Quasi-periodic water waves

Abstract

We present the result and the ideas of the recent paper (Berti and Montalto, Quasi-periodic standing wave solutions of gravity-capillary water waves, http://arxiv.org/abs/1602.02411, 2016) concerning the existence of Cantor families of small-amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean, with infinite depth, in irrotational regime, under the action of gravity and surface tension at the free boundary. These quasi-periodic solutions are linearly stable.

1 Introduction

An important research stream of the last decades has been the development of Kolmogorov–Arnold–Moser (KAM) theory for partial differential equations (PDEs). Actually, many PDEs arising in Physics are infinite-dimensional Hamiltonian or reversible dynamical systems

$$\begin{aligned} u_t = X (u), \quad u \in {\mathcal {H}}, \end{aligned}$$

(1.1)

defined on some phase space ${\mathcal {H}}$ of functions $u := u(x)$. Main examples are the nonlinear wave, Klein–Gordon and Schrödinger equations, in one and more space dimension, the KdV (Korteweg de Vries) equation, the water waves equations for fluids. The Hamiltonian and the reversible structure of such equations eliminates dissipative phenomena, like “friction”. When such PDEs are defined on a bounded domain, like a compact interval $x \in [0, \pi ]$, or $x \in \mathbb T^d := (\mathbb R\slash 2 \pi \mathbb Z)^d$ (periodic boundary conditions), or, more in general, x belongs to a compact manifold, their dynamics is expected to have a “recurrent” behaviour in time, and it is natural to expect the existence of many time-periodic and quasi-periodic solutions.

We recall that a quasi-periodic solution of the Eq. (1.1) with $\nu$ frequencies is a smooth solution defined for all times, of the form

$$\begin{aligned} u (t) = U (\omega t ) \in {\mathcal {H}} \qquad \mathrm{where} \qquad \mathbb T^\nu \ni \varphi \mapsto U( \varphi ) \in \mathcal {H} \end{aligned}$$

is $2 \pi$-periodic in the angular variables $\varphi := (\varphi _1, \ldots , \varphi _\nu )$ and the frequency vector $\omega \in \mathbb R^\nu$ is irrational, namely $\omega \cdot k \ne 0$, $\forall k \in \mathbb Z^\nu \setminus \{0\}$. In such a case, the linear flow $\{ \omega t \}_{t \in \mathbb R}$ is dense on $\mathbb T^\nu$ and the torus-manifold

$$\begin{aligned} {\mathcal {T}} := U ( \mathbb T^\nu ) \subset \mathcal {H} \end{aligned}$$

is invariant under the flow $\Phi ^t$ of (1.1). Denoting by $\Psi ^t_\omega : \mathbb T^\nu \rightarrow \mathbb T^\nu$ the linear flow

$$\begin{aligned} \Psi ^t_\omega (\varphi ) := \varphi + \omega t, \quad \varphi \in \mathbb T^\nu , \end{aligned}$$

the search of a quasi-periodic solution amounts to look for U such that

$$\begin{aligned} \Phi ^t \circ U = U \circ \Psi ^t_\omega . \end{aligned}$$

(1.2)

Note that (1.2) only requires that the flow $\Phi ^t$ is defined and smooth on the compact manifold ${\mathcal {T}} := U ( \mathbb T^\nu )$. This remark is important because, for PDEs, the flow could be ill-posed in a neighborhood of the torus ${\mathcal {T}}$. From a functional point of view, (1.2) is equivalent to the equation

$$\begin{aligned} \omega \cdot \partial _\varphi U ( \varphi ) - X ( U ( \varphi )) = 0, \quad \forall \varphi \in \mathbb T^\nu . \end{aligned}$$

When $\nu = 1$, the solution u(t) is periodic in time, with period $2 \pi \slash \omega$.

In the seventies, existence of periodic solutions for semilinear wave equations has been obtained by Rabinowitz [31] and Brezis–Coron–Nirenberg [12] (followed by many others) via minimax variational methods. These proofs work only to find periodic orbits with a rational frequency, because the other periods give rise to a “small divisors” problem. A fortiori, they cannot be extended for the search of quasi-periodic solutions.

Independently of these global results, local bifurcation results of periodic and quasi-periodic solutions were also proved. We cite the pioneering bifurcation results of periodic solutions by Rabinowitz [32, 33] for fully nonlinear forced wave equations with a small dissipation term. Then, we refer to the beginning of KAM theory for PDEs started with works of Kuksin [27] and Wayne [37], concerning also quasi-periodic solutions, and subsequently developed in the nineties by Craig–Wayne [16], Bourgain [11], Pöeschel [30], among others. All the latter results hold for 1-dimensional semilinear wave and Schrödinger equations. In the last years, KAM theory has been considerably further extended, mainly in two important directions:

• 1. Wave and Schrödinger equations in higher space dimensions,

• 2. quasi-linear PDEs, namely equations of the form $u_t = L (u) + N(u)$ where L is a linear differential (or pseudo-differential) operator of order $\mathrm{ord}(L)$ and the nonlinearity N depends on the derivatives $\partial _{x}^{\alpha } u$ of the same order $|\alpha | = \mathrm{ord}(L)$.

We have no space to report, here, the complete literature concerning KAM theory for PDEs for which we refer to the recent survey paper [6]. Here, we want to present the recent KAM result in [7] for the water waves equations, which are, indeed, a quasi-linear system.

