\infty a_k \left( 1 - r^{2 k} \right) \cos kt . \end{aligned}$$
(21)
Now, we see that another relation is equivalent to condition (11) written in terms of the last two expressions, namely
$$\begin{aligned} \int _{-\pi }^\pi \!\! v (t) x' (t) \, \mathrm {d}t = 0 \Longleftrightarrow a_0 = \frac{1}{2} \sum _{k=1}^\infty k \, a_k^2 \left( 1 - r^{4 k} \right) . \end{aligned}$$
(22)
It follows from the last equality that \(a_0 > 0\) in the nontrivial case. Then, equality (20) shows that the value of r is related not only to the depth h, but also to a particular solution of problem P\((Q_0,h)\).
It is worth mentioning that both expressions (21) are similar to those for the infinite depth (cf. [36], Section 3.7, where Babenko’s results are outlined), and in that case, a consequence is the following expression for the derivative: \(x_t = - 1 - \mathcal {C} \eta _t\). Here
$$\begin{aligned} (\mathcal {C} v) (t) = \frac{1}{2 \pi } \int _{-\pi }^\pi v (\tau ) \cot \frac{t-\tau }{2} \mathrm {d}\tau \end{aligned}$$
is the \(2 \pi \)-periodic Hilbert transform; this formula is an alternative to (2).
The crucial point for obtaining a similar relation in the case of finite depth is to introduce the operator \(\mathcal {B}_r = \mathcal {C} + \mathcal {K}_r\) for \(r \in (0, 1)\), where
$$\begin{aligned} (\mathcal {K}_r v) (t) = \frac{2}{\pi } \int _{-\pi }^\pi v (\tau ) K_r (t-\tau ) \, \mathrm {d}\tau \quad \text{ with } \quad \ K_r (t-\tau ) = \sum _{n=1}^\infty \frac{r^{2 n}}{1 - r^{2 n}} \sin (t-\tau ).\nonumber \\ \end{aligned}$$
(23)
It is straightforward to calculate that \(\mathcal {B}_r\) can also be defined on \(L^2 (-\pi , \pi )\) by linearity from relations (4). Combining these formulae and (21) yields that
$$\begin{aligned} x_t = - 1 - \mathcal {B}_r v_t \quad \text{ for } \quad t \in (- \pi , \pi ). \end{aligned}$$
(24)
An important fact about the operator \(\mathcal {B}_r\) is that it is a conjugation in the following sense. If F(u) is analytic in \(A_r\) and \({\text {Im}}F\) vanishes identically on \(\{ |u|=r ; {\text {Re}}u \ne -r \}\), then
$$\begin{aligned} {\text {Re}}F (\mathrm {e}^{\mathrm {i}t}) + [ \mathcal {B}_r ({\text {Im}}F) ] (t) = 0 \quad \text{ for } \text{ all } \quad t \in (- \pi , \pi ) . \end{aligned}$$
(25)
Let us calculate the derivative \(z_\varphi \) of the mapping inverse to the complex potential. In view of the first equality (17), we have
$$\begin{aligned} z_\varphi = z_u \, u_\varphi = - \mathrm {i}z_u \, \mathrm {e}^{-\mathrm {i}w} w_\varphi = - \mathrm {i}u \, z_u. \end{aligned}$$
Combining this and (19), we obtain that
$$\begin{aligned} z_\varphi = 1 + \sum _{k=1}^\infty k a_k \left( u^k + r^{2 k} u^{-k} \right) , \end{aligned}$$
(26)
and the function on the right-hand side is analytic in \(A_r\). Since \(z_\varphi \) does not vanish in the closure of \(A_r\), we have that \(z_\varphi ^{-1} = |\nabla \varphi |^2 \, \overline{z_\varphi }\) is also analytic in \(A_r\). Moreover, the Bernoulli equation (15) implies that
$$\begin{aligned} z_\varphi ^{-1} = (\mu - 2 v) (x_\varphi - \mathrm {i}y_\varphi ) = (\mu - 2 v) (\mathrm {i}v_t - x_t) \quad \text{ when } \quad u = \mathrm {e}^{-\mathrm {i}t} \end{aligned}$$
(27)
(cf. formula (3.38) in [36]). Here, the second equality is a consequence of the Cauchy–Riemann equations. Then, equality (24) yields that
$$\begin{aligned} z_\varphi ^{-1} = (\mu - 2 v) (1 + \mathcal {B}_r v_t + \mathrm {i}v_t) \quad \text{ for } \ t \in (-\pi , \pi ) . \end{aligned}$$
It follows from previous considerations that the constant in the Laurent expansion of \(z_\varphi ^{-1}\) is equal to \(\mu \). Furthermore, \({\text {Im}}\{z_\varphi ^{-1} - \mu \}\) vanishes identically on \(\{ |u|=r ; {\text {Re}}u \ne -r \}\), which allows us to apply formula (25) to the function \(z_\varphi ^{-1} - \mu \), which is equal to
$$\begin{aligned} (\mu - 2 v) \, \mathcal {B}_r (v') - 2 v + \mathrm {i}(\mu - 2 v) v' \quad \text{ when } \ u = \mathrm {e}^{\mathrm {i}t} . \end{aligned}$$
Here, again \('\) stands for differentiation with respect to t. Thus, we arrive at
$$\begin{aligned} (\mu - 2 v) \, \mathcal {B}_r (v') - 2 v + \mathcal {B}_r \, [(\mu - 2 v) v'] = 0 \quad \text{ for } \quad t \in (-\pi , \pi ), \end{aligned}$$
which simplifies to Babenko’s equation for waves on water of finite depth:
$$\begin{aligned} \mu \, \mathcal {B}_r (v') = v + v \, \mathcal {B}_r (v') + \mathcal {B}_r \, (v' v) \quad \text{ for } \quad t \in (-\pi , \pi ) . \end{aligned}$$
(28)
In Sect. 3.2, we show how solutions of this equation define periodic waves.

