Traveling quasi-periodic water waves with constant vorticity

We prove the first bifurcation result of time quasi-periodic traveling waves solutions for space periodic water waves with vorticity. In particular we prove existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restricting the surface tension to a Borel set of asymptotically full Lebesgue measure.


Introduction and main result
The search for traveling surface waves in inviscid fluids is a very important problem in fluid mechanics, widely studied since the pioneering work of Stokes [38] in 1847. The existence of steady traveling waves, namely solutions which look stationary in a moving frame, either periodic or localized in space, is nowadays well understood in many different situations, mainly for bidimensional fluids.
On the other hand, the natural question regarding the existence of time quasi-periodic traveling waves -which can not be reduced to steady solutions in a moving frame-has been not answered so far. This is the goal of the present paper. We consider space periodic waves. Major difficulties in this project concern the presence of 'small divisors" and the quasi-linear nature of the equations. Related difficulties appear in the search of time periodic standing waves which have been constructed in the last years in a series of papers by Iooss, Plotnikov, Toland [34,25,22,23] for pure gravity waves, by Alazard-Baldi [1] in presence of surface tension and subsequently extended to time quasi-periodic standing waves solutions by Berti-Montalto [6] and Baldi-Berti-Haus-Montalto [2]. Standing waves are not traveling as they are even in the space variable. We also mention that all these last results concern irrotational fluids.
In this paper we prove the first existence result of time quasi-periodic traveling wave solutions for the gravity-capillary water waves equations with constant vorticity for bidimensional fluids. The small amplitude solutions that we construct exist for any value of the vorticity (so also for irrotational fluids), any value of the gravity and depth of the fluid, and provided the surface tension is restricted to a Borel set of asymptotically full measure, see Theorem 1.5. For irrotational fluids the traveling wave solutions that we construct do not clearly reduce to the standing wave solutions in [6]. We remark that, in case of non zero vorticity, one can not expect the bifurcation of standing waves since they are not allowed by the linear theory.
Before presenting in detail our main result, we introduce the water waves equations.
The water waves equations. We consider the Euler equations of hydrodynamics for a 2dimensional perfect, incompressible, inviscid fluid with constant vorticity γ, under the action of gravity and capillary forces at the free surface. The fluid fills an ocean with depth h ą 0 (eventually infinite) and with space periodic boundary conditions, namely it occupies the region D η,h :" px, yq P TˆR :´h ď y ă ηpt, xq ( , T :" T x :" R{p2πZq . (1.1) fluid with constant vorticity v x´uy " γ , the velocity field is the sum of the Couette flowˆ´γ y 0˙, which carries all the vorticity γ of the fluid, and an irrotational field, expressed as the gradient of a harmonic function Φ, called the generalized velocity potential. Denoting by ψpt, xq the evaluation of the generalized velocity potential at the free interface ψpt, xq :" Φpt, x, ηpt, xqq, one recovers Φ by solving the elliptic problem ∆Φ " 0 in D η,h , Φ " ψ at y " ηpt, xq , Φ y Ñ 0 as y Ñ´h . (1. 2) The third condition in (1.2) means the impermeability property of the bottom Imposing that the fluid particles at the free surface remain on it along the evolution (kinematic boundary condition), and that the pressure of the fluid plus the capillary forces at the free surface is equal to the constant atmospheric pressure (dynamic boundary condition), the time evolution of the fluid is determined by the following system of equations (see [8,42]) Here g is the gravity, κ is the surface tension coefficient, which we assume to belong to an interval rκ 1 , κ 2 s with κ 1 ą 0, and Gpηq is the Dirichlet-Neumann operator Gpηqψ :" Gpη, hqψ :" a 1`η 2 x pB n Φq| y"ηpxq " p´Φ x η x`Φy q| y"ηpxq . (1.4) The water waves equations (1.3) are a Hamiltonian system that we describe in Section 2.1, and enjoy two important symmetries. First, they are time reversible: we say that a solution of (1.3) is reversible if ηp´t,´xq " ηpt, xq , ψp´t,´xq "´ψpt, xq . (1.5) Second, since the bottom of the fluid domain is flat, the equations (1.3) are invariant by space translations. We refer to Section 2.1 for more details.
Let us comment shortly about the phase space of (1.3). As Gpηqψ is a function with zero average, the quantity ş T ηpxq dx is a prime integral of (1.3). Thus, with no loss of generality, we restrict to interfaces with zero spatial average ş T ηpxq dx " 0. Moreover, since Gpηqr1s " 0, the vector field on the right hand side of (1.4) depends only on η and ψ´1 2π ş T ψ dx. As a consequence, the variables pη, ψq of system (1.3) belong to some Sobolev space H s 0 pTqˆ9 H s pTq for some s large. Here H s 0 pTq, s P R, denotes the Sobolev space of functions with zero average H s 0 pTq :" ! u P H s pTq : and 9 H s pTq, s P R, the corresponding homogeneous Sobolev space, namely the quotient space obtained by identifying all the H s pTq functions which differ only by a constant. For simplicity of notation we shall denote the equivalent class rψs " tψ`c, c P Ru, just by ψ.
Linear water waves. When looking to small amplitude solutions of (1.3), a fundamental role is played by the system obtained linearizing (1.3) at the equilibrium pη, ψq " p0, 0q, namely # B t η " Gp0qψ B t ψ "´pg´κB 2 x qη`γB´1 x Gp0qψ . (1.6) The Dirichlet-Neumann operator at the flat surface η " 0 is the Fourier multiplier with symbol (1.8) As we will show in Section 2.2, all reversible solutions (see (1.5)) of (1.6) arê ηpt, xq ψpt, xq˙" ÿ nPNˆM n ρ n cospnx´Ω n pκqtq P n ρ n sinpnx´Ω n pκqtqÿ nPNˆM n ρ´n cospnx`Ω´npκqtq P´nρ´n sinpnx`Ω´npκqtq˙, (1.9) where ρ n ě 0 are arbitrary amplitudes and M n and P˘n are the real coefficients Gj p0q j 2¸1 4 , j P Zzt0u , P˘n :" γ 2 M n n˘M´1 n , n P N . (1.10) Note that the map j Þ Ñ M j is even. The frequencies Ω˘npκq in (1.9) are Ω j pκq :" d´κ j 2`g`γ 2 4 G j p0q j 2¯G j p0q`γ 2 G j p0q j , j P Zzt0u . (1.11) Note that the map j Þ Ñ Ω j pκq is not even due to the vorticity term γG j p0q{j, which is odd in j. Note that Ω j pκq actually depends also on the depth h, the gravity g and the vorticity γ, but we highlight in (1.11) only its dependence with respect to the surface tension coefficient κ, since in this paper we shall move just κ as a parameter to impose suitable non-resonance conditions, see Theorem 1.5. Other choices are possible. All the linear solutions (1.9), depending on the irrationality properties of the frequencies Ω˘npκq and the number of non zero amplitudes ρ˘n ą 0, are either time periodic, quasi-periodic or almost-periodic. Note that the functions (1.9) are the linear superposition of plane waves traveling either to the right or to the left. Remark 1.1. Actually, (1.9) contains also standing waves, for example when the vorticity γ " 0 (which implies Ω´npκq " Ω n pκq, P´n "´P n ) and ρ´n " ρ n , giving solutions even in x. This is the well known superposition effect of waves with the same amplitude, frequency and wavelength traveling in opposite directions.
(1.12) Remark 1.3. If ν " 1, such functions are time periodic and indeed stationary in a moving frame with speed ω 1 {j 1 . On the other hand, if the number of frequencies ν is ě 2, the waves (1.12) cannot be reduced to steady waves by any appropriate choice of the moving frame.
Let us make some comments. 1) Theorem 1.5 holds for any value of the vorticity γ, so in particular it guarantees existence of quasi-periodic traveling waves also for irrotational fluids, i.e. γ " 0. In this case the solutions (1.19) do not reduce to those in [6], which are standing, i.e. even in x. If the vorticity γ ‰ 0, one does not expect the existence of standing wave solutions since the water waves vector field (1.3) does not leave invariant the subspace of functions even in x.
2) Theorem 1.5 produces time quasi-periodic solutions of the Euler equation with a velocity field which is a small perturbation of the Couette flowˆ´γ y 0˙. Indeed, from the solution pηpt, xq, ψpt, xqq in (1.19), one recovers the generalized velocity potential Φpt, x, yq by solving the elliptic problem (1.2) and finally constructs the velocity fieldˆu pt, x, yq vpt, x, yq˙"ˆ´γ y 0˙`∇ Φpt, x, yq.
The time quasi-periodic potential Φpt, x, yq has size Op a |ξ|q, as ηpt, xq and ψpt, xq. 3) In the case ν " 1 the solutions constructed in Theorem 1.5 reduce to steady periodic traveling waves, which can be obtained by an application of the Crandall-Rabinowitz theorem, see e.g. [30,41,43]. 4) Theorem 1.5 selects initial data giving raise to global in time solutions (1.19) of the water waves equations (1.3). So far, no results about global existence for (1.3) with periodic boundary conditions are known. The available results concern local well posedness with a general vorticity, see e.g. [10], and a ε´2 existence for initial data of size ε in the case of constant vorticity [21]. 5) With the choice (1.15)-(1.16) the unperturbed frequency vector Ωpκq " pΩ σana pκqq a"1,...,ν is diophantine for most values of the surface tension κ and for all values of vorticity, gravity and depth. It follows by the more general results of Sections 4 and 5.2. This may not be true for an arbitrary choice of the linear frequencies Ω j pκq, j P Zzt0u. For example, in the case h "`8, the vector Ωpκq "`Ω´n 3 pκq, Ω´n 2 pκq, Ω´n 1 pκq, Ω n1 pκq, Ω n2 pκq, Ω n3 pκqȋ s resonant, for all the values of κ, also taking into account the restrictions on the indexes for the search of traveling waves, see Section 3.4. Indeed, recalling (1.11) and that, for h "`8, G j p0, hq " |j|, we have, for ℓ "`´ℓ n3 ,´ℓ n2 ,´ℓ n1 , ℓ n1 , ℓ n2 , ℓ n3˘t hat the system Ωpκq¨ ℓ " γpℓ n1`ℓn2`ℓn3 q " 0 , n 1 ℓ n1`n2 ℓ n2`n3 ℓ n3 " 0 , has integer solutions. In this case the possible existence of quasi-periodic solutions of the water waves system (1.3) depends on the frequency modulation induced by the nonlinear terms. 6) Comparison with [6]. There are significant differences with respect to [6], which proves the existence of quasi-periodic standing waves for irrotational fluids, not only in the result -the solutions of Theorem 1.5 are traveling waves of fluids with constant vorticity-but also in the techniques.
(1) The first difference -which is a novelty of this paper-is a new formulation of degenerate KAM theory exploiting "momentum conservation", namely the space invariance of the Hamilton equations. The degenerate KAM theory approach for PDEs has been developed in [3], and then [6], [2], in order to prove the non-trivial dependence of the linear frequencies with respect to a parameter -in our case the surface tension κ-, see the "Transversality" Proposition 4.5. A key assumption used in [3], [6], [2] is that the linear frequencies are simple (because of Dirichlet boundary conditions in [3] and Neumann boundary conditions in [6], [2]). This is not true for traveling waves (e.g. in case of zero vorticity one has Ω j pκq " Ω´jpκq identically in κ). In order to deal with these resonances we strongly exploit the invariance of the equations (1.3) under space translations, which ultimately imply the restrictions to the indexes (4.8)- (4.10). In this way, assuming that the moduli of the tangential sites are all different as in (1.15), cfr. with item 5), we can remove some otherwise possibly degenerate case. This requires to keep trace along all the proof of the "momentum conservation property" that we characterize in different ways in Section 3.4. The momentum conservation law has been used in several KAM results for semilinear PDEs since the works [16,17], [28,35], see also [31,20,15] and references therein. The present paper gives a new application in the context of degenerate KAM theory (with additional difficulties arising by the quasi-linear nature of the water waves equations).
(2) Other significant differences with respect to [6] arise in the reduction in orders (Section 7) of the quasi-periodic linear operators obtained along the Nash-Moser iteration. In particular we mention that we have to preserve the Hamiltonian nature of these operators (at least until Section 7.4). Otherwise it would appear a time dependent operator at the order |D| 1{2 , of the form iapϕqH|D| 1 2 , with apϕq P R independent of x, compatible with the reversible structure, which can not be eliminated. Note that the operator iapϕqH|D| 1 2 is not Hamiltonian (unless apϕq " 0). Note also that the above difficulty was not present in [6] dealing with standing waves, because an operator of the form iapϕqH|D| 1 2 does not map even functions into even functions. In order to overcome this difficulty we have to perform always symplectic changes of variables (at least until Section 7.4), and not just reversible ones as in [6,2]. We finally mention that we perform as a first step in Section 7.1 a quasi-periodic time reparametrization to avoid otherwise a technical difficulty in the conjugation of the remainders obtained by the Egorov theorem in Section 7.3. This difficulty was not present in [6], since it arises conjugating the additional pseudodifferential term due to vorticity, see Remark 7.5. 7) Another novelty of our result is to exploit the momentum conservation also to prove that the obtained quasi-periodic solutions are indeed quasi-periodic traveling waves, according to Definition 1.2. This requires to check that the approximate solutions constructed along the Nash-Moser iteration of Section 9 (and Section 6) are indeed traveling waves. Actually this approach shows that the preservation of the momentum condition along the Nash-Moser-KAM iteration is equivalent to the construction of embedded invariant tori which support quasi-periodic traveling waves, namely of the form upϕ, xq " U pϕ´ xq (see Definition 3.1), or equivalently, in action-angle-normal variables, which satisfy (3.52). We expect this method can be used to obtain quasi-periodic traveling waves for other PDE's which are translation invariant.
Literature. We now shortly describe the literature regarding the existence of time periodic or quasi-periodic solutions of the water waves equations, focusing on the results more related to Theorem 1.5. We describes only results concerning space periodic waves, that we divide in three distinct groups: piq steady traveling solutions, piiq time periodic standing waves, piiiq time quasi-periodic standing waves.
This distinction takes into account not only the different shapes of the waves, but also the techniques for their construction. (i) Time and space periodic traveling waves which are steady in a moving frame. The literature concerning steady traveling wave solutions is huge, and we refer to [7] for an extended presentation. Here we only mention that, after the pioneering work of Stokes [38], the first rigorous construction of small amplitude space periodic steady traveling waves goes back to the 1920's with the papers of Nekrasov [33], Levi-Civita [27] and Struik [39], in case of irrotational bidimensional flows under the action of pure gravity. Later Zeidler [45] considered the effect of capillarity. In the presence of vorticity, the first result is due to Gerstner [18] in 1802, who gave an explicit example of periodic traveling wave, in infinite depth, and with a particular non-zero vorticity. One has to wait the work of Dubreil-Jacotin [14] in 1934 for the first existence results of small amplitude, periodic traveling waves with general (Hölder continuous, small) vorticity, and, later, the works of Goyon [19] and Zeidler [46] in the case of large vorticity. More recently we point out the works of Wahlén [41] for capillary-gravity waves and non-constant vorticity, and of Martin [30] and Walhén [42] for constant vorticity. All these results deal with 2d water waves, and can ultimately be deduced by the Crandall-Rabinowitz bifurcation theorem from a simple eigenvalue.
We also mention that these local bifurcation results can be extended to global branches of steady traveling waves by applying the methods of global bifurcation theory. We refer to Keady-Norbury [29], Toland [40], McLeod [32] for irrotational flows and Constantin-Strauss [9] for fluids with non-constant vorticity.
In the case of three dimensional irrotational fluids, bifurcation of small amplitude traveling waves periodic in space has been proved in Reeder-Shinbrot [36], Craig-Nicholls [11,12] for both gravity-capillary waves (by variational bifurcation arguments ła Weinstein-Moser) and by 24] for gravity waves (this is a small divisor problem). These solutions, in a moving frame, look steady bi-periodic waves. (ii) Time periodic standing waves. Bifurcation of time periodic standing water waves were obtained in a series of pioneering papers by Iooss, Plotnikov and Toland [34,25,22,23] for pure gravity waves, and by Alazard-Baldi [1] for gravity-capillary fluids. Standing waves are even in the space variable and so they do not travel in space. There is a huge difference with the results of the first group: the construction of time periodic standing waves involves small divisors. Thus the proof is based on Nash-Moser implicit function techniques and not only on the classical implicit function theorem. (iii) Time quasi-periodic standing waves. The first results in this direction were obtained very recently by Berti-Montalto [6] for the gravity-capillary system and by Baldi-Berti-Haus-Montalto [2] for the gravity water waves. Both papers deal with irrotational fluids.
2 Hamiltonian structure and linearization at the origin In this section we describe the Hamiltonian structure of the water waves equations (1.3), their symmetries and the solutions of the linearized system (1.6) at the equilibrium.