1.1 Capillary–gravity standing wave solutions

Consider a 2-dimensional ocean, with infinite depth, filled by an incompressible fluid, in irrotational regime, under the action of gravity and capillarity at the surface. The fluid satisfies periodic boundary conditions and occupies the free boundary region

$$\begin{aligned} {\mathcal {D}}_\eta := \left\{ (x,y)\in \mathbb T\times \mathbb R\,:\, y<\eta (t,x), \quad \mathbb T:= \mathbb R\slash (2 \pi \mathbb Z) \right\} . \end{aligned}$$

Since the velocity field is irrotational, it is the gradient of a velocity potential $\Phi (t, x, y)$. The incompressibility condition means that $\Phi$ is an harmonic function on ${\mathcal {D}}_\eta$. In this context, the Euler equation for the motion of the fluid reduces to the Bernoulli equation. The water waves equations are

(1.3)

where g is the acceleration of gravity, $\kappa \in [\kappa _1, \kappa _2 ]$, $\kappa _1 > 0$, is the surface tension coefficient and

$$\begin{aligned} \frac{\eta _{xx}}{(1 + \eta _x^2)^{3/2}} = \partial _x \left( \frac{\eta _x}{\sqrt{1+ \eta _x^2}} \right) \end{aligned}$$

is the mean curvature of the free surface. The unknowns of the problem are the free surface $y = \eta (x)$ and the velocity potential $\Phi : {\mathcal {D}}_\eta \rightarrow \mathbb R$. The first equation in (1.3) is the Bernoulli condition (also called dynamics condition) according to which the jump of pressure across the free surface is proportional to the mean curvature. The last equation in (1.3) (also called Kinematic condition) expresses that the velocity of the free surface coincides with the one of the fluid particles, and therefore, the fluid particles on the free surface $y = \eta (x, t)$ remain on it along the fluid evolution. In the sequel, we shall assume (with no loss of generality) that the gravity constant $g = 1$.

Following Zakharov [35] and Craig–Sulem [14], the evolution problem (1.3) may be written as an infinite-dimensional Hamiltonian system. At each time $t \in \mathbb R$, the profile $\eta ( t, x )$ of the fluid and the value

$$\begin{aligned} \psi (t,x)=\Phi (t,x,\eta (t,x)) \end{aligned}$$

of the velocity potential $\Phi$ restricted to the free boundary uniquely determine the velocity potential $\Phi$ in the whole ${\mathcal {D}}_\eta$, solving (at each t) the elliptic problem

$$\begin{aligned}&\Delta \Phi =0 \ \text {in } {\mathcal {D}}_\eta , \ \Phi (x + 2 \pi , y) = \Phi (x, y), \, \ \Phi \arrowvert _{y=\eta }=\psi ,\ \nabla \Phi (x, y) \rightarrow 0 \text { as } y\rightarrow -\infty . \end{aligned}$$

As proved in [14, 35], system (1.3) is then equivalent to the system

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t \eta = G(\eta )\psi ,\\ \partial _t\psi + \eta + \frac{1}{2} \psi _x^2 -\frac{1}{2} \frac{\left( G(\eta )\psi + \eta _x \psi _x \right) ^2}{1+ \eta _x^2} = \kappa \frac{\eta _{xx}}{(1 + \eta _x^2)^{3/2}} \end{array} \right. \end{aligned}$$

(1.4)

where $G(\eta )$ is the so-called Dirichlet–Neumann operator defined by

$$\begin{aligned} G(\eta )\psi (x) := \sqrt{1+\eta _x^2}\, \partial _n \Phi \arrowvert _{y=\eta (x)} =(\partial _y \Phi )(x,\eta (x))-\eta _x (x) \, (\partial _x \Phi )(x,\eta (x)) \end{aligned}$$

(we denote by $\eta _x$ the space derivative $\partial _x \eta$.) The operator $G(\eta )$ is linear in $\psi$, self-adjoint with respect to the $L^2$ scalar product and semi-positive definite, actually its Kernel are only the constants. It depends in an analytic way with respect to the free boundary $\eta (x)$ and its derivative with respect to $\eta$ is

$$\begin{aligned} d_\eta G(\eta ) [\hat{\eta } ] \psi = - G(\eta ) (B \hat{\eta }) -\partial _x (V \hat{\eta }) \end{aligned}$$

(1.5)

where

$$\begin{aligned} B:= B(\eta ,\psi ) := \frac{\eta _x \psi _x + G(\eta )\psi }{ 1 + \eta _x^2 }, \qquad V := V(\eta ,\psi ) := \psi _x - B \eta _x. \end{aligned}$$

(1.6)

The vector $(V,B) = \nabla _{x,y} \Phi$ is the velocity field evaluated at the free surface $y = \eta (x)$. It is well known since Calderon that the Dirichlet–Neumann operator $G(\eta )$ is a pseudo-differential operator with principal symbol |D|; actually, $G(\eta ) - |D| \in OPS^{-\infty }$, if $\eta$ is ${\mathcal {C}}^\infty$.