3.2 Solutions of Babenko’s Equation Define Periodic Waves

Let us outline a procedure for obtaining a solution of problem (12)–(15) from that of Babenko’s equation; that is, if Eq. (28) with \(r \in (0, 1)\) is satisfied by some \(\mu > 0\) and an even function v (the existence of such pairs—at least in the form (41)—follows from Theorem 2), then one can find the following:
  1. (1)

    a \(2 \, \pi \)-periodic symmetric curve with zero mean and a negative number \(-h\), which define the wave profile and the level of horizontal bottom, respectively, thus giving a one-period water domain, say \(\Omega \), in the (xy)-plane;

     
  2. (2)

    a function \(\psi \) harmonic in \(\Omega \) and vanishing on its top side and two positive constants serving as the terms on the right-hand sides of the boundary conditions (13) and (15).

     
Suppose we have an even, \(2 \pi \)-periodic solution v of Eq. (28), whose Fourier coefficients we denote by \(b_0, b_1, b_2, \dots \) to distinguish these coefficients from those in (19), and let the periodic extension of v to \(\mathbb {R}\) be real analytic. Using these coefficients, we define the following holomorphic function on \(A_r\):
$$\begin{aligned} z (u) = \mathrm {i}\left[ \log u - B + \sum _{k=1}^\infty b_k \left( u^k - r^{2 k} u^{-k} \right) \right] . \end{aligned}$$
(29)
Here, B is a real number that will be determined below in terms of the Fourier coefficients of v. Let us consider the images that correspond under this mapping to the curves and segments of \(\partial A_r\). Firstly, we see that \(z (\mathrm {e}^{\mathrm {i}t}) = x (t) + \mathrm {i}y (t)\) for \(t \in (-\pi , \pi )\), where
$$\begin{aligned} x (t) = - t - \sum _{k=1}^\infty b_k \left( 1 + r^{2 k} \right) \sin kt , \quad \ y (t) = - B + \sum _{k=1}^\infty b_k \left( 1 - r^{2 k} \right) \cos kt.\nonumber \\ \end{aligned}$$
(30)
Since this curve given parametrically serves as the upper part of \(\Gamma \), we require its mean value to vanish. This gives that
$$\begin{aligned} B = \frac{1}{2} \sum _{k=1}^\infty k b_k^2 \left( 1 - r^{4 k} \right) , \end{aligned}$$
(31)
where the series converges because the Fourier coefficients of the real-analytic v decay faster than any power of k.
Now we are in a position to determine the mean depth of flow h. In view of symmetry, we have that z(r) is the midpoint of the bottom; that is, \(z (r) = - \mathrm {i}h\). Then putting \(u=r\) into (29), we find that
$$\begin{aligned} h = B - \log r = \frac{1}{2} \sum _{k=1}^\infty k b_k^2 \left( 1 - r^{4 k} \right) - \log r , \end{aligned}$$
(32)
and so h is positive; here the last equality is a consequence of (31). Thus, the second expression (30) takes the following form:
$$\begin{aligned} y (t) = - (h + \log r) + \sum _{k=1}^\infty b_k \left( 1 - r^{2 k} \right) \cos kt . \end{aligned}$$
(33)
Hence, the curve \(z_s = \{ x = x (t) = - t - (\mathcal {B}_r \, y) (t) , \ \ y = y (t) ; \ \ t \in [-\pi , \pi ] \}\) has the zero mean value. Here, the first formula in (4) is applied to express x(t) in terms of y(t).
Furthermore, we have
$$\begin{aligned} z (|u| \mathrm {e}^{\pm \mathrm {i}\pi })= & {} \mp \pi + \mathrm {i}\left[ \log |u| - h + \sum _{k=1}^\infty (-1)^k b_k \left( u^{2 k} - r^{2 k} \right) \! / |u|^{k} \right] \quad \text{ for } \quad u \in [-1, -r] , \end{aligned}$$
(34)
thus obtaining two vertical segments \(z_-\) and \(z_+\) on the lines \(x = -\pi \) and \(x = \pi \), respectively.
Taking into account (31) and (32), we see that
$$\begin{aligned} z (r \mathrm {e}^{\mathrm {i}t}) = - \mathrm {i}h - t - 2 \sum _{k=1}^\infty b_k r^k \sin k t \quad \text{ for } \quad t \in [-\pi , \pi ] \end{aligned}$$
(35)
on the inner circumference. This defines a horizontal segment \(z_b\) on the line \(y = -h\).

It is clear that the curve \(\Gamma = z_+ \cup z_s \cup z_- \cup z_b\) constructed above is closed and one can check (for example, numerically) that the set \(\Omega \) enclosed within \(\Gamma \) is a domain. The next step is to show that z(u), defined with the help of the Fourier coefficients of v, is a conformal mapping of \(A_r\) onto \(\Omega \). For this purpose, one can use the boundary correspondence principle; its form relevant for our case (see, for example, [22], Chapter 5, Theorem 1.3) is formulated for the convenience of the reader.

Theorem 1

(The boundary correspondence principle) Let D and \(D^*\) be two bounded simply connected domains with piecewise smooth boundaries and let f be holomorphic in D and continuous in \(\bar{D}\). If f(p) parametrises \(\partial D^*\) counterclockwise and p is a counterclockwise parametrisation of \(\partial D\), then f is a conformal mapping of D onto \(D^*\).