Hamiltonian structure
The Hamiltonian formulation of the water waves equations (1.3) with non-zero constant vorticity was obtained by Constantin-Ivanov-Prodanov [8] and Wahlén [42] in the case of finite depth. For irrotational flows it reduces to the classical Craig-Sulem-Zakharov formulation in [44], [13].
On the phase space H 1 0 pTqˆ9 H 1 pTq, endowed with the non canonical Poisson tensor we consider the Hamiltonian Such Hamiltonian is well defined on H 1 0 pTqˆ9 H 1 pTq since Gpηqr1s " 0 and ş T Gpηqψ dx " 0. It turns out [8,42] that equations (1.3) are the Hamiltonian system generated by Hpη, ψq with respect to the Poisson tensor J M pγq, namely where p∇ η H, ∇ ψ Hq P 9 L 2 pTqˆL 2 0 pTq denote the L 2 -gradients. Remark 2.1. The non canonical Poisson tensor J M pγq in (2.1) has to be regarded as an operator from (subspaces of) pL 2 0ˆ9 L 2 q˚" 9 L 2ˆL2 0 to L 2 0ˆ9 L 2 , that is The operator B´1 x maps a dense subspace of L 2 0 in 9 L 2 . For sake of simplicity, throughout the paper we may omit this detail. Above the dual space pL 2 0ˆ9 L 2 q˚with respect to the scalar product in L 2 is identified with 9 L 2ˆL2 0 . The Hamiltonian (2.2) enjoys several symmetries which we now describe.
the Hamiltonian (2.2) satisfies H˝τ ς " H for any ς P R, or, equivalently, the water waves vector field X defined in the right hand side on (1.