The Eq. (1.4) are the Hamiltonian system (see [14, 35])

$$\begin{aligned} \partial _t \eta = \nabla _\psi H (\eta , \psi ), \quad \partial _t \psi = - \nabla _\eta H (\eta , \psi ) \end{aligned}$$

$$\begin{aligned} \partial _t u = J \nabla _u H (u), \quad u := \begin{pmatrix} \eta \\ \psi \\ \end{pmatrix}, \quad J := \begin{pmatrix} 0 &{}\quad \mathrm{Id} \\ - \mathrm{Id} &{}\quad 0 \\ \end{pmatrix}, \end{aligned}$$

(1.7)

where $\nabla$ denotes the $L^2$-gradient, and the Hamiltonian

$$\begin{aligned} H(\eta , \psi ) := \frac{1}{2} (\psi , G(\eta ) \psi )_{L^2(\mathbb T_x)} + \int _{\mathbb T} \frac{\eta ^2}{2} \, \mathrm{d}x + \kappa \int _{\mathbb T} \sqrt{1 + \eta _x^2} \, \mathrm{d}x \end{aligned}$$

(1.8)

is the sum of the kinetic energy

$$\begin{aligned} K := \frac{1}{2} (\psi , G(\eta ) \psi )_{L^2(\mathbb T_x)} = \frac{1}{2} \int _{{\mathcal {D}}_\eta } | \nabla \Phi |^2 (x,y) \mathrm{d}x \mathrm{d}y, \end{aligned}$$

the potential energy and the energy of the capillarity forces (area surface integral) expressed in terms of the variables $(\eta , \psi )$. In light of (1.7), the variables $(\eta , \psi )$ are symplectic “Darboux coordinates”. The symplectic structure induced by (1.7) is the standard Darboux 2-form

$$\begin{aligned} {\mathcal {W}}(u_1, u_2):= ( u_1, J u_2 )_{L^2(\mathbb T_x)} = ( \eta _1, \psi _2 )_{L^2 (\mathbb T_x)} - ( \psi _1, \eta _2 )_{L^2 (\mathbb T_x)} \end{aligned}$$

(1.9)

for all $u_1 = (\eta _1, \psi _1)$, $u_2 = (\eta _2, \psi _2)$.

The water-waves system (1.4)–(1.7) exhibits several symmetries. First of all, the mass

$$\begin{aligned} \int _\mathbb T\eta \, \mathrm{d}x \end{aligned}$$

is a prime integral of (1.4). Moreover,

$$\begin{aligned} \partial _t \int _\mathbb T\psi \, \mathrm{d}x = - \int _{\mathbb T} \eta \, \mathrm{d}x - \int _{\mathbb T} \nabla _\eta K \, \mathrm{d}x = - \int _{\mathbb T} \eta \, \mathrm{d}x \end{aligned}$$

because $\int _{\mathbb T} \nabla _\eta K \, \mathrm{d}x = 0$. This follows because $\mathbb R\ni c \mapsto K (c+ \eta , \psi )$ is constant (the bottom of the ocean is at $- \infty$) and so, $0 = d_\eta K (\eta , \psi ) [1] = (\nabla _\eta K, 1 )_{L^2(\mathbb T)}$. As a consequence, the subspace

$$\begin{aligned} \int _{\mathbb T} \eta \, \mathrm{d}x = \int _{\mathbb T} \psi \, \mathrm{d}x = 0 \end{aligned}$$

(1.10)

is invariant under the evolution of (1.4) and we shall restrict to solutions satisfying (1.10).

Also, the subspace of functions which are even in x,

$$\begin{aligned} \eta (x) = \eta (-x), \quad \psi (x) = \psi (- x ), \end{aligned}$$

(1.11)

is invariant under (1.4). Thus, we restrict $(\eta , \psi )$ to the phase space of $2 \pi$-periodic even functions with zero mean, i.e., which admit the Fourier expansion

$$\begin{aligned} \eta ( x) = \sum \limits _{j \ge 1} \eta _j \cos (j x ), \quad \psi ( x) = \sum \limits _{j \ge 1} \psi _j \cos (j x ) . \end{aligned}$$

(1.12)

In this case also, the velocity potential $\Phi (x,y)$ is even and $2 \pi$-periodic in x, and so the x-component of the velocity field $v = (\Phi _x, \Phi _y)$ vanishes at $x = k \pi$, $\forall k \in \mathbb Z$. Hence, there is no flux of fluid through the lines $x = k \pi$, $k \in \mathbb Z$, and a solution of (1.4) satisfying (1.11) physically describes the motion of a liquid confined between two walls.

Another important symmetry of the capillary water waves system is reversibility, namely the Eqs. (1.4)–(1.7) are reversible with respect to the involution $\rho : (\eta , \psi ) \mapsto (\eta , - \psi )$, or, equivalently, the Hamiltonian is even in $\psi$:

$$\begin{aligned} H \circ \rho = H, \quad H( \eta , \psi ) = H ( \eta , - \psi ), \quad \rho : (\eta , \psi ) \mapsto (\eta , - \psi ). \end{aligned}$$

(1.13)

As a consequence, it is natural to look for solutions of (1.4) satisfying

$$\begin{aligned} u(-t ) = \rho u(t), \quad \hbox {i.e.} \quad \eta (-t, x) = \eta (t, x), \ \psi (-t,x ) = - \psi (t,x ), \ \forall t, x \in \mathbb R, \end{aligned}$$

(1.14)

namely $\eta$ is even in time and $\psi$ is odd in time. Solutions of the water-waves equations (1.4) satisfying (1.12) and (1.14) are called capillary–gravity standing water waves.