According to this theorem, z(u) maps \(A_r\) onto \(\Omega \) conformally, provided one can show (for example, numerically) that the map \(\partial A_r \ni u \mapsto z \in \Gamma \) is a homeomorphism. Moreover, condition (11) is fulfilled for \(\eta (x) = y (t (x))\); here t(x) is the inverse of \(x = - t - (\mathcal {B}_r \, y) (t)\), which exists if the curve \(z_s\) is not self-intersecting. Thus, the curve \(y = \eta (x)\) defines the upper side of \(\Omega \).
Fig. 1

A sketch of the annular domain \(A_r\) with several points on its boundary marked in counterclockwise order

Fig. 2

The curve \(\Gamma \) corresponding to \(\partial A_r\) through the mapping z(u) defined by (29) and (31), where the sequence \(\{b_k\}_{k=0}^\infty \) consists of the Fourier coefficients of v. The latter solves (28) with \(r=4/5\) and \(\mu \approx 0.32671\), and \((\mu , v)\) belongs to the branch bifurcating from \(\mu _1\). The marked points on \(\Gamma \) correspond to the numbered points on \(\partial A_r\) in Fig. 1. The mean depth of the one-wave domain \(\Omega \) is \(h \approx 0.22739\), while the wave amplitude is approximately equal to 0.17326

Figures 1, 2, 3, 4 and 5 illustrate how the boundary correspondence principle works numerically in recovering Stokes waves from solutions of (28). We consider the equation with \(r=4/5\) and take the solution \((\mu , v)\) with \(\mu \approx 0.32671\). This solution belongs to the branch bifurcating from \(\mu _1\) (equal to 0.219512195122 for \(r=4/5\)), and the value of \(\mu \) is close to the critical one on this branch (see Figs. 6 and 9). Substituting the Fourier coefficients of v into (29) and (31), we define z(u) which is holomorphic in \(A_r\) and maps \(\overline{A_r}\) onto \({\overline{\Omega }}\) continuously; the latter set is the closure of the prospective one-wave domain. To demonstrate that z(u) is a conformal mapping, we choose several points on \(\partial A_r\), numbering them counterclockwise (see Fig. 1), and calculate their images on \(\Gamma \), assigning to each the same number as the object point has on \(\partial A_r\). It happens that the images are also numbered counterclockwise in agreement with the boundary correspondence principle (see Fig. 2). To be sure that the counterclockwise boundary correspondence is not violated between the chosen points, we provide three additional figures.
Fig. 3

The graph of (36) with \(r=4/5\); its monotonicity confirms that the boundary correspondence is not violated on the bottom part of \(\Gamma \)

Fig. 4

The graph of (37) with \(r=4/5\); its monotonicity confirms that the boundary correspondence is not violated on the right-hand side of \(\Gamma \)

Fig. 5

The graph of the first function in (30) with \(r=4/5\); its monotonicity confirms that the boundary correspondence is not violated on the left-hand half of the upper part of \(\Gamma \)

In Fig. 3, the graph of
$$\begin{aligned} x_h (t) = - t - 2 \sum _{k=1}^\infty b_k r^k \sin k t \end{aligned}$$
(36)
is plotted for \(r=4/5\) and t varying from 0 to \(\pi \) (this parametrises the upper half of the inner circumference clockwise, provided it is considered a part of \(\partial A_r\); see Fig. 1). According to (35), this gives the left-hand half of the bottom shown in Fig. 2, also parametrised clockwise. Since (36) is a monotone function, there is no violation of the boundary correspondence on the bottom, because by symmetry it is sufficient to check this on its right-hand half only.
In Fig. 4, the graph of
$$\begin{aligned} y_+ (u) = \log |u| - h + \sum _{k=1}^\infty (-1)^k b_k \left( u^{2 k} - r^{2 k} \right) \! / |u|^{k} \end{aligned}$$
(37)
is plotted for \(r=4/5\) and u varying from \(-r\) to \(-1\) (this parametrises the lower side of the cut counterclockwise, provided it is considered a part of \(\partial A_r\); see Fig. 1). According to (34), this gives the right-hand side of \(\Gamma \) shown in Fig. 2. Since (37) is a monotone function, there is no violation of the boundary correspondence on the right-hand side of \(\Gamma \).

Finally, the graph of the first function in (30) is plotted in Fig. 5 for \(r=4/5\) and t varying from 0 to \(\pi \) (this parametrises the upper half of the exterior circumference of \(\partial A_r\) counterclockwise; see Fig. 1). According to the first equation (30), this parametrises the left-hand part of the upper side of \(\Gamma \) shown in Fig. 2. Since (30) is a monotone function, there is no violation of the boundary correspondence on this part of \(\Gamma \).