3) satisfies
X˝τ ς " τ ς˝X , @ς P R . (2.8) In order to verify this property, note that the Dirichlet-Neumann operator satisfies Wahlén coordinates. The variables pη, ψq are not Darboux coordinates, in the sense that the Poisson tensor (2.1) is not the canonical one for values of the vorticity γ ‰ 0. Wahlén [42] noted that in the variables pη, ζq, where ζ is defined by By (2.12), the symplectic form of (2.14) is the standard one, where J´1 is the symplectic operator regarded as a map from L 2 0ˆ9 L 2 into 9 L 2ˆL2 0 . Note that JJ´1 " Id L 2 0ˆ9 L 2 and J´1J " Id 9 L 2ˆL2 0 . The Hamiltonian vector field X H pη, ζq in (2.14) is characterized by the identity dHpη, ζqrp us " W`X H pη, ζq, p u˘, @p u :"ˆp η p ζ˙. The corresponding solutions pηpt, xq, ψpt, xqq of (1.3) induced by (2.11) are reversible as well.
We finally note that the transformation W defined in (2.11) commutes with the translation operator τ ς , therefore the Hamiltonian H in (2.13) is invariant under τ ς , as well as H in (2.2). By Noether theorem, the horizontal momentum ş T ζη x dx is a prime integral of (2.14).

Linearization at the equilibrium
In this section we study the linear system (1.6) and prove that its reversible solutions have the form (1.9). In view of the Hamiltonian (2.2) of the water waves equations (1.3), also the linear system (1.6) is Hamiltonian and it is generated by the quadratic Hamiltonian Thus, recalling (2.3), the linear system (1.6) is The linear operator Ω L acts from (a dense subspace) of L 2 0ˆ9 L 2 to 9 L 2ˆL2 0 . In the Wahlén coordinates (2.11), the linear Hamiltonian system (1.6), i.e. (2.18), transforms into the linear Hamiltonian system B tˆη ζ˙" JΩ Wˆη ζ˙, The linear operator Ω W acts from (a dense subspace) of L 2 0ˆ9 L 2 to 9 L 2ˆL2 0 . The linear system (2.19) is the Hamiltonian system obtained by linearizing (2.14) at the equilibrium pη, ζq " p0, 0q. We want to transform (2.19) in diagonal form by using a symmetrizer and then introducing complex coordinates. We first conjugate (2.19) under the symplectic transformation (with respect to the standard symplectic form W in (2.15)) of the phase spacê η ζ˙" Mˆu vẇ here M is the diagonal matrix of self-adjoint Fourier multipliers where rζs is the element in 9 H 1 with representant ζpxq. Note that Ωpκ, Dq is the Fourier multiplier with symbol tΩ´jpκqu jPZzt0u .
Back to the variables pη, ψq with the change of coordinates (2.11) one obtains formula (1.9).
Decomposition of the phase space in Lagrangian subspaces invariant under (2.19). We express the Fourier coefficients z j P C in (2.27) as In the new coordinates pα j , β j q jPZzt0u , we write (2.32) as (recall that M j " M´j) (2.34) The symplectic form (2.15) then becomes Each 2-dimensional subspace in the sum (2.33), spanned by pα j , β j q P R 2 is therefore a symplectic subspace. The quadratic Hamiltonian H L in (2.20) reads In view of (2.33), the involution S defined in (2.4) reads We may also enumerate the independent variables pα j , β j q jPZzt0u as`α´n, β´n, α n , β n˘, n P N. Thus the phase space H :" L 2 0ˆ9 L 2 of (2.14) decomposes as the direct sum "ˆη ζ˙"ˆM n pα n cospnxq´β n sinpnxqq M´1 n pβ n cospnxq`α n sinpnxqq˙, pα n , β n q P R 2 * , (2.38) V n,´: " "ˆη ζ˙"ˆM n pα´n cospnxq`β´n sinpnxqq M´1 n pβ´n cospnxq´α´n sinpnxqq˙, which are invariant for the linear Hamiltonian system (2.19), namely JΩ W : V n,σ Þ Ñ V n,σ (for a proof see e.g. remark 2.10). The symplectic projectors Π Vn,σ , σ P t˘u, on the symplectic subspaces V n,σ are explicitly provided by (2.33) and (2.34) with j " nσ. Note that the involution S defined in (2.4) and the translation operator τ ς in (2.7) leave the subspaces V n,σ , σ P t˘u, invariant.

Tangential and normal subspaces of the phase space
We decompose the phase space H of (2.14) into a direct sum of tangential and normal Lagrangian subspaces H ⊺ S`,Σ and H = S`,Σ . Note that the main part of the solutions (1.19) that we shall obtain in Theorem 1.5 is the component in the tangential subspace H ⊺ S`,Σ , whereas the component in the normal subspace H = S`,Σ is much smaller. Recalling the definition of the sets S`and Σ defined in (1.15) respectively (1.16), we split where H ⊺ S`,Σ is the finite dimensional tangential subspace V n a ,σa (2.41) and H = S`,Σ is the normal subspace defined as its symplectic orthogonal The symplectic projections Π ⊺ S`,Σ and Π = S`,Σ satisfy the following properties: Lemma 2.5. We have that Proof. Since the subspaces H ⊺ :" H ⊺ S`,Σ and H = :" H = S`,Σ are symplectic orthogonal, we have, recalling (2.15), that pJ´1v, wq L 2 " pJ´1w, vq L 2 " 0, @v P H ⊺ , @w P H = .
Thus, using the projectors Π ⊺ :" Π ⊺ S`,Σ , Π = :" Π = S`,Σ , we have that and, taking adjoints, ppΠ = q˚J´1Π ⊺ v, wq L 2 " ppΠ ⊺ q˚J´1Π = w, vq L 2 " 0 for any v, w P H, so that Now inserting the identity Π = " Id´Π ⊺ in (2.45), we get proving the second identity of (2.43). The first identity of (2.43) follows applying J to the left and to the right of the second identity. The identity (2.44) follows in the same way.
Note that the restricted symplectic form W| H = S`,Σ is represented by the symplectic structure where Π L 2 = is the L 2 -projector on the subspace H = S`,Σ . Indeed We also denote the associated (restricted) Poisson tensor In the next lemma we prove that J´1 = and J = are each other inverses.
is the associated Liouville 1-form (the operator J´1 = is defined in (2.46)). Finally, given a Hamiltonian K : T νˆRνˆH= S`,Σ Ñ R, the associated Hamiltonian vector field (with respect to the symplectic form (2.54)) is where ∇ w K denotes the L 2 gradient of K with respect to w P H = S`,Σ . Indeed, the only nontrivial component of the vector field X K is the last one, which we denote by rX K s w P H = S`,Σ . It fulfills and (2.56) follows by Lemma 2.6. We remark that along the paper we only consider Hamiltonians such that the L 2 -gradient ∇ w K defined by (2.57), as well as the Hamiltonian vector field Π = S`,Σ J∇ w K, maps spaces of Sobolev functions into Sobolev functions (not just distributions), with possible loss of derivatives.
Tangential and normal subspaces in complex variables. Each 2-dimensional symplectic subspace V n,σ , n P N, σ "˘1, defined in (2.38)-(2.39) is isomorphic, through the linear map MC defined in (2.32), to the complex subspace Denoting by Π j the L 2 -projection on H j , we have that Π Vn,σ " MC Π j pMCq´1. Thus MC is an isomorphism between the tangential subspace H ⊺ S`,Σ defined in (2.41) and and between the normal subspace H = S`,Σ defined in (2.42) and Denoting by Π ⊺ S , Π K S0 , the L 2 -orthogonal projections on the subspaces H S and H K S0 , we have that The following lemma, used in Section 5, is an easy corollary of the previous analysis.
Lemma 2.9. We have that pv ⊺ , Ω W wq L 2 " 0, for any v ⊺ P H ⊺ S`,Σ and w P H = S`,Σ . Proof. Write v ⊺ " MCz ⊺ and MCz K with z ⊺ P H S and z K P H K S0 . Then, by (2.22) and (2.25), 10. The same proof of Lemma 2.9 actually shows that pv n,´σ , Ω W v n,σ q L 2 " 0 for any v n,˘σ P V n,˘σ , for any n P N, σ "˘1. Thus Wpv n,´σ , JΩ W v n,σ q " pv n,´σ , J´1JΩ W v n,σ q L 2 " 0 which shows that JΩ W maps V n,σ in itself. Notation. For a À s b means that a ď Cpsqb for some positive constant Cpsq. We denote N :" t1, 2, . . .u and N 0 :" t0u Y N.