Existence of small-amplitude time-periodic pure gravity (without surface tension) standing wave solutions has been proved by Iooss, Plotnikov, Toland in [23], see also [19, 20], and in [29] in finite depth. Existence of time-periodic capillary–gravity standing wave solutions has been recently proved by Alazard–Baldi [1]. The above results are proved via a Lyapunov–Schmidt decomposition combined with a Nash–Moser iterative scheme.

In [7], we have extended this result proving also the existence of time quasi-periodic capillary–gravity standing wave solutions of (1.4) as well as their linear stability. This is the result of Theorem 1.1. The reducibility of the linearized equations at the quasi-periodic solutions is not only an interesting dynamical information but it is also the key for the existence proof.

We also mention that existence of small-amplitude 2-d travelling gravity water wave solutions dates back to Levi-Civita [24] (standing waves are not traveling because they are even in space, see (1.11)). Existence of small-amplitude 3-d traveling gravity–capillary water wave solutions with space-periodic boundary conditions has been proved by Craig–Nicholls [13] (it is not a small divisor problem) and by Iooss–Plotnikov [21, 22] in the case of zero surface tension (in such a case, it is a small divisor problem).

The first existence results of quasi-periodic solutions of PDEs with unbounded perturbations (i.e., the nonlinearity contains derivatives) have been obtained by Kuksin [28] for KdV, see also Kappeler–Pöschel [26], by Liu–Yuan [25], Zhang–Gao–Yuan [36] for derivative NLS, by Berti–Biasco–Procesi [8]-[9] for derivative NLW. All these previous results still refer to semilinear perturbations, i.e., the order of the derivatives in the nonlinearity is strictly lower than the order of the constant coefficient (integrable) linear differential operator.

For quasi-linear, also fully nonlinear, perturbations, the first KAM results have been recently proved by Baldi–Berti–Montalto in [2,3,4] for Hamiltonian perturbations of Airy, KdV and mKdV equations. These techniques have been applied by Feola–Procesi [18] also to quasi-linear perturbations of 1-d Schrödinger equations.

The gravity–capillary water-waves system (1.4) is, indeed, a quasi-linear PDE. In suitable complex coordinates (having introduced the good unknown of Alinhac), it can be written in the symmetric form

$$\begin{aligned} {\mathtt u}_t = {\mathrm i} T(D) {\mathtt u} + N({\mathtt u}, \bar{\mathtt u} ), \quad {\mathtt u} \in \mathbb C, \end{aligned}$$

where

$$\begin{aligned} T(D) := |D|^{1/2} (1 - \kappa \partial _{xx})^{1/2} \end{aligned}$$

is the Fourier multiplier which describes the linear dispersion relation of the water-waves equations linearized at $(\eta , \psi ) = 0$ (see (1.15)–(1.18)), and the nonlinearity $N ({\mathtt u}, \bar{\mathtt u} )$ depends on the highest order term $| D |^{3/2} {\mathtt u}$ as well.

1.2 Main result

We look for small-amplitude quasi-periodic solutions of (1.4), and therefore, it is of main importance the dynamics of the linearized system at the equilibrium $(\eta , \psi ) = (0,0)$ (flat ocean and fluid at rest), namely

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t \eta = G(0)\psi ,\\ \partial _t\psi + \eta = \kappa \eta _{xx} \end{array} \right. \end{aligned}$$

(1.15)

where $G(0) = |D_x|$ is the Dirichlet–Neumann operator at the flat surface $\eta = 0$, namely

$$\begin{aligned} |D_x| \cos (jx) = |j| \cos (jx), \quad |D_x| \sin (jx) = |j| \sin (jx), \ \forall j \in \mathbb Z. \end{aligned}$$

In compact Hamiltonian form, the system (1.15) reads

$$\begin{aligned} \partial _t u = J \Omega u , \quad \Omega := \begin{pmatrix} 1 - \kappa \partial _{xx} &{}\quad 0 \\ 0 &{}\quad G(0) \\ \end{pmatrix}, \end{aligned}$$

(1.16)

which is the Hamiltonian system generated by the quadratic Hamiltonian (see (1.8))

$$\begin{aligned} H_{L} := \frac{1}{2} ( u, \Omega u )_{L^2 (\mathbb T_x)} = \frac{1}{2} (\psi , G(0) \psi )_{L^2(\mathbb T_x)} + \frac{1}{2} \int _{\mathbb T} \left( \eta ^2 + \kappa \eta _x^2\right) \mathrm{d}x. \end{aligned}$$

(1.17)

The standing wave solutions of the linear system (1.15) are

$$\begin{aligned} \eta (t,x) = \mathop \sum \limits _{ j \ge 1} a_j \cos (\omega _j t) \cos (jx), \quad \psi (t,x) = - \mathop \sum \limits _{ j \ge 1} a_j j^{-1} \omega _j \sin (\omega _j t) \cos (jx), \end{aligned}$$

where $a_j \in \mathbb R$, and the linear frequencies of oscillations are

$$\begin{aligned} \omega _j := \omega _j (\kappa ) := \sqrt{ j (1+ \kappa j^2) }, \quad j \ge 1. \end{aligned}$$

(1.18)

Fix an arbitrary finite subset ${\mathbb S}^+ \subset \mathbb N^+ := \{1,2, \ldots \}$ (tangential sites) and consider the linear standing-wave solutions

$$\begin{aligned} \eta (t,x)= & {} \sum _{ j \in {\mathbb S}^+} \sqrt{\xi _j} \cos (\omega _j t) \cos (jx), \nonumber \\ \psi (t,x)= & {} - \sum _{ j \in {\mathbb S}^+} \sqrt{\xi _j} j^{-1} \omega _j \sin (\omega _j t) \cos (jx), \ \xi _j > 0, \end{aligned}$$