It remains to check that \(\Omega \) is a one-wave domain for some Stokes wave; that is, there exists a stream function \(\psi \) defined on \({\overline{\Omega }}\) that satisfies conditions (13)–(15) with some constant serving as the right-hand side term in (13). For this purpose, we map \(\Omega \) conformally onto an auxiliary rectangle
$$\begin{aligned} R^* = \left\{ (\varphi ^*, \psi ^*) : -\pi< \varphi ^*< \pi , -\psi _0< \psi ^* < 0 \right\} \end{aligned}$$
so that the images of \(z_s\) and \(z_b\) are the top and bottom parts of \(\partial R^*\), respectively, where the value \(\psi _0 > 0\) will be chosen later. Thus, there are harmonic functions \(\varphi ^*\) and \(\psi ^*\) defined on \(\Omega \), and for every \(\psi _0\) the image of \(R^*\) under the mapping \(\mathrm {e}^{-\mathrm {i}(\varphi ^* + \mathrm {i}\psi ^*)}\) is the annular domain \(A_\rho \) with some \(\rho \). It is clear that the value of \(\rho \) decreases from unity to zero as \(\psi _0\) characterising \(R^*\) increases from zero to infinity. Requiring \(\rho \) to be equal to r, we fix the value of \(\psi _0\), thus determining \(\psi ^*\), which in turn gives the constant value \(-Q_*\) that appears on the right-hand side of condition (13); here the sign is chosen so that \(Q_*\) is positive. It should be noted that this procedure guarantees that condition (14) is also fulfilled. It remains to use \(\varphi ^*\) and \(\psi ^*\) for determining \(\psi \) so that it satisfies condition (15) along with (13) and (14).
Using the Fourier coefficients \(b_1, b_2, \dots \) of v in formula (26), we obtain a function \(z_{\varphi ^*}\) which is holomorphic in \(A_r\) and nonvanishing there. According to equation (28), we have that
$$\begin{aligned} \left[ \{1 - 2 \mu ^{-1} y (u)\} \overline{z_{\varphi ^*} (u)} - 1 \right] _{|u|=1} \end{aligned}$$
is the limit as \(|u| \rightarrow 1\) of some holomorphic function given on \(A_r\) whose imaginary part is equal to zero on \(\partial A_r \cap \{ |u| = r \}\). Moreover, the same limit property holds for \(z_{\varphi ^*}\) itself, and so it is also true for the function whose limit as \(|u| \rightarrow 1\) is equal to
$$\begin{aligned} \left[ \{1 - 2 \mu ^{-1} y (u)\} |z_{\varphi ^*} (u)|^2 \right] _{|u|=1} . \end{aligned}$$
Therefore, we have that
$$\begin{aligned} 1 - 2 \mu ^{-1} \eta (x) = q^2 |\nabla \psi ^* (x, \eta (x))|^2 , \quad x \in (-\pi , \pi ) , \end{aligned}$$
with some \(q > 0\) and \(\eta \) defined above. For \(\psi = q \sqrt{\mu }\, \psi ^*\) the last relation coincides with (15).

Thus, the triple \((\mu , \eta , \psi )\) satisfies problem P\((Q_0,h)\) with h defined by (32), where \(Q_0 = q \sqrt{\mu }\, Q_*\) and \(Q_*\) depends on r implicitly as described above. This completes the description of the procedure for obtaining a solution of problem (12)–(15) from the given solution of Babenko’s equation.

3.3 Local Branches of Solutions of Babenko’s Equation

To show the existence of small solutions of Eq. (28), bifurcating from the zero solution, we apply the Crandall–Rabinowitz theory in the form presented in [14], Section 8.4, and dealing with bifurcation from simple eigenvalues of the linearised equation (see [21] for the original formulation).

Theorem 2

Let \({{\mathcal {X}}}\), \({{\mathcal {Y}}}\) be real Banach spaces with the continuous embedding \({{\mathcal {X}}} \subset {{\mathcal {Y}}}\), and let \(U \subset \mathbb {R}\times {{\mathcal {X}}}\) be open and such that \(\{ (\mu , 0) : \mu \in \mathbb {R}\} \subset U\). If a map \({{\mathcal {F}}} (\mu , v): U \mapsto {{\mathcal {Y}}}\) is analytic on U with
$$\begin{aligned} {{\mathcal {F}}} (\mu , 0) = 0 \quad and \quad {{\mathcal {F}}}_v (\mu , 0) = \mu {{\mathcal {I}}} - A \quad for \ all \ \mu \in \mathbb {R}, \end{aligned}$$
where \({{\mathcal {I}}}\) is the embedding operator, then every simple eigenvalue \(\mu ^*\) of A is a bifurcation point. Moreover, there exist \(\varepsilon > 0\) and an analytic branch of solutions
$$\begin{aligned} \{ (\mu (s), \, v (s)) : |s| < \varepsilon \} \subset \mathbb {R}\times {{\mathcal {X}}} \end{aligned}$$
bifurcating from \((\mu ^*, 0)\). For pairs belonging to this curve,
$$\begin{aligned} \mu (s) = \mu ^* + o (s) \quad and \quad v (s) = s \, v^{*} + o (s) \quad when \ 0< |s| < \varepsilon . \end{aligned}$$
Here, \(v^{*}\) is the element generating the null space of \(\mathcal{F}_v (\mu ^*, 0)\).

A real-valued function v belongs to the Sobolev space \(H_\mathrm{per}^1 (-\pi , \pi )\), provided it is absolutely continuous on \([-\pi , \pi ]\), \(v (-\pi ) = v (\pi )\), and its weak derivative \(v'\) belongs to \(L^2_\mathrm{per} (-\pi , \pi )\). Let \({\hat{H}}_\mathrm{per}^1 (-\pi , \pi )\) be the subspace of \(H_\mathrm{per}^1 (-\pi , \pi )\) consisting of even functions.

In terms of the map \({{\mathcal {F}}}: \mathbb {R}\times {\hat{H}}_\mathrm{per}^1 (-\pi , \pi ) \rightarrow {\hat{L}}^2_\mathrm{per} (-\pi , \pi )\) (the latter space consists of even functions) defined by
$$\begin{aligned} {{\mathcal {F}}} (\mu , v) = \mu \mathcal {B}_r (v') - v + v \, \mathcal {B}_r (v') + \mathcal {B}_r \, (v' v) , \end{aligned}$$
(38)
equation (28) takes the following form:
$$\begin{aligned} {{\mathcal {F}}} (\mu , v) = 0 , \quad (\mu , v) \in \mathbb {R}\times {\hat{H}}_\mathrm{per}^1(-\pi ,\pi ) . \end{aligned}$$
(39)
To apply Theorem 2 to this equation, thus obtaining local branches of Stokes-wave solutions of small amplitude, we just notice that \({{\mathcal {F}}} (\mu , v)\) is analytic because its partial Fréchet derivatives vanish identically for all orders greater than 3.
Furthermore, \({{\mathcal {F}}}_{v} (\mu , 0) = \mu \mathcal {B}_r \, (\mathrm {d}/ \mathrm {d}t) - I\), where I is the identity operator. All characteristic values of \(\mathcal {B}_r \, (\mathrm {d}/ \mathrm {d}t)\) are simple, being equal to
$$\begin{aligned} \mu _n = \frac{1 - r^{2n}}{n (1 + r^{2n})} , \quad n=1,2,\dots , \end{aligned}$$
(40)
and having \(v_n (t) = \cos n t\) as the corresponding eigenfunctions. Then, Theorem 2 yields the following.