Functional setting
Along the paper we consider functions upϕ, xq P L 2`Tν`1 , C˘depending on the space variable x P T " T x and the angles ϕ P T ν " T ν ϕ (so that T ν`1 " T ν ϕˆT x ) which we expand in Fourier series as upϕ, xq " ÿ jPZ u j pϕqe i jx " ÿ ℓPZ ν ,jPZ u ℓ,j e ipℓ¨ϕ`jxq . We also consider real valued functions upϕ, xq P R, as well as vector valued functions upϕ, xq P C 2 (or upϕ, xq P R 2 ). When no confusion appears, we denote simply by L 2 , L 2 pT ν`1 q, L 2 x :" L 2 pT x q, L 2 ϕ :" L 2 pT ν q either the spaces of real/complex valued, scalar/vector valued, L 2 -functions. In this paper a crucial role is played by the following subspace of functions of pϕ, xq.
Definition 3.1. (Quasi-periodic traveling waves) Let  :" p 1 , . . . ,  ν q P Z ν be the vector defined in (2.53). A function upϕ, xq is called a quasi-periodic traveling wave if it has the form upϕ, xq " U pϕ´ xq where U : T ν Ñ C K , K P N, is a p2πq ν -periodic function.
Comparing with Definition 1.2, we find convenient to call quasi-periodic traveling wave both the function upϕ, xq " U pϕ´ xq and the function of time upωt, xq " U pωt´ xq.
Quasi-periodic traveling waves are characterized by the relation upϕ´ ς,¨q " τ ς u @ς P R , where τ ς is the translation operator in (2.7). Product and composition of quasi-periodic traveling waves is a quasi-periodic traveling wave. Expanded in Fourier series as in (3.1), a quasi-periodic traveling wave has the form upϕ, xq " ÿ ℓPZ ν ,jPZ,j` ¨ℓ"0 u ℓ,j e ipℓ¨ϕ`jxq , namely, comparing with Definition 3.1, The traveling waves upϕ, xq " U pϕ´ xq where U p¨q belongs to the Sobolev space H s pT ν , C K q in (1.14) (with values in C K , K P N), form a subspace of the Sobolev space where xℓ, jy :" maxt1, |ℓ|, |j|u. Note the equivalence of the norms (use (3.4)) }u} H s pT ν ϕˆT x q » s }U } H s pT ν q . For s ě s 0 :" " ν`1 2 ‰`1 P N one has H s pT ν`1 q Ă CpT ν`1 q, and H s pT ν`1 q is an algebra. Along the paper we denote by } } s both the Sobolev norms in (1.14) and (3.5).
For K ě 1 we define the smoothing operator Π K on the traveling waves and Π K K :" Id´Π K . Note that, writing a traveling wave as in (3.4), the projector Π K in (3.6) is equal to Whitney-Sobolev functions. Along the paper we consider families of Sobolev functions λ Þ Ñ upλq P H s pT ν`1 q and λ Þ Ñ U pλq P H s pT ν q which are k 0 -times differentiable in the sense of Whitney with respect to the parameter λ :" pω, κq P F Ă R νˆr κ 1 , κ 2 s where F Ă R ν`1 is a closed set. The case that we encounter is when ω belongs to the closed set of Diophantine vectors DCpυ, τ q defined in (1.13). We refer to Definition 2.1 in [2], for the definition of a Whitney-Sobolev function u : F Ñ H s where H s may be either the Hilbert space H s pT νˆT q or H s pT ν q. Here we mention that, given υ P p0, 1q, we can identify a Whitney-Sobolev function u : F Ñ H s with k 0 derivatives with the equivalence class of functions f P W k0,8,υ pR ν`1 , H s q{ " with respect to the equivalence relation f " g when B j λ f pλq " B j λ gpλq for all λ P F , |j| ď k 0´1 , with equivalence of the norms }u} k0,υ s,F " ν,k0 }u} W k 0 ,8,υ pR ν`1 ,H s q :" The key result is the Whitney extension theorem, which associates to a Whitney-Sobolev function u : F Ñ H s with k 0 -derivatives a function r u : R ν`1 Ñ H s , r u in W k0,8 pR ν`1 , H s q (independently of the target Sobolev space H s ) with an equivalent norm. For sake of simplicity in the notation we often denote } } k0,υ s,F " } } k0,υ s . Thanks to this equivalence, all the tame estimates which hold for Sobolev spaces carry over for Whitney-Sobolev functions. For example the following classical tame estimate for the product holds: (see e.g. Lemma 2.4 in [2]): for all s ě s 0 ą pν`1q{2, }uv} k0,υ s ď Cps, k 0 q}u} k0,υ s }v} k0,υ s0`C ps 0 , k 0 q}u} k0,υ s0 }v} k0,υ s .
Linear operators. Along the paper we consider ϕ-dependent families of linear operators A : T ν Þ Ñ LpL 2 pT x qq, ϕ Þ Ñ Apϕq, acting on subspaces of L 2 pT x q, either real or complex valued. We also regard A as an operator (which for simplicity we denote by A as well) that acts on functions upϕ, xq of space and time, that is pAuqpϕ, xq :" pApϕqupϕ,¨qq pxq . The action of an operator A as in (3.12) on a scalar function upϕ, xq P L 2 expanded as in (3.1) is We identify an operator A with its matrix`A j 1 j pℓ´ℓ 1 q˘j ,j 1 PZ,ℓ,ℓ 1 PZ ν , which is Töplitz with respect to the index ℓ. In this paper we always consider Töplitz operators as in (3.12), (3.13).

Real operators. A linear operator
Equivalently A is real if it maps real valued functions into real valued functions. We represent a real operator acting on pη, ζq belonging to (a subspace of) L 2 pT x , R 2 q by a matrix where A, B, C, D are real operators acting on the scalar valued components η, ζ P L 2 pT x , Rq. The change of coordinates (2.24) transforms the real operator R into a complex one acting on the variables pz, zq, given by the matrix A matrix operator acting on the complex variables pz, zq of the form (3.15), we call it real. We shall also consider real operators R of the form (3.15) acting on subspaces of L 2 .

Pseudodifferential calculus
In this section we report fundamental notions of pseudodifferential calculus, following [6].

Definition 3.3. (ΨDO)
A pseudodifferential symbol apx, jq of order m is the restriction to RˆZ of a function apx, ξq which is C 8 -smooth on RˆR, 2π-periodic in x, and satisfies |B α x B β ξ apx, ξq| ď C α,β xξy m´β , @α, β P N 0 . We denote by S m the class of symbols of order m and S´8 :" X mě0 S m . To a symbol apx, ξq in S m we associate its quantization acting on a 2π-periodic function upxq " ř jPZ u j e ijx as rOppaquspxq :" ÿ jPZ apx, jqu j e ijx .
We denote by OPS m the set of pseudodifferential operators of order m and OPS´8 :" Ş mPR OPS m . For a matrix of pseudodifferential operators When the symbol apxq is independent of ξ, the operator Oppaq is the multiplication operator by the function apxq, i.e. Oppaq : upxq Þ Ñ apxqupxq. In such a case we also denote Oppaq " apxq.
We shall use the following notation, used also in [1,6,2]. For any m P Rzt0u, we set where χ is an even, positive C 8 cut-off satisfying (3.10). We also identify the Hilbert transform H, acting on the 2π-periodic functions, defined by with the Fourier multiplier Opp´i sign pξqχpξqq. Similarly we regard the operator as the Fourier multiplier B´1 x " Op`´i χpξqξ´1˘and the projector π 0 , defined on the 2π-periodic functions as π 0 u :" 1 2π with the Fourier multiplier Op`1´χpξq˘. Finally we define, for any m P Rzt0u, xDy m :" π 0`| D| m :" Op`p1´χpξqq`χpξq|ξ| m˘.
Along the paper we consider families of pseudodifferential operators with a symbol apλ; ϕ, x, ξq which is k 0 -times differentiable with respect to a parameter λ :" pω, κq in an open subset We recall the pseudodifferential norm introduced in Definition 2.11 in [6].
where }Apλq} m,s,α :" max 0ďβďα sup ξPR }B β ξ apλ,¨,¨, ξq} s xξy´m`β. For a matrix of pseudodifferential operators A P OPS m as in (3.18), we define }A} Given a function apλ; ϕ, xq P C 8 which is k 0 -times differentiable with respect to λ, the weighted norm of the corresponding multiplication operator is Composition of pseudodifferential operators. If Oppaq, Oppbq are pseudodifferential operators with symbols a P S m , b P S m 1 , m, m 1 P R, then the composition operator OppaqOppbq is a pseudodifferential operator Oppa#bq with symbol a#b P S m`m 1 . It admits the asymptotic expansion: for any N ě 1 The following result is proved in Lemma 2.13 in [6].
Moreover, for any integer N ě 1, the remainder R N :" Oppr N q in (3.23) satisfies The commutator between two pseudodifferential operators Oppaq P OPS m and Oppbq P OPS m 1 is a pseudodifferential operator in OPS m`m 1´1 with symbol a ‹ b P S m`m 1´1 , namely rOppaq, Oppbqs " Op pa ‹ bq, that admits, by (3.23), the expansion a ‹ b "´i ta, bu`r r 2 pa, bq , r r 2 pa, bq :" r 2 pa, bq´r 2 pb, aq P S m`m 1´2 , where ta, bu :" B ξ aB x b´B x aB ξ b , (3.26) is the Poisson bracket between apx, ξq and bpx, ξq. As a corollary of Lemma 3.5 we have: Lemma 3.6. (Commutator) Let A " Oppaq and B " Oppbq be pseudodifferential operators with symbols apλ; ϕ, x, ξq P S m , bpλ; ϕ, x, ξq P S m 1 , m, m 1 P R. Then the commutator rA, Bs :" Finally we consider the exponential of a pseudodifferential operator of order 0. The following lemma follows as in Lemma 2.12 of [5] (or Lemma 2.17 in [6]). Lemma 3.7. (Exponential map) If A :" Oppapλ; ϕ, x, ξqq is in OP S 0 , then e A is in OP S 0 and for any s ě s 0 , α P N 0 , there is a constant Cps, αq ą 0 so that The same holds for a matrix A of the form (3.18) in OPS 0 .
Dirichlet-Neumann operator. We finally remind the following decomposition of the Dirichlet-Neumann operator proved in [6], in the case of infinite depth, and in [2], for finite depth.