(1.19)

which are Fourier supported in ${\mathbb S}^+$. The main result of [7] proves that such linear standing-wave solutions can be continued to solutions of the nonlinear water-waves Hamiltonian system (1.4) for most values of the surface tension parameter $\kappa \in [\kappa _1, \kappa _2]$. Theorem 1.1 below states the existence of quasi-periodic solutions

$$\begin{aligned} u (\tilde{\omega }t, x) = (\eta , \psi )( \tilde{\omega }t, x) \end{aligned}$$

of (1.4), with frequency $\tilde{\omega }:= ( \tilde{\omega }_j )_{j \in {\mathbb S}^+}$ (to be determined), close to the solutions (1.19) of (1.15), for most values of the surface tension parameter $\kappa \in [ \kappa _1, \kappa _2 ]$.

Let $\nu := | {\mathbb S}^+|$ denote the cardinality of ${\mathbb S}^+$. The function $u (\varphi , x) = (\eta , \psi )(\varphi , x)$, $\varphi \in \mathbb T^\nu$, belongs to the Sobolev spaces of $(2\pi )^{\nu +1}$-periodic real functions

$$\begin{aligned} H^s(\mathbb T^{\nu +1}, \mathbb R^2) := \left\{ u = (\eta , \psi ) : \eta , \psi \in H^s \right\} \end{aligned}$$

$$\begin{aligned} H^s := H^s(\mathbb T^{\nu +1}, \mathbb R)= & {} \left\{ f = \sum _{(\ell ,j) \in \mathbb Z^{\nu +1}} \widehat{f}_{\ell j} \, e^{{\mathrm i} (\ell \cdot \varphi + jx)} : \right. \nonumber \\&\left. \Vert f \Vert _s^2 := \sum _{(\ell ,j) \in \mathbb Z^{\nu +1}} | \widehat{f}_{\ell j}|^2 \langle \ell ,j \rangle ^{2s} < + \infty \right\} \end{aligned}$$

(1.20)

where $\langle \ell ,j \rangle := \max \{ 1, |\ell |, |j| \}$ with $| \ell | := \max _{i = 1, \ldots , \nu } |\ell _i|$. For

$$\begin{aligned} s \ge s_0 := \left[ \frac{\nu +1}{2}\right] +1 \in \mathbb N, \end{aligned}$$

the Sobolev spaces $H^s \subset L^\infty ( \mathbb T^{\nu +1})$ are an algebra with respect to the product of functions.

Theorem 1.1

(KAM for capillary–gravity periodic standing water waves [7]). For every choice of finitely many tangential sites ${\mathbb S}^+ \subset \mathbb N^+$, there exists $\bar{s} > s_0$, $\varepsilon _0 \in (0,1)$ such that for every $|\xi | \le \varepsilon _0^2$, $\xi := (\xi _j)_{j \in {\mathbb S}^+ }$, there exists a Cantor-like set $\mathcal {G}\subset [\kappa _1,\kappa _2]$ with asymptotically full measure as $\xi \rightarrow 0$, i.e.

$$\begin{aligned} \lim _{\xi \rightarrow 0} | \mathcal {G}| = \kappa _2- \kappa _1, \end{aligned}$$

such that, for any surface tension coefficient $\kappa \in \mathcal {G}$, the capillarity-gravity system (1.4) has a time quasi-periodic standing wave solution $u( \tilde{\omega }t, x ) = (\eta ( \tilde{\omega }t, x), \psi ( \tilde{\omega }t, x) )$, with Sobolev regularity $(\eta , \psi ) (\varphi , x) \in H^{\bar{s}} ( \mathbb T^\nu \times \mathbb T, \mathbb R^2)$, of the form

$$\begin{aligned} \begin{aligned} \eta ( \tilde{\omega }t,x)&= \mathop \sum \limits _{ j \in {\mathbb S}^+} \sqrt{\xi _j} \cos ({\tilde{\omega }}_j t) \cos (jx) + o( \sqrt{|\xi |} ), \\ \psi (\tilde{\omega },x)&= - \mathop \sum \limits _{ j \in {\mathbb S}^+} \sqrt{\xi _j} j^{-1} \omega _j \sin ({\tilde{\omega }}_j t) \cos (jx)+ o( \sqrt{|\xi |} ) \end{aligned} \end{aligned}$$

(1.21)

with a Diophantine frequency vector $\tilde{\omega }:= \tilde{\omega }(\kappa , \xi ) \in \mathbb R^\nu$ satisfying ${\tilde{\omega }}_j - \omega _j (\kappa ) \rightarrow 0$, $j \in {\mathbb S}^+$, as $\xi \rightarrow 0$. The terms $o( \sqrt{|\xi |} )$ are small in $H^{\bar{s}} ( \mathbb T^\nu \times \mathbb T, \mathbb R^2)$. Also, these quasi-periodic solutions are linearly stable.

Let us make some comments.