Theorem 3

For every \(n=1,2,\dots \), there exists \(\varepsilon _n > 0\) such that for \(0< |s| < \varepsilon _n\) there is a family \(\big ( \mu _n^{(s)}, \, v_n^{(s)} \big )\) of solutions to Eq. (39). Together with the bifurcation point \((\mu _n , 0)\), where \(\mu _n\) is given by formula (40), the points of this family form an analytic curve
$$\begin{aligned} C_n = \big \{ \big ( \mu _n^{(s)}, \, v_n^{(s)} (t) \big ) : |s| < \varepsilon _n \big \} \subset \mathbb {R}\times {\hat{H}}_\mathrm{per}^1 , \quad n=1,2,\dots . \end{aligned}$$
Moreover, the asymptotic formulae
$$\begin{aligned} \mu _n^{(s)} = \mu _n + o (s) , \quad v_n^{(s)} (t) = s \cos n t + o (s) \end{aligned}$$
(41)
hold for these solutions as \(|s| \rightarrow 0\).
Fig. 6

The branch of solutions of Eq. (28) with \(r=4/5\), bifurcating from the zero solution at \(\mu _1 (4/5) = 0.219512195122\). The upper bound mentioned prior to Definition 1 is also included

Theorem 3 is illustrated in Fig. 6, where we have a plot of the bifurcation branch \(C_1\) in terms of \(\mu \) and the norm of solution \(\Vert v\Vert _\infty \) in the space \(L^\infty (-\pi , \pi )\). The plotted branch bifurcating from \(\mu _1 (4/5)\) has no secondary bifurcation points like those on the analogous branch for Eq. (1); see [12, 13] for a rigorous proof and detailed discussion. Moreover, it exhibits the phenomenon of a turning point at the largest value of \(\mu \) attained on \(C_1\), occurring high on the branch; see further details in Sect. 4.3. (The fastest traveling wave of given period corresponds to this point.) By means of a different method, this property was demonstrated in [19], whereas our method shows that it also takes place for equation (1) on the branch bifurcating from \(\mu _1 (0)\). This phenomenon is related to the ‘Tanaka instability’ found numerically in [43], and later investigated analytically in [39].

4 Numerical Solution of Babenko’s Equation

In this section, we describe a numerical method for solving Eq. (28) in the class of even periodic functions on \((-\pi , \pi )\). The existence of small solutions of this kind is proved in Sect. 3.3, whereas general solutions are discussed in Sect. 5. The essence of our method is to calculate the Fourier coefficients \(b_0, b_1, \ldots \) of the solution, which allows us to recover the conformal mapping z(u) (see Sect. 3.2), thus demonstrating numerically the equivalence of Babenko’s equation and problem P\((Q_0,h)\).

4.1 Transformation of (28) to a Form Convenient for Discretization

Let \(r \in [0, 1)\) be fixed. Then \(J_r = {\mathcal {B}}_r \mathrm {d}/ \mathrm {d}t\) is a self-adjoint operator on \(L^2_\mathrm{per} (-\pi , \pi )\) of \(2 \pi \) periodic square integrable functions. Its domain is \(H_\mathrm{per}^1 (-\pi , \pi )\) (see Sect. 3.3 for the definition), and it can also be defined by linearity from \(J_r \cos n t = \lambda _n \cos nt\) for \(n=0,1,\dots \) and \(J_r \sin n t = \lambda _n \sin nt\) for \(n=1,2,\dots \); here, the eigenvalues are \(\lambda _n = \mu _n^{-1}\) for \(n \ge 1\) and \(\lambda _0 = 0\); cf. (40). Since the corresponding eigenfunctions form a basis in \(L^2_\mathrm{per} (-\pi , \pi )\), the following spectral decomposition holds:
$$\begin{aligned} J_r = \sum _{n = 1}^{\infty } \lambda _n ( {\hat{P}}_n + {\tilde{P}}_n ) . \end{aligned}$$
(42)
Here, \({\hat{P}}_n\)\(({\tilde{P}}_n)\) is the projector onto the subspace spanned by \(\cos nt\) (\(\sin nt\), respectively).
Seeking solutions of (28) in \({\hat{H}}_\mathrm{per}^1 (-\pi , \pi )\), it is convenient to write the equation in an equivalent form to accelerate numerical calculations. This form arises after replacing \(J_r\) in (28) by the right-hand side of (42) with \({\tilde{P}}_n\) omitted, which is possible in view of the bijection between \({\hat{H}}_\mathrm{per}^1 (-\pi , \pi )\) and the Sobolev space \(H^1 (0, \pi )\); indeed, for every \(w \in H^1 (0, \pi )\) its even extension v belongs to \({\hat{H}}_\mathrm{per}^1 (-\pi , \pi )\) and vice versa. Therefore, it is convenient to set \({\mathcal {J}}_r = \sum _{n = 1}^{\infty } \lambda _n P_n\), where \(P_n\) is the projector onto the subspace of \(L^2 (0, \pi )\) spanned by \(\cos nt\). Then, \({\mathcal {J}}_r\) is defined on \(H^1 (0, \pi )\) and \({\mathcal {J}}_r w = J_r v (= {\mathcal {B}}_r v')\) almost everywhere on \((0, \pi )\), and so (28) takes the equivalent form
$$\begin{aligned} \mu {\mathcal {J}}_r w = w + w {\mathcal {J}}_r w + \frac{1}{2} {\mathcal {J}}_r (w^2) , \quad t \in (0, \pi ) , \end{aligned}$$
(43)
where w(t) is sought in \(H^1 (0, \pi )\). After reducing the interval to \((0, \pi )\), only half as many Fourier harmonics are needed, which significantly accelerates the computations in solving Eq. (43) numerically. For this purpose, a modified version of the software SpecTraVVave is applicable; the latter is available freely at the site indicated in [32], while its detailed description can be found in [23].
For a reason explained below, we amend (43) further; namely, we set \(\mu _0 = 1\) and set \({\mathcal {L}}_r = \sum _{n = 0}^{\infty } \mu _n P_n\). Hence, \({\mathcal {L}}_r\) is invertible and \({{\mathcal {L}}_r}^{-1} = P_0 + {\mathcal {J}}_r\); that is, \({\mathcal {L}}_r {\mathcal {J}}_r = I - P_0\), where I is the identity operator. Applying \({\mathcal {L}}_r\) to both sides of (43), we obtain the following equation:
$$\begin{aligned} \mu (I - P_0) w = {\mathcal {L}}_r w + {\mathcal {L}}_r ( w {\mathcal {J}}_r w ) + \frac{1}{2} (I - P_0) w^2 , \quad t \in (0, \pi ) . \end{aligned}$$
(44)
It should be noted that the unbounded operator \({\mathcal {J}}_r\) is present in the nonlinear part of the last equation only, and so one can expect that (44) should demonstrate better numerical stability. Finally, Eqs. (44) and (28) are equivalent in the following sense. The sets \(\{ b_n (w) \}_{n=0}^\infty \) and \(\{ b_n (v) \}_{n=0}^\infty \) of the Fourier coefficients coincide for solutions of (44) and (28), respectively, provided the value of \(\mu \) is the same for both solutions.