D k 0 -tame and modulo-tame operators
We present the notion of tame and modulo tame operators introduced in [6]. Let A :" Apλq be a linear operator as in (3.12), k 0 -times differentiable with respect to the parameter λ in the open set Λ 0 Ă R ν`1 .
We say that M A psq is a tame constant of the operator A. The constant M A psq " M A pk 0 , σ, sq may also depend on k 0 , σ but we shall often omit to write them. When the "loss of derivatives" σ is zero, we simply write D k0 -tame instead of D k0 -0-tame. For a matrix operator as in (3.15), we denote the tame constant M R psq :" max tM R1 psq, M R2 psqu.
Note that the tame constants M A psq are not uniquely determined. An immediate consequence of (3.34) is that }A} LpH s 0`σ ,H s 0 q ď 2M A ps 0 q. Also note that, representing the operator A by its matrix elements pA j 1 j pℓ´ℓ 1 qq ℓ,ℓ 1 PZ ν ,j,j 1 PZ as in (3.13), we have for all |k| ď k 0 , j 1 P Z, The class of D k0 -σ-tame operators is closed under composition.
Lemma 3.12. (Composition, Lemma 2.20 in [6]) Let A, B be respectively D k0 -σ A -tame and D k0 -σ B -tame operators with tame constants respectively M A psq and M B psq. Then the composed operator A˝B is D k0 -pσ A`σB q-tame with tame constant It is proved in Lemma 2.22 in [6] that the action of a D k0 -σ-tame operator Apλq on a Sobolev function u " upλq P H s`σ is bounded by Pseudodifferential operators are tame operators. We use in particular the following lemma: Lemma 3.13. (Lemma 2.21 in [6]) Let A " apλ; ϕ, x, Dq P OPS 0 be a family of pseudodifferential operators satisfying }A} k0,υ 0,s,0 ă 8 for s ě s 0 . Then A is D k0 -tame with a tame constant M A psq satisfying, for any s ě s 0 , The same statement holds for a matrix operator R as in (3.15).
In view of the KAM reducibility scheme of Section 8 we also consider the stronger notion of D k0 -modulo-tame operator, that we need only for operators with loss of derivative σ " 0. We first recall the notion of majorant operator : given a linear operator A acting as in (3.13), we define the majorant operator |A| by its matrix elements p|A The constant M 7 A psq is called a modulo-tame constant for the operator A. For a matrix of operators as in (3.15), we denote the modulo-tame constant M 7 A psq. In view of the next lemma, given a linear operator A acting as in (3.13), we define the operator The same statement holds for matrix operators A, B as in (3.15).
By Lemma 3.15 we deduce the following result, cfr. Lemma 2.20 in [5].
Lemma 3.16. (Exponential) Let A and xB ϕ y b A be D k0 -modulo-tame and assume that M 7 A ps 0 q ď 1. Then the operators e˘A´Id and xB ϕ y b e˘A´Id are D k0 -modulo-tame with modulo-tame constants satisfying Given a linear operator A acting as in (3.13), we define the smoothed operator Π N A, N P N whose matrix elements are We also denote Π K N :" Id´Π N . It is proved in Lemma 2.27 in [6] that The same estimate holds with a matrix operator R as in (3.15).

Hamiltonian and Reversible operators
In this paper we shall exploit both the Hamiltonian and reversible structure along the reduction of the linearized operators, that we now present. Hamiltonian operators. A matrix operator R as in (3.14) is Hamiltonian if the matrix s self-adjoint, namely B˚" B, C˚" C, A˚"´D and A, B, C, D are real.
Correspondingly, a matrix operator as in (3.15) is Hamiltonian if where the symplectic 2-form W is defined in (2.15).
Reversible and reversibility preserving operators. Let S be an involution as in (2.4) acting on the real variables pη, ζq P R 2 , or as in (2.51) acting on the action-angle-normal variables pθ, I, wq, or as in (2.29) acting in the pz, zq complex variables introduced in (2.24).
• reversibility preserving if Rp´ϕq˝S " S˝Rpϕq for all ϕ P T ν .
Since in the complex coordinates pz, zq the involution S defined in (2.4) reads as in (2.29), an operator Rpϕq as in (3.15) is reversible, respectively anti-reversible, if, for any i " 1, 2, where, with a small abuse of notation, we still denote pSuqpxq " up´xq. Moreover, recalling that in the Fourier coordinates such involution reads as in (2.30), we obtain the following lemma.
• reversibility preserving if, for any i " 1, 2, Note that the composition of a reversible operator with a reversibility preserving operator is reversible. The flow generated by a reversibility preserving operator is reversibility preserving. If Rpϕq is reversibility preserving, then pω¨B ϕ Rqpϕq is reversible.
We shall say that a linear operator of the form ω¨B ϕ`A pϕq is reversible if Apϕq is reversible. Conjugating the linear operator ω¨B ϕ`A pϕq by a family of invertible linear maps Φpϕq, we get the transformed operator The conjugation of a reversible operator with a reversibility preserving operator is reversible.
A reversibility preserving operator maps reversible, respectively anti-reversible, functions into reversible, respectively anti-reversible, functions.
Lemma 3.22. Let X be a reversible vector field, according to (2.5), and upϕ, xq be a reversible quasi-periodic function. Then the linearized operator d u Xpupϕ,¨qq is reversible, according to Definition 3.17.
Finally we note the following lemma. Proof. The involution S defined in (2.4) maps V n,˘i nto itself, acting as in (2.36). Then, by the decomposition (2.33), each projector Π Vn,σ commutes with S.

Momentum preserving operators
The following definition is crucial in the construction of traveling waves.
where the translation operator τ ς is defined in (2.7). A linear matrix operator Apϕq of the form (3.14) or (3.15) is momentum preserving if each of its components is momentum preserving.
Momentum preserving operators are closed under several operations.
has a unique propagator Φ t pϕq for any t P r0, 1s. Then Φ t pϕq is momentum preserving.
We shall say that a linear operator of the form ω¨B ϕ`A pϕq is momentum preserving if Apϕq is momentum preserving. In particular, conjugating a momentum preserving operator ω¨B ϕ`A pϕq by a family of invertible linear momentum preserving maps Φpϕq, we obtain the transformed operator ω¨B ϕ`A`p ϕq in (3.46) which is momentum preserving. Lemma 3.27. Let X be a vector field translation invariant, according to (2.8). Let u be a quasiperiodic traveling wave. Then the linearized operator d u Xpupϕ,¨qq is momentum preserving.
We now provide a characterization of the momentum preserving property in Fourier space.
Lemma 3.28. Let ϕ-dependent family of operators Apϕq, ϕ P T ν , is momentum preserving if and only if the matrix elements of Apϕq, defined by (3.13), fulfill Proof. By (3.13) we have, for any function upxq, Therefore (3.48) is equivalent to (3.50).
We characterize the symbol of a pseudodifferential operator which is momentum preserving: Lemma 3.29. A pseudodifferential operator Apϕ, x, Dq " Oppapϕ, x, ξqq is momentum preserving if and only if its symbol satisfies Proof. If the symbol a satisfies (3.51), then, for all ς P R, proving that τ ς˝A pϕ, x, Dq " Apϕ´ ς, x, Dq˝τ ς . The vice versa follows using that apϕ, x, ξq " e´i ξx Apϕ, x, Dqre iξx s.
Lemma 3.30. If βpϕ, xq is a quasi-periodic traveling wave, then the operator Bpϕq defined in (3.29) is momentum preserving.
We also note the following lemma. Proof. Recall that the translation τ ς maps V n,˘i nto itself, acting as in (2.37). Consider the L 2 -orthogonal decomposition H " H = ' H K = , setting H = :" H = S`,Σ for brevity: Applying Thus also the L 2 -orthogonal subspace H K = is invariant under the action of τ ς and we conclude, by the uniqueness of the orthogonal decomposition, that The next lemma concerns the Dirichlet-Neumann operator.
Proof. It follows by (2.9) and the characterization in (3.2) of the quasi-periodic traveling wave ηpϕ, xq.

Transversality of linear frequencies
In this section we extend the KAM theory approach of [6], [3] in order to deal with the linear frequencies Ω j pκq defined in (1.11). The main novelty is the use of the momentum condition in the proof of Proposition 4.5. We shall also exploit that the tangential sites S :" t  1 , . . . ,  ν u Ă Zzt0u defined in (2.48), have all distinct modulus | a | " n a , see assumption (1.15).
We first introduce the following definition.
, the scalar function f¨c is not identically zero on the whole interval rκ 1 , κ 2 s.
From a geometric point of view, if f is non-degenerate it means that the image of the curve f prκ 1 , κ 2 sq Ă R N is not contained in any hyperplane of R N .
We shall use in the sequel that the maps κ Þ Ñ Ω j pκq are analytic in rκ 1 , κ 2 s. We decompose Note that the dependence on κ of Ω j pκq enters only through ω j pκq, because Gj p0q j is independent of κ. Note also that j Þ Ñ ω j pκq is even in j, whereas the component due to the vorticity j Þ Ñ γ Gjp0q j is odd. Moreover this term is, in view of (1.8), uniformly bounded in j.