• 1. No global-in-time existence results concerning the initial value problem of the water waves equations (1.4) under periodic boundary conditions are known so far. The present Nash–Moser-KAM iterative procedure selects many values of the surface tension parameter $\kappa \in [\kappa _1, \kappa _2]$ which give rise to the quasi-periodic solutions (1.21), which are defined for all times. Clearly, by a Fubini-type argument, it also results that, for most values of $\kappa \in [\kappa _1, \kappa _2]$, there exist quasi-periodic solutions of (1.4) for most values of the amplitudes $| \xi | \le \varepsilon _0^2$. The fact that we find quasi-periodic solutions restricting to a proper subset of parameters is not a technical issue. The capillarity–gravity water-waves equations (1.4) are not expected to be integrable (albeit a rigorous proof is still lacking): yet, the third-order Birkhoff normal form possesses multiple resonant triads (Wilton ripples), see Craig–Sulem [15].

• 2. In the proof of Theorem 1.1, all the estimates depend on the surface tension coefficient $\kappa > 0$ and the result does not hold at the limit of zero surface tension $\kappa \rightarrow 0$. Because of capillarity, the linear frequencies (1.18) grow asymptotically $\sim \sqrt{\kappa } j^{3/2}$ as $j \rightarrow + \infty$. Without surface tension, the linear frequencies grow asymptotically as $\sim j^{1/2}$ and a different proof is required.

• 3. The quasi-periodic solutions (1.21) are mainly supported in Fourier space on the tangential sites ${\mathbb S}^+$. The dynamics of the water waves equations (1.4) restricted to the symplectic subspaces

  $$\begin{aligned} H_{{\mathbb S}^+} \!:= & {} \left\{ v = \sum _{j \in {\mathbb S}^+} \begin{pmatrix} \eta _j \\ \psi _j \\ \end{pmatrix} \cos (jx) \right\} , \nonumber \\ H_{{\mathbb S}^+}^\bot \!:= & {} \left\{ z = \sum _{j \in \mathbb N\setminus {\mathbb S}^+} \begin{pmatrix} \eta _j \\ \psi _j \\ \end{pmatrix} \cos (jx) \in H^1_0(\mathbb T_x) \right\} . \end{aligned}$$

  (1.22)

  is quite different. We call $v \in H_{{\mathbb S}^+}$ the tangential variable and $z \in H_{{\mathbb S}^+}^\bot$ the normal one. On the finite-dimensional subspace $H_{{\mathbb S}^+},$ we shall describe the dynamics by introducing the action–angle variables $(\theta , I) \in \mathbb T^\nu \times \mathbb R^\nu$ as in (2.2). The quasi-periodic solutions (1.21) of (1.4) are, therefore, close to $\mathbb T^\nu \times \{ \xi \} \times \{ z = 0 \}$, $\xi \in \mathbb R_+^\nu$. On the infinite-dimensional subspace $H_{{\mathbb S}^+}^\bot$, the solution stays, forever, close to the elliptic equilibrium $z = 0$, in some Sobolev norm. A first key observation is that, for most values of the surface tension parameter $\kappa \in [ \kappa _1, \kappa _2 ]$, the unperturbed linear frequencies (1.18), regrouped on the tangential and normal components

  $$\begin{aligned} \qquad {\mathbf {\omega }} (\kappa ) := ( \omega _j (\kappa ) )_{j \in {\mathbb S}^+}, \quad {\mathbf {\Omega }} (\kappa ) := ( \Omega _j (\kappa ) )_{j \in \mathbb N^+ \setminus {\mathbb S}^+} := ( \omega _j (\kappa ) )_{j \in \mathbb N^+ \setminus {\mathbb S}^+},\nonumber \\ \end{aligned}$$

  (1.23)

  are Diophantine, namely

  $$\begin{aligned} |{\mathbf {\omega }} (\kappa ) \cdot \ell | \ge \frac{\gamma }{| \ell |^\tau }, \quad \forall \ell \in \mathbb Z^\nu \setminus \{0\}, \end{aligned}$$

  and also satisfy stronger non-resonance properties, the so-called first- and second-order Melnikov non-resonance conditions, see (2.14), which are non-resonance conditions between the tangential and the normal frequencies. We shall prove this fact by degenerate KAM theory, see Sect. 2.2. For such values of $\kappa \in [\kappa _1, \kappa _2 ]$, the solutions (1.19) of the linear equation (1.15) are already sufficiently good approximate quasi-periodic solutions of the nonlinear water-waves system (1.4). Since the parameter space $[\kappa _1, \kappa _2 ]$ is fixed, the small divisor constant $\gamma$ can be taken $\gamma = o( \varepsilon ^a)$ with $a > 0$ small as needed, see (2.15). As a consequence, for proving the continuation of (1.19) to solutions of the nonlinear water-waves system (1.4), all the terms which are at least quadratic in (1.4) are already perturbative (i.e. in (2.1), it is sufficient to regard the vector field $\varepsilon X_{P_{\varepsilon }}$ as a perturbation of the linear vector field $J \Omega$).