For Eq. (44), the existence of small solutions follows from its equivalence to (28). It can also be established directly with the help of the Crandall–Rabinowitz theorem, see Sect. 3.3, which yields the asymptotic formulae (41) for the branch of solutions of (44) bifurcating from \(\mu _n\) and trivial w. This can serve as an initial guess in the numerical procedure.

4.2 Discretization of Equation (44)

We use the standard cosine collocation method, according to which solutions of (44) are sought in the form of linear combinations of \(\cos mx\), \(m = 0, 1, \dots \)—a basis in \(L^2(0, \pi )\). For the discretization, the subspace \(\mathcal S_N\) spanned by the first N cosines is used, which is defined by their values at the collocation points \(x_n = \pi \frac{2n - 1}{2N}\) for \(n = 1, \ldots , N\). Thus, for any \(f \in H^1 (0, \pi )\) the vector \(f^N\) given by its coordinates
$$\begin{aligned} f^N_n = \sum _{k=0}^{N-1} (P_k f) (x_n) , \quad n = 1, \dots , N , \end{aligned}$$
is considered. The operator \({\mathcal {L}}_r^N\), discretizing \(\mathcal L_r\), is defined as follows:
$$\begin{aligned} ( {\mathcal {L}}_r^N f^N )_n = \sum _{k=0}^{N-1} (P_k {\mathcal {L}}_r f) (x_n) , \quad n = 1, \dots , N . \end{aligned}$$
Furthermore, \({\mathcal {J}}_r^N\) and \(P_0^N\) are introduced as the discretizations of \({\mathcal {J}}_r\) and \(P_0\), respectively.
These definitions are correct because \(f^N\) defines the function f with values \(f(x_n) = f^N_n\) uniquely up to a projection on the subspace orthogonal to \({\mathcal {S}}_N\). It is clear that each of these discrete operators is a composition of the discrete cosine transform, some diagonal matrix and the inverse discrete cosine transform. The diagonal matrix for \({\mathcal {L}}_r^N\) is \(\{ 1, \ldots , \mu _{N-1} \}\), whereas the diagonal matrix for \({\mathcal {J}}_r^N\) is \(\{ 0, \lambda _1, \ldots , \lambda _{N-1} \}\), and \(\{ 1, 0, \ldots , 0 \}\) is the diagonal matrix for \(P_0^N\). The discrete analogue of (44) is as follows:
$$\begin{aligned} {\mathcal {L}}_r^N w^N - \mu \left( I - P_0^N \right) w^N + \mathcal L_r^N \left( w^N {\mathcal {J}}_r^N w^N \right) + \frac{1}{2} \left( I - P_0^N \right) \left( w^N \right) ^2 = 0 . \end{aligned}$$
(45)
Solving this equation, one finds that that solutions \((\mu , w^N)\) form curves in an auxiliary \((\mu , a)\)-plane, where
$$\begin{aligned} a = \Vert w^N \Vert = \max _n |w^N_n| . \end{aligned}$$
Therefore, it is convenient to parametrise these curves in order to make computations more effective. Due to the presence of a parameter, say \(\theta \), we have \(\mu = \mu (\theta )\) and \(a = a (\theta )\) on each curve of solutions. Substituting \(\mu (\theta )\) into (45) instead of \(\mu \), we complement this algebraic system by the equation
$$\begin{aligned} \max _{n = 1, \ldots , N} |w^N_n| = a(\theta ) . \end{aligned}$$
(46)
The resulting system (45), (46) has \(N + 1\) equations for the unknowns \(\theta , w^N_1, \ldots , w^N_N\).
Fig. 7

The rate of decay for the computed Fourier coefficients of the endpoint solution on the branch \(C_{31}\) with \(r=4/5\) (see Fig. 10 for this branch and Fig. 12 for the wave profile corresponding to this solution). The oscillations at the left end are due to the fact that the wave profile has two equal steep but smooth crests per wavelength complementing the angular peak in the middle