Lemma 4.2. (Non-degeneracy-I)
The following frequency vectors are non-degenerate: Recalling (4.1), we have that, for any j P Z, Moreover B κ λ j pκq "´2λ j pκq 2 , for any j P Z, and therefore, for any n P N, B n κ r ω j pκq " r c n λ j pκq n r ω j pκq , r c n :" c 1¨. . .¨c n , c n :" 3´2n . We now prove items 2 and 3, i.e. the non-degeneracy of the vector` Ωpκq, r Ω j pκq˘P R ν`1 for any j P ZzpS Y p´Sqq, where r Ω j pκq is defined in (4.2). Items 1 and 4 follow similarly. For this purpose, by analyticity, it is sufficient to find one value of κ P rκ 1 , κ 2 s so that the determinant of the pν`1qˆpν`1q matrix is not zero. We actually show that det Apκq ‰ 0 for any κ P rκ 1 , κ 2 s. By (4.2)-(4.4) and the multilinearity of the determinant function, we get Since Bpκq is a Vandermorde matrix, we conclude that Now, the fact that det Apκq ‰ 0 for any κ P rκ 1 , κ 2 s is a consequence from the following Claim: For any p, p 1 P t 1 , . . . ,  ν , ju, p ‰ p 1 , one has λ p pκq ‰ λ p 1 pκq for any κ P rκ 1 , κ 2 s.
Proof of the Claim: If p 1 " 0 and p ‰ 0, the claim follows because, by (4.3), Consider now the case p, p 1 ‰ 0. We now prove that the map p Þ Ñ λ p pκq is strictly monotone on p0,`8q. In case of finite depth, G p p0q " p tanhphpq, and The function f pyq :" 3 tanhpyq´p1´tanh 2 pyqqy is positive for any y ą 0. Indeed f pyq Ñ 0 as y Ñ 0, and it is strictly monotone increasing for y ą 0, since f 1 pyq " 2p1´tanh 2 pyqqp1`y tanhpyqq ą 0.
We deduce that B p λ p pκq ą 0, also if the depth h "`8. Since the function p Þ Ñ λ p pκq is even we have proved that that it is strictly monotone decreasing on p´8, 0q and increasing in p0,`8q. Thus, if λ p pκq " λ p 1 pκq then p "´p 1 . But this case is excluded by the assumption (1.15) and the condition j R S Y p´Sq, which together imply |p| ‰ |p 1 |.
Note that in items 3 and 4 of Lemma 4.2 we require that j and j 1 do not belong to t0u Y S Y p´Sq. In order to deal in Proposition 4.5 when j and j 1 are in S Y p´Sq, we need also the following lemma. It is actually a direct consequence of the proof of Lemma 4.2, noting that Ω j pκq´ω j pκq is independent of κ. Lemma 4.3. (Non-degeneracy-II) Let ωpκq :"`ω  1 pκq, . . . , ω  ν pκq˘. The following vectors are non-degenerate: For later use, we provide the following asymptotic estimate of the linear frequencies.
where, for any n P N 0 , there exists a constant C n,h ą 0 such that Proof. By (4.1) we deduce (4.5) with Then (4.6) follows exploiting that (both for finite and infinite depth) the quantities |j|pG j p0q´|j|q and G j p0q{|j| are uniformly bounded in j, see (1.8).
The next proposition is the key of the argument. We remind that  " p 1 , . . . ,  ν q denotes the vector in Z ν of tangential sites introduced in (2.53).
Proposition 4.5. (Transversality) There exist m 0 P N and ρ 0 ą 0 such that, for any κ P rκ 1 , κ 2 s, the following hold: (4.10) We call ρ 0 the amount of non-degeneracy and m 0 the index of non-degeneracy.
Proof. We prove separately (4.7)-(4.10). In this proof we set for brevity K :" rκ 1 , κ 2 s. Proof of (4.7). By contradiction, assume that for any m P N there exist κ m P K and ℓ m P Z ν zt0u such thatˇˇˇB The sequences pκ m q mPN Ă K and pℓ m { xℓ m yq mPN Ă R ν zt0u are both bounded. By compactness, up to subsequences κ m Ñ κ P K and ℓ m { xℓ m y Ñ c ‰ 0. Therefore, in the limit for m Ñ`8, by (4.11) we get B n κ Ωpκq¨c " 0 for any n P N 0 . By the analyticity of Ωpκq, we deduce that the function κ Þ Ñ Ωpκq¨c is identically zero on K, which contradicts Lemma 4.2-1. Proof of (4.8). We divide the proof in 4 steps.
Step 3. We consider first the case when the sequence pℓ m q mPN Ă Z ν is bounded. Up to subsequences, we have definitively that ℓ m " ℓ P Z ν . Moreover, since j m and ℓ m satisfy (4.12), also the sequence pj m q mPN is bounded and, up to subsequences, definitively j m "  P S c 0 . Therefore, in the limit m Ñ 8, from (4.13) we obtain B n κ` Ωpκq¨ℓ`Ω  pκq˘| κ"κ " 0 , @ n P N 0 , ¨ℓ` " 0 .
Step 4. We consider now the case when the sequence pℓ m q mPN is unbounded. Up to subsequences  { xℓ m y P R. Note that d is finite because j m and ℓ m satisfy (4.12). Therefore (4.13) becomes, in the limit m Ñ 8, B n κ` Ωpκq¨c`d ? κ˘| κ"κ " 0 , @ n P N 0 .
Step 1. By Lemma 4.4, for any κ P K, In this case (4.9) is already fulfilled with n " 0. Thus we restrict to indexes ℓ P Z ν and j, j 1 P S c 0 , such thaťˇ| Furthermore we may assume j m ‰ j 1 m because the case j m " j 1 m is included in (4.7).
Step 2. By contradiction, we assume that, for any m P N, there exist κ m P K, ℓ m P Z ν and j m , j 1 m P S c 0 , satisfying (4.16), such that, for any 0 ď n ď m, #ˇˇB  Up to subsequences κ m Ñ κ P K and ℓ m { xℓ m y Ñ c P R ν .
Step 3. We start with the case when pℓ m q mPN Ă Z ν is bounded. Up to subsequences, we have definitively that If j m "´j 1 m we deduce by the momentum relation that |j m | " |j 1 m | ď Cxℓ m y ď C, and we conclude that in any case the sequences pj m q mPN and pj 1 m q mPN are bounded. Up to subsequences, we have definitively that j m "  and j 1 m "  1 , with ,  1 P S c 0 and such that Therefore (4.17) becomes, in the limit m Ñ 8, By analyticity, we obtain that We distinguish several cases: • Let ,  1 R´S and || ‰ | 1 |. By (4.19) the vector p Ωpκq, Ω  pκq, Ω  1 pκqq is degenerate with c :" pℓ, 1,´1q ‰ 0, contradicting Lemma 4.2-4.
• Let  1 R´S and  P´S. With no loss of generality suppose  "´ 1 . In view of (4.1), the first equation in (4.19) implies that, for any κ P K By Lemma 4.3 the vector` ωpκq, ω  1 pκq, 1˘is non-degenerate, which is a contradiction.

Nash-Moser theorem and measure estimates
Under the rescaling pη, ζq Þ Ñ pεη, εζq, the Hamiltonian system (2.14) transforms into the Hamiltonian system generated by H ε pη, ζq :" ε´2Hpεη, εζq " H L pη, ζq`εP ε pη, ζq , where H is the water waves Hamiltonian (2.13) expressed in the Wahlén coordinates (2.11), H L is defined in (2.20) and We now study the Hamiltonian system generated by the Hamiltonian H ε pη, ζq, in the actionangle and normal coordinates pθ, I, wq defined in Section 2.3. Thus we consider the Hamiltonian H ε pθ, I, wq defined by where A is the map defined in (2.50). The associated symplectic form is given in (2.54). By Lemma 2.9 (see also (2.35), (2.49)), in the variables pθ, I, wq the quadratic Hamiltonian H L defined in (2.20) simply reads, up to a constant, where Ωpκq P R ν is defined in (1.18) and Ω W in (2.19). Thus the Hamiltonian H ε in (5.2) is We look for an embedded invariant torus of the Hamiltonian vector field X Hε :" pB I H ε ,´B θ H ε , Π = S`,Σ J∇ w H ε q filled by quasi-periodic solutions with Diophantine frequency vector ω P R ν (which satisfies also first and second order Melnikov non-resonance conditions, see (5.14)-(5.16)).