Linear stability. The quasi-periodic solutions $u( \tilde{\omega }t) = (\eta ( \tilde{\omega }t), \psi ( \tilde{\omega }t) )$ found in Theorem 1.1 are linearly stable. This is not only a dynamically relevant information but also an essential ingredient of the existence proof (it is not necessary for time-periodic solutions as in [1, 19, 20, 23]). Let us state precisely the result. Around each invariant torus, there exist symplectic coordinates

$$\begin{aligned} (\phi , y, w) = (\phi , y, \eta , \psi ) \in \mathbb T^\nu \times \mathbb R^\nu \times H_{{\mathbb S}^+}^\bot \end{aligned}$$

(see [10]) in which the water-waves Hamiltonian reads

$$\begin{aligned}&\omega \cdot y + \frac{1}{2} K_{2 0}(\phi ) y \cdot y + \left( K_{11}(\phi ) y , w \right) _{L^2(\mathbb T_x)} + \frac{1}{2} \left( K_{02}(\phi ) w , w \right) _{L^2(\mathbb T_x)}\\&\quad +\, K_{\ge 3}(\phi , y, w) \end{aligned}$$

where $K_{\ge 3}$ collects the terms at least cubic in the variables (y, w). In these coordinates, the quasi-periodic solution reads $t \mapsto (\omega t , 0, 0 )$ (for simplicity, we denote the frequency $\tilde{\omega }$ of the quasi-periodic solution by $\omega$) and the corresponding linearized water-waves equations are

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{ \phi } = K_{20}(\omega t)[ y] + K_{11}^T(\omega t)[ w] \\ \dot{ y} = 0 \\ \dot{ w} = J K_{02}(\omega t)[ w] + J K_{11}(\omega t)[ y]\, . \end{array}\right. } \end{aligned}$$

Thus, the actions $y (t) = y(0)$ do not evolve in time, and the third equation reduces to the PDE

$$\begin{aligned} \dot{w} = J K_{02}(\omega t)[ w] + J K_{11}(\omega t)[ y (0)]. \end{aligned}$$

(1.24)

The self-adjoint operator $K_{02} (\omega t)$ is, up to a finite-dimensional remainder, the restriction to $H_{{\mathbb S}^+}^\bot$ of the linearized water-waves vector field $\partial _u \nabla H (u (\omega t ))$, which is explicitly computed in (2.19).

Denote $H^{s}_\bot := H^{s}_\bot (\mathbb T_x ) := H^s (\mathbb T_x) \cap H_{{\mathbb S}}^\bot$ (real or complex valued). We prove the existence of bounded and invertible “symmetrizer” maps $\mathbf{W}_{m, \infty }(\varphi )$, $m = 1,2$ such that $\forall \varphi \in \mathbb T^\nu$, $s \ge s_0$,

$$\begin{aligned}&\mathbf{W}_{m, \infty }(\varphi ) : \nonumber \\&\quad \left( H^s(\mathbb T_x, \mathbb C) \times H^s(\mathbb T_x, \mathbb C) \right) \cap H_{{\mathbb S}_+}^\bot \rightarrow \big ( H^s(\mathbb T_x, \mathbb R) \times H^{s - \frac{1}{2}}(\mathbb T_x, \mathbb R) \big ) \cap H_{{\mathbb S}_+}^\bot , \nonumber \\ \end{aligned}$$

(1.25)

$$\begin{aligned}&\mathbf{W}_{m, \infty }^{-1}(\varphi ) : \nonumber \\&\quad \big ( H^s(\mathbb T_x, \mathbb R) \times H^{s - \frac{1}{2}}(\mathbb T_x, \mathbb R) \big ) \cap H_{{\mathbb S}_+}^\bot \rightarrow \big ( H^s(\mathbb T_x, \mathbb C) \times H^s(\mathbb T_x, \mathbb C) \big ) \cap H_{{\mathbb S}_+}^\bot ,\nonumber \\ \end{aligned}$$

(1.26)

and such that, under the quasi-periodic-in-time change of variables

$$\begin{aligned} w = (\eta , \psi ) = \mathbf{W}_{1, \infty } (\omega t) w_\infty , \quad w_\infty = ( {\mathtt w}_\infty , \overline{\mathtt w}_\infty ), \end{aligned}$$

the Eq. (1.24) transforms into the diagonal system

$$\begin{aligned} \partial _t{ w_\infty }= & {} - {\mathrm i} \mathbf{D}_\infty w_\infty + f_\infty (\omega t), \nonumber \\ f_\infty (\omega t):= & {} \mathbf{W}_{2, \infty }(\varphi ) (\omega t)^{- 1} J K_{11}(\omega t)[ y(0)] = \begin{pmatrix} {\mathtt f}_\infty (\omega t) \\ \overline{\mathtt f}_\infty (\omega t) \end{pmatrix} \end{aligned}$$

(1.27)

where, denoting ${\mathbb S}_0 := {\mathbb S}_+ \cup (- {\mathbb S}_+) \cup \{ 0 \} \subseteq \mathbb Z$,

$$\begin{aligned} \mathbf{D}_\infty := \begin{pmatrix} D_\infty &{}\quad 0 \\ 0 &{}\quad - D_\infty \end{pmatrix}, \quad D_\infty := \mathrm{diag}_{j \in {\mathbb S}_0^c} \{ \mu _j^\infty \}, \quad \mu _j^\infty \in \mathbb R, \end{aligned}$$

(1.28)

is a Fourier multiplier operator of the form

$$\begin{aligned} \mu _j^{\infty } := \mathtt m_3^{\infty } \sqrt{|j|(1 + \kappa j^2)} + {\mathtt m}_1^{\infty } |j|^{\frac{1}{2}} + r_j^\infty , \ j \in {\mathbb S}_0^c, \quad r_j^\infty = r_{- j}^\infty , \end{aligned}$$

(1.29)

where, for some ${\mathtt a} > 0$,

$$\begin{aligned} \mathtt m_3^{\infty } = 1 + O(\varepsilon ^{\mathtt a}), \ {\mathtt m}_1^{\infty } = O(\varepsilon ^{\mathtt a} ), \ \sup _{j \in {\mathbb S}_0^c} | r_j^{\infty }| = O(\varepsilon ^{\mathtt a} ), \ \forall |k| \le k_0, \end{aligned}$$

see (2.12)–(2.13), (2.15) and $k_0 \in \mathbb N$ is a constant which depends only on the linear frequencies $\omega _j (\kappa )$ defined in (1.18). The ${\mathrm i} \mu _j^{\infty }$ are the Floquet exponents of the quasi-periodic solution. The fact that they are purely imaginary is a consequence of the reversible structure of the water-waves equations.