4.3 On Numerical Resolution and Accuracy

The Newton iteration method is applied for computing solutions of system (45), (46). Firstly, we find a solution bifurcating from the trivial one, and the Crandall–Rabinowitz asymptotic formula (41) is used for determining an initial approximation of the solution. Then, solutions that have larger values of \(\Vert w \Vert _{\infty } = \Vert v \Vert _{\infty }\) are computed along the parametrised bifurcation curve. Since the maximum of the free surface elevation is attained at \(x = 0\), which corresponds to \(t = 0\), the norm is equal to
$$\begin{aligned} v(0) = w(0) = \sum _{n=0}^{\infty } a_n (w) , \end{aligned}$$
where the initial N Fourier coefficients are calculated via the discrete cosine Fourier transform, whereas the rest are taken equal to zero; that is, \(a_{N} = a_{N+1} = \cdots = 0\). This allows us to compute the norm of the solution with high accuracy, provided N is sufficiently large. We illustrate this by plotting how the Fourier coefficients decay with N for the endpoint solution on the branch \(C_{31}\) computed for \(r=4/5\); see Fig. 7. This branch presented in Figs. 10 and 12 gives the wave profile corresponding to this particular solution.

Taking into account that the coefficients decay exponentially, it is reasonable to suggest that \(\sum _{n \ge N} |a_n|< |a_{N-1}| < 10^{-8}\), and so we take \(N = 2048\), which corresponds to 4096 modes per wavelength. Furthermore, the tolerance used in our computations is \(10^{-11}\); that is, to compute the coefficients \(a_k\), \(k = 0, \ldots , N-1\), and \(\mu \) we repeat Newton’s procedure until \(|a_k^{(i)} - a_k^{(i-1)}| < 10^{-11}\) for two successive iterations. Thus, the tolerance and N used in this example give the solution correct to eight digits after the decimal point, and within this accuracy, the obtained solution is indistinguishable from that of extreme form; indeed, we have \(\mu \approx 0.24984594\), which differs from the limiting value by less than \(10^{-7}\). Therefore, our choice of the tolerance and N seems optimal because it yields a balance between accuracy and computation time. Indeed, taking \(N = 1024\), one obtains a solution that is correct to only five digits after the decimal point, whereas the computation time increases unreasonably for \(N = 4096\).

More details about the realisation of the algorithm and, in particular, about the specific parametrisation of bifurcation curves used in the process of computing them can be found in [23]. The advantages of the algorithm are as follows: it is easy to implement and runs very fast. Of course, more iterations or/and more Fourier modes (larger N) are needed to achieve the required accuracy for solutions close to extreme ones. Another substantial element of our approach is that we rely on the ‘trial and error’ method; it is hard to exclude it in dealing with the location of turning points on bifurcation curves and secondary bifurcation points.

It was mentioned above that a modified version of the software SpecTraVVave [32] was used. Originally, the available software was designed to treat equations of the form
$$\begin{aligned} \mu w = {\mathcal {L}}_r w + f (w) + B , \end{aligned}$$
where the nonlinear term f(w) is local, while B is a constant obtained by normalization of a solution; see [23]. Thus, we had to make some changes to treat equation (44). Moreover, tools for handling subharmonic bifurcations and following global bifurcation branches were added; the modified software is available from the authors.
Fig. 8

The solution branches \(C_2\), \(C_3\) and \(C_4\) for Eq. (28) with \(r=0\), bifurcating from the zero solution at \(\mu _2 (0) = 1/2\), \(\mu _3 (0) = 1/3\) and \(\mu _4 (0) = 1/4\), respectively. The secondary solution branches are denoted by \(C_{21}\), \(C_{31}\) and \(C_{41}\), respectively. The upper bound mentioned prior to Definition 1 is also included

Fig. 9

The solution branch \(C_1\) for Eq. (28) with \(r=4/5\) in a vicinity of the turning point, whose characteristics are as follows: \(\mu \approx 0.32671\) and the \(L^\infty \)-norm of the solution is approximately equal to 0.15862. The bold dot marks the solution plotted in Fig. 2. The upper bound mentioned prior to Definition 1 is also included

4.4 Bifurcation Curves for Equation (44)

We begin with the results of a test calculation in which the algorithm described in Sect. 4.2 is applied to Eq. (44) with \(r = 0\), thus giving bifurcation curves for Eq. (1). The curves plotted in Fig. 8 show the bifurcations from a trivial solution and the first three secondary bifurcations for this case; the curve \(C_1\) is omitted because its behaviour is similar to that presented in Fig. 6, including the presence of a turning point. The secondary bifurcation branches \(C_{21}\), \(C_{31}\) and \(C_{41}\) bifurcate from \(C_2\), \(C_3\) and \(C_4\), respectively, at the points where \(\mu \) is approximately equal to 0.58768, 0.39172 and 0.29389, respectively. These values are in good agreement with those presented by Aston [3]; see Table 1 in his paper.