(5.5)
If F pi, αq " 0, then the embedding ϕ Þ Ñ ipϕq is an invariant torus for the Hamiltonian vector field X Hα , filled with quasi-periodic solutions with frequency ω.
The norm of the periodic components of the embedded torus Ipϕq :" ipϕq´pϕ, 0, 0q :" pΘpϕq, Ipϕq, wpϕqq , Θpϕq :" θpϕq´ϕ , and m 0 P N is the index of non-degeneracy provided by Proposition 4.5, which only depends on the linear unperturbed frequencies. Thus, k 0 is considered as an absolute constant and we will often omit to write the dependence of the various constants with respect to k 0 . We look for quasi-periodic solutions of frequency ω belonging to a δ-neighbourhood (independent of ε) Ω :" ω P R ν : dist`ω, Ωrκ 1 , κ 2 s˘ă δ ( , δ ą 0 , of the curve Ωrκ 1 , κ 2 s defined by (1.18).
We remind that the conditions on the indexes in (5.15)-(5.16) (where  P Z ν is the vector in (2.53)) are due to the fact that we look for traveling wave solutions. These restrictions are essential to prove the measure estimates of the next section.

Measure estimates
By (5.10), the function α 8 p¨, κq from Ω into its image α 8 pΩ, κq is invertible and Then, for any β P α 8 pC υ 8 q, Theorem 5.2 proves the existence of an embedded invariant torus filled by quasi-periodic solutions with Diophantine frequency ω " α´1 8 pβ, κq for the Hamiltonian Consider the curve of the unperturbed tangential frequency vector Ωpκq in (1.18). In Theorem 5.3 below we prove that for "most" values of κ P rκ 1 , κ 2 s the vector pα´1 8 p Ωpκq, κq, κq is in C υ 8 , obtaining an embedded torus for the Hamiltonian H ε in (5.2), filled by quasi-periodic solutions with Diophantine frequency vector ω " α´1 8 p Ωpκq, κq, denoted r Ω in Theorem 1.5. Thus εApi 8 p r Ωtqq, where A is defined in (2.50), is a quasi-periodic traveling wave solution of the water waves equations (2.14) written in the Wahlén variables. Finally, going back to the original Zakharov variables via (2.10) we obtain solutions of (1.3). This proves Theorem 1.5 together with the following measure estimate.
In order to estimate the measure of the sets (5.27)-(5.30) that are nonempty, the key point is to prove that the perturbed frequencies satisfy estimates similar to (4.7)-(4.10).
Lemma 5.5. (Perturbed transversality) For ε P p0, ε 0 q small enough and for all κ P rκ 1 , κ 2 s, We recall that ρ 0 is the amount of non-degeneracy that has been defined in Proposition 4.5.

Approximate inverse
In order to implement a convergent Nash-Moser scheme that leads to a solution of F pi, αq " 0, where F pi, αq is the nonlinear operator defined in (5.5), we construct an almost approximate right inverse of the linearized operator d i,α F pi 0 , α 0 qrp ı, p αs " ω¨B ϕ p ı´d i X Hα pi 0 pϕqq rp ıs´pp α, 0, 0q .
We closely follow the strategy presented in [4] and implemented for the water waves equations in [6,2]. The main novelty is to check that this construction preserves the momentum preserving properties needed for the search of traveling waves. Therefore, along this section we shall focus on this verification. The estimates are very similar to those in [6,2]. First of all, we state tame estimates for the composition operator induced by the Hamiltonian vector field X P " pB I P,´B θ P, Π = S`,Σ J∇ w P q in (5.5).
Along this section, we assume the following hypothesis, which is verified by the approximate solutions obtained at each step of the Nash-Moser Theorem 9.2.
In the sequel we denote by σ " σpν, τ q constants, which may increase from lemma to lemma, which represent "loss of derivatives".
On an exact solution (that is Z " 0), the terms K 00 , K 01 in the Taylor expansion (6.18) vanish and K 10 " ω. More precisely, arguing as in Lemma 5.4 in [2], we have Lemma 6.4. There is σ :" σpν, τ q ą 0, such that, for all s ě s 0 , Under the linear change of variables the linearized operator d i,α F pi δ q is approximately transformed into the one obtained when one linearizes the Hamiltonian system (6.21) at pφ, y, wq " pϕ, 0, 0q, differentiating also in α at α 0 and changing B t ù ω¨B ϕ , namelÿ p φ p y p w p α‹ ‹ ‚ Þ Ñ¨ω¨B ϕ p φ´B φ K 10 pϕqr p φs´B α K 10 pϕqrp αs´K 20 pϕqp y´rK 11 pϕqs J p w ω¨B ϕ p y`B φφ K 00 pϕqr p φs`B α B φ K 00 pϕqrp αs`rB φ K 10 pϕqs J p y`rB φ K 01 pϕqs J p w ω¨B ϕ p w´J =`Bφ K 01 pϕqr p φs`B α K 01 pϕqrp αs`K 11 pϕqp y`K 02 pϕqp w˘‹ ‚. (6.22) In order to construct an "almost approximate" inverse of (6.22), we need that is "almost invertible" (on traveling waves) up to remainders of size OpN´a n´1 q, where, for n P N 0 N n :" K p n , K n :" K χ n 0 , χ " 3{2 . (6.24) The pK n q ně0 is the scale used in the nonlinear Nash-Moser iteration of Section 9 and pN n q ně0 is the one in the reducibility scheme of Section 8. Let H s = pT ν`1 q :" H s pT ν`1 q X H = S`,Σ . (AI) Almost invertibility of L ω : There exist positive real numbers σ, µpbq, a, p, K 0 and a subset Λ o Ă DCpυ, τ qˆrκ 1 , κ 2 s such that, for all pω, κq P Λ o , the operator L ω may be decomposed as where, for every traveling wave function g P H s`σ = pT ν`1 , R 2 q and for every pω, κq P Λ o , there is a traveling wave solution h P H s = pT ν`1 , R 2 q of L ă ω h " g satisfying, for all s 0 ď s ď S,

(6.26)
In addition, if g is anti-reversible, then h is reversible. Moreover, for any s 0 ď s ď S, for any traveling wave h P H = S`,Σ , the operators R ω , R K ω satisfy the estimates This assumption shall be verified by Theorem 8.10 at each n-th step of the Nash-Moser nonlinear iteration.
In order to solve (6.34), we choose p α such that the average in ϕ of the right hand side is zero. By Lemma 6.4 and (6.1), the ϕ-average of the matrix M 1 satisfies xM 1 y ϕ " Id`Opευ´1q. Then, for ευ´1 small enough, xM 1 y ϕ is invertible and xM 1 y´1 ϕ " Id`Opευ´1q. Thus we define p α :"´xM 1 y´1 ϕ`x g 1 y ϕ`x M 2 g 2 y ϕ`x M 3 g 3 y ϕ˘, (6.35) and the solution of equation (6.34) Finally the property p φpϕ´ ςq " p φpϕq for any ς P R follows by (6.20), (6.32) and the fact that p w in (6.33) is a traveling wave. This proves that p p φ, p y, p wq is a traveling wave variation, i.e. (6.30) holds. Moreover, using (6.29), (6.19), Lemma 3.23, the fact that J and S anti-commutes and (AI), one checks that p p φ, p y, p wq is reversible, i.e.