Now, we turn to numerical results obtained for Eq. (28) with \(r=4/5\). The solution branch \(C_1\) is presented in Fig. 6, and some of its characteristics are described below Theorem 3. In particular, it is pointed out that it has a turning point, and so we give an enlarged plot of the curve \(C_1\) in a vicinity of this point; see Fig. 9, where a bold dot marks one of two solutions corresponding to \(\mu \approx 0.32671\). The wave profile corresponding to this solution is plotted in Fig. 2, where some of its characteristics are given; moreover, its \(L^\infty \)-norm is approximately equal to 0.15862.
Fig. 10

The branch \(C_3\) of solutions of Eq. (28) with \(r=4/5\), bifurcating from the zero solution at \(\mu _3 (4/5) = 0.194868414381\). The secondary solution branch \(C_{31}\) bifurcates from \(C_3\) at \(\mu \approx 0.25298\). The dots mark those solutions on \(C_3\) and \(C_{31}\) whose wave profiles are plotted in Figs. 11 and 12, respectively. The upper bound mentioned prior to Definition 1 is also included

The last example concerns the solution branch \(C_3\) for Eq. (28) with \(r=4/5\). It is presented in Fig. 10, where one observes the presence of a turning point as well as the secondary bifurcation. Indeed, the branch \(C_{31}\) bifurcates from \(C_3\) at the point where \(\mu \) is approximately equal to 0.25298, and shortly after that \(C_3\) has its turning point. The algorithm proposed in Sect. 4.2 allows us to solve (44) up to both critical values on \(C_3\) and \(C_{31}\); see Figs. 11 and 12, respectively, for the plots of wave profiles corresponding to these solutions.
Fig. 11

The wave profile of the extreme form corresponding to the endpoint solution on the branch \(C_3\) for Eq. (28) with \(r=4/5\). The characteristics of this wave are as follows: \(\mu \approx 0.25175\); the profile’s crests (troughs) are at \(y = y_c \approx 0.12777\) (\(y = y_t \approx -0.03312\), respectively)

Fig. 12

The wave profile of the extreme form corresponds to the endpoint solution on the branch \(C_{31}\) for Eq. (28) with \(r=4/5\). Its characteristics are as follows: \(\mu \approx 0.24985\) the profile’s smooth crests (troughs) are at \(y = \tilde{y}_c \approx 0.10406\) (\(y = y_t \approx -0.03310\), respectively), whereas the peaks are at \(y = {\hat{y}}_c \approx 0.12608\)

In Fig. 11, the wave profile corresponds to the endpoint solution on the branch \(C_3\) (\(\mu \approx 0.25175\) for this solution of Eq. (28) with \(r=4/5\)). Like a small-amplitude wave characterised by the second formula (41), this profile has wavelength \(2 \pi / 3\), and so three wave periods are plotted. Moreover, this Stokes wave has the extreme form; that is, the tangents to two smooth arcs form the angle \(2 \pi / 3\) at every crest. The tangency is demonstrated with sufficient accuracy in the figure, where the angle inscribed into the wave profile has sides \(y = y_c \pm x / \sqrt{3}\) with \(y_c = y (0)\); see (33) for y(t) and the first formula in (30) for x(t), which describe the profile parametrically. Of course, the same tangency occurs at every crest.

In Fig. 12, the wave profile corresponds to the endpoint solution on the branch \(C_{31}\) (\(\mu \approx 0.0.24985\) for this solution of Eq. (28) with \(r=4/5\)). The profile has wavelength \(2 \pi \), and so two wave periods are plotted. Thus, period tripling occurs as \(C_{31}\) bifurcates from the branch \(C_3\); an analogous effect is described in [50] for waves on infinitely deep water (see, in particular, Fig. 3 on p. 25 of that paper). Moreover, the wave is symmetric with respect to the vertical through the highest, midperiod crest. The latter has the extreme form like each crest in Fig. 11, whereas the wave profile is smooth at the other two crests per period.

5 Concluding Remarks

We have considered the nonlinear problem describing steady gravity waves on water of finite depth. This problem is reduced to a single pseudodifferential operator Eq. (28) (Babenko’s equation), which generalises the well-known Eq. (1) describing waves on infinitely deep water. Local bifurcation is investigated analytically with the help of the Crandall–Rabinowitz theorem, while a combination of analytical and numerical methods is applied to demonstrate that the initial, free-boundary problem and Babenko’s equation are equivalent in the following sense. For every solution of the initial problem one of its components, namely, the free-surface elevation, is a solution of Babenko’s equation for some value of its parameter; this value is determined by the solution of the free-boundary problem. In contrast, every solution of Babenko’s equation defines a solution of some free-boundary problem through a certain procedure.

Also, we outline an algorithm that allows us to solve Babenko’s equation numerically using a modification of the free software SpecTraVVave; see [32]. It should be emphasised that the developed numerical procedure is not only very fast, but remarkable for its high accuracy. The latter is essential in computing solutions corresponding to extreme wave profiles, thus allowing us to plot the global bifurcation branches presented in Sect. 4.3.

This paper is just an initial step in studies of Babenko’s equation both analytically and numerically. Firstly, it is desirable to prove rigorously that every solution of Babenko’s equation defines a solution of the free-boundary problem that describes steady waves on a flow of finite depth with certain characteristics. Secondly, it is natural to show that the profiles of waves below the highest (which has the extreme form, being nonsmooth at its highest point) are real analytic curves. Thirdly, one has to demonstrate the absence of subharmonic bifurcations in a neighbourhood of every point where bifurcation from the zero solution occurs. Finally, a global Stokes-wave theory should be developed and used for proving that there exist subharmonic bifurcations on branches of smooth waves close to the highest wave. All these results have been established for waves on infinitely deep water on the basis of Eq. (1); see [12, 13].

An interesting direction for further numerical investigations is to find higher bifurcations that might exist for waves on water of finite depth, as happens in the case of deep water, as was shown in [3], where just several isolated points of higher bifurcations are listed in Table 1. Since the algorithm based on Eq. (28) and realised using the software SpecTraVVave is a rather robust tool, one could apply it for calculating branching bifurcation curves that have more than one bifurcation point.

Notes

Acknowledgements

The authors are grateful to Henrik Kalisch, without whose support the paper would not appear. E. D. acknowledges the support from the Norwegian Research Council. Numerous comments by the referees are gratefully acknowledged; the presentation is substantially improved due to them.

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Authors and Affiliations

  1. 1.Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussian Federation
  2. 2.Department of MathematicsUniversity of BergenBergenNorway

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