(6.40)
Moreover, the first three components of T 0 g form a reversible traveling wave variation (i.e. satisfy (6.37) and (6.30)). Finally, T 0 is an almost approximate right inverse of d i,α F pi 0 q, namely where, for any traveling wave variation g, for all s 0 ď s ď S, }Pg} k0,υ s À S υ´1´}F pi 0 , α 0 q} k0,υ s0`σ }g} k0,υ s`σ Proof. We claim that the first three components of T 0 g form a reversible traveling wave variation. Indeed, differentiating (6.10) it follows that DG δ pϕ, 0, 0q, thus pDG δ pϕ, 0, 0qq´1, is reversibility and momentum preserving (cfr. (3.54)). In particular these operators map an (anti)-reversible, respectively traveling, waves variation into a (anti)-reversible traveling waves variation (cfr. Lemma 3.34). Moreover, by Proposition 6.5, the operator D´1 maps an anti-reversible traveling wave into a vector whose first three components form a reversible traveling wave. This proves the claim. We now prove that the operators P, P ω and P K ω are defined on traveling waves. They are computed e.g. in Theorem 5.6 of [2]. To define them, introduce first the linear operators R Z r p φ, p y, p w, p αs :"¨´B φ K 10 pϕ, α 0 qr p φs B φφ K 00 pϕ, α 0 qr p φs`rB φ K 10 pϕ, α 0 qs J p y`rB φ K 01 pϕ, α 0 qs J p ẃ Next, we denote by Π the projection pp ı, p αq Þ Ñ p ı, by u δ pϕq " pϕ, 0, 0q the trivial torus, and by E, It is then proved in Theorem 5.6 of [2] that P :" E˝T 0 , P ω :" E ω˝T0 , P K ω :" E K ω˝T0 . A direct inspection of these formulas shows that P, P ω and P K ω are defined on traveling wave variations. In particular, note that the operators R ω , R K ω in (6.45) are defined only if p w is a traveling wave, because the operators R ω , R K ω defined in (AI) act only on a traveling wave. However, note that, if g is a traveling wave variation, the third component of D r G δ pu δ q´1T 0 g is a traveling wave and therefore the operators E ω , E K ω in (6.46) are well defined. The estimates (6.41)-(6.44) are proved as in Theorem 5.6 of [2], using Lemma 6.5.
Remark 7.2. From now on we consider the operator L in (7.12) acting on (a dense subspace of) the whole L 2 pTqˆL 2 pTq. In particular we extend the operator B´1 x to act on the whole L 2 pTq as in (3.20). In Sections 7.1-7.6 we are going to make several transformations, whose aim is to conjugate L to a constant coefficients Fourier multiplier, up to a pseudodifferential operator of order zero plus a remainder that satisfies tame estimates, both small in size, see L 9 in (7.168). Finally, in Section 7.7 we shall conjugate the restricted operator L ω in (7.1).
Notation. In (7.12) and hereafter any function a is identified with the corresponding multiplication operators h Þ Ñ ah, and, where there is no parenthesis, composition of operators is understood. For example, B x cB x means: h Þ Ñ B x pcB x hq. Lemma 7.3. The functions pη, ζq " T δ pϕq and B, r V , c defined in (7.11), (7.13) are quasi-periodic traveling waves. The functions pη, ζq " T δ pϕq are pevenpϕ, xq, oddpϕ, xqq, B is oddpϕ, xq, r V is evenpϕ, xq and c is evenpϕ, xq. The Hamiltonian operator L is reversible and momentum preserving.
Proof. The function pη, ζq " T δ pϕq is a quasi-periodic traveling wave and, using also Lemmata 3.32 and 3.26, we deduce that B, r V , c are quasi-periodic traveling waves. Since pη, ζq " T δ pϕq is reversible, we have that pη, ζq is pevenpϕ, xq, oddpϕ, xqq. Therefore, using also (2.6), we deduce that B is oddpϕ, xq, r V is evenpϕ, xq and c is evenpϕ, xq. By Lemmata 3.22 and 3.27, the operator L in (7.9) evaluated at the reversible quasi-periodic traveling wave W T δ pϕq is reversible and momentum preserving.
For the sequel we will always assume the following ansatz (satisfied by the approximate solutions obtained along the nonlinear Nash-Moser iteration of Section 9): for some constants µ 0 :" µ 0 pτ, νq ą 0, υ P p0, 1q, (cfr. Lemma 6.2) In order to estimate the variation of the eigenvalues with respect to the approximate invariant torus, we need also to estimate the variation with respect to the torus ipϕq in another low norm } } s1 for all Sobolev indexes s 1 such that s 1`σ0 ď s 0`µ0 , for some σ 0 :" σ 0 pτ, νq ą 0 . Thus, by (7.14), we have s1`σ0 ď 1 . The constants µ 0 and σ 0 represent the loss of derivatives accumulated along the reduction procedure of the next sections. What is important is that they are independent of the Sobolev index s. In the following sections we shall denote by σ :" σpτ, ν, k 0 q ą 0, σ N pq 0 q :" σ N pq 0 , τ, ν, k 0 q, σ M :" σ M pk 0 , τ, νq ą 0, ℵ M pαq constants (which possibly increase from lemma to lemma) representing losses of derivatives along the finitely many steps of the reduction procedure.
Remark 7.4. In the next sections µ 0 :" µ 0 pτ, ν, M, αq ą 0 will depend also on indexes M, α, whose maximal values will be fixed depending only on τ and ν (and k 0 which is however considered an absolute constant along the paper). In particular M is fixed in (8.5), whereas the maximal value of α depends on M , as explained in Remark 7.14.
We finally recall that I 0 " I 0 pω, κq is defined for all pω, κq P R νˆr κ 1 , κ 2 s and that the functions B, r V and c appearing in L in (7.12) are C 8 in pϕ, xq, as u " pη, ζq " T δ pϕq is.

(7.19)
Remark 7.5. We perform as a first step the time reparametrization (7.17) of L, with a function ppϕq which will be fixed only later in Step 4 of Section 7.3, to avoid otherwise a technical difficulty in the conjugation of the remainders obtained by the Egorov theorem in Step 1 of Section 7.3. We need indeed to apply the Egorov Proposition 3.9 for conjugating the additional pseudodifferential term in (7.12) due to vorticity.

(7.24)
The operator L 0 is Hamiltonian, reversible and momentum preserving.
Remark 7.7. The map P is not reversibility and momentum preserving according to Definitions 3.17, respectively 3.24, but maps (anti)-reversible, respectively traveling, waves, into (anti)reversible, respectively traveling, waves. Note that the multiplication operator for the function ρpϑq, which satisfies (7.24), is reversibility and momentum preserving according to Definitions 3.17 and 3.24.

Linearized good unknown of Alinhac
We conjugate the linear operator L 0 in (7.21), where we rename ϑ with ϕ, by the multiplication matrix operator obtaining (in view of (3.46)) where a is the function a :" r V B x`ρ pω¨B ϕ Bq . The matrix Z amounts to introduce, as in [26] and [6,2], a linearized version of the "good unknown of Alinhac".

Symmetrization and reduction of the highest order
The aim of this long section is to conjugate the Hamiltonian operator L 1 in (7.25) to the Hamiltonian operator L 5 in (7.89) whose coefficient m 3 2 of the highest order is constant. This is achieved in several steps. All the transformations of this section are symplectic.
Recalling the expansion (3.32) of the Dirichlet-Neumann operator, we first write is a small remainder in OPS´8.
Step 2: We now conjugate the operator L 2 in (7.54) with the multiplication matrix operator where qpϕ, yq is a real function, close to 1, to be determined. The maps Q and Q´1 are symplectic (cfr. (3.42)). We have that where A :" q´1`´γ 2 Gp0qB´1 y`a 1 B y`a4˘q`ρ q´1pω¨B ϕ qq , (7.61) B :"´q´1a 2 Gp0qa 2 q´1 , (7.62) We choose the function q so that the coefficients of the highest order terms of the off-diagonal operators B and C satisfy q´2a 2 2 " q 2 a 2 2 a 3 " m 3 2 pϕq , (7.65) with m 3 2 pϕq independent of x. This is achieved choosing and, recalling (7.56), the function β so that p1`β x pϕ, xqq 3 cpϕ, xq " mpϕq , (7.67) with mpϕq independent of x (the function c is defined in (7.22)). The solution of (7.67) is {3´1¯.
Remark 7.11. The number of regularizing iterations M P N will be fixed by the KAM reduction scheme in Section 8, see (8.5). Note that it is independent of the Sobolev index s.
So far the operator L 6 of Lemma 7.10 depends on two indexes M, N which provide respectively the order of the regularizing off-diagonal remainder R p´M,oq 6 and of the smoothing tame operator T 6,N . From now on we fix N " M . The goal of this section is to transform the operator L 6 in (7.108), with N " M (cfr. (7.124)), into the operator L 8 in (7.146) whose coefficient in front of B x is a constant. We first eliminate the x-dependence and then the ϕ-dependence.
Time reduction. In order to remove the ϕ-dependence of the coefficient xa pdq 1 y x pϕq of the first order term of the operator L 7 in (7.136), we conjugate L 7 with the map pVuqpϕ, xq :" upϕ, x`̺pϕqq, , (7.142) where ̺pϕq is a real periodic function to be chosen, see (7.145). Note that V is a particular case of the transformation E in (7.34) for a function βpϕ, xq " ̺pϕq, independent of x. We have that whereas the Fourier multipliers are left unchanged and a pseudodifferential operator of symbol apϕ, x, ξq transforms as V´1Oppapϕ, x, ξqqV " Oppapϕ, x´̺pϕq, ξqq . Note that (7.144) holds for any ω P DCpυ, τ q. We sum up these two transformations into the following lemma.

Reduction of the order 1/2
The goal of this section is to transform the operator L 8 in (7.146) into the operator L 9 in (7.168) whose coefficient in front of |D| 1{2 is a constant. We eliminate the x-dependence and, in view of the property (7.147), we obtain that this transformation removes also the ϕ-dependence.
Summing up, we have obtained the following lemma.

Proof of Theorem 8.2
The proof of Theorem 8.2 is inductive. We first show that pS1q n -pS3q n hold when n " 0.

(8.41)
The operator X is reversibility and momentum preserving.
Proof. Recalling (8.35), we have that r j :"´ipR pdq K q j j p0q, for all j P S c 0 . By the reversibility of R pdq K and (3.44) we deduce that r j P R. Recalling the definition of M 7 ps 0 q in (8.18) (with s " s 0 ) and Definition 3.14, we have, for all 0 ď |k| ď k 0 , }|B k λ R pdq K |h} s0 ď 2υ´| k| M 7 ps 0 q }h} s0 , and therefore |B k λ pR pdq K q j j p0q| À υ´| k| M 7 ps 0 q . Hence (8.45) follows. The last bound for |r j pi 1 q´r j pi 2 q| follows analogously.
9 Proof of Theorem 5.2 Theorem 5.2 is a consequence of Theorem 9.2 below. We consider the finite dimensional subspaces of traveling wave variations E n :" Ipϕq " pΘ, I, wqpϕq such that (3.53) holds : Θ " Π n Θ , I " Π n I , w " Π n w ( where Π n w :" Π Kn w are defined as in (3.6) with K n in (6.24), and we denote with the same symbol Π n gpϕq :" ř |ℓ|ďKn g ℓ e iℓ¨ϕ . Note that the projector Π n maps (anti)-reversible traveling variations into (anti)-reversible traveling variations.