The second equation of system (1.27) is actually the complex conjugated of the first one, and (1.27) reduces to the infinitely many decoupled scalar equations

$$\begin{aligned} \partial _t {\mathtt w}_{\infty , j} = - {\mathrm i} \mu _j^\infty {\mathtt w}_{\infty , j} + {\mathtt f}_{\infty , j} (\omega t), \quad \forall j \in {\mathbb S}_0^c. \end{aligned}$$

By variation of constants, the solutions are

$$\begin{aligned} {\mathtt w}_{\infty , j} (t) = c_j e^{ - {\mathrm i} \mu _j^\infty t} + {\mathtt v}_{\infty , j} (t) , \quad {\mathtt v}_{\infty , j} (t) := \sum _{\ell \in \mathbb Z^\nu } \frac{ {\mathtt f}_{\infty , j, \ell } \, e^{{\mathrm i} \omega \cdot \ell t } }{ {\mathrm i} (\omega \cdot \ell + \mu _j^\infty ) } , \quad \forall j \in {\mathbb S}^c_0.\nonumber \\ \end{aligned}$$

(1.30)

Since the Melnikov conditions (2.14) hold at a solution, then ${\mathtt v}^{\infty }_j (t)$ in (1.30) is well defined. Moreover, (1.25) implies $\Vert f_\infty (\omega t)\Vert _{H^s_x \times H^s_x} \le C |y (0)|$. As a consequence, the Sobolev norm of the solution of (1.27) with initial condition $w_{\infty } (0) \in H^{{\mathfrak s}_0} (\mathbb T_x)$, ${\mathfrak s}_0 < s$ (in a suitable range of values), satisfies

$$\begin{aligned} \Vert w_\infty (t) \Vert _{H^{{\mathfrak s}_0}_x \times H^{{\mathfrak s}_0}_x} \le C(s) (| y (0) | + \Vert w_\infty (0) \Vert _{H^{{\mathfrak s}_0}_x \times H^{{\mathfrak s}_0}_x}), \end{aligned}$$

and, for all $t \in \mathbb R$, using (1.25), (1.26), we get

$$\begin{aligned} \Vert (\eta , \psi )(t) \Vert _{H^{{\mathfrak s}_0}_x \times H^{ {\mathfrak s}_0 -\frac{1}{2}}_x} \le \Vert (\eta (0),\psi (0)) \Vert _{H^{{\mathfrak s}_0}_x \times H^{ {\mathfrak s}_0 - \frac{1}{2}}_x} \end{aligned}$$

which proves the linear stability of the quasi-periodic solution. Note that the profile $\eta \in H^{{\mathfrak s}_0} (\mathbb T_x)$ is more regular than the velocity potential $\psi \in H^{ {\mathfrak s}_0 - \frac{1}{2}}(\mathbb T_x)$, as it is expected in the presence of surface tension.

Clearly, a crucial point is the diagonalization of (1.24) into (1.28). With respect to [1], this requires to analyze more in detail the pseudo-differential nature of the operators obtained after each conjugation and to implement a KAM scheme with second-order Melnikov non-resonance conditions, as we shall explain in detail below. We now present the main ideas of the proof.

2 Ideas of the proof

We prove Theorem 1.1 by a Nash–Moser iterative scheme in Sobolev spaces formulated as a “Theorem of hypothetic conjugation” á la Herman (Sect. 2.1) plus a degenerate KAM theory argument to perform the measure estimates in $\kappa$ (Sect. 2.2). The core of the Nash–Moser scheme is to prove that the linearized operators obtained at any approximate solution are invertible, with an inverse that satisfies tame estimates in Sobolev spaces. We explain how to prove this property in Sect. 2.3.

First of all, instead of working in a shrinking neighborhood of the origin, it is a convenient device to rescale the variable $u \mapsto \varepsilon u$ with $u = O (1)$, writing (1.4), (1.7) as

$$\begin{aligned} \partial _t u = J \Omega u + \varepsilon X_{P_\varepsilon } (u) \end{aligned}$$

(2.1)

where $J \Omega$ is defined in (1.16) and $X_{P_\varepsilon } (u)$ is the Hamiltonian vector field generated by the Hamiltonian

$$\begin{aligned} {\mathcal {H}}_\varepsilon (u) := \varepsilon ^{-2} H(\varepsilon u ) = H_L (u) + \varepsilon P_\varepsilon (u) \end{aligned}$$

where H is the water-waves Hamiltonian (1.8) and $H_L$ is defined in (1.17).

We decompose the phase space as in (1.22),

$$\begin{aligned} H_{0, \mathrm{even}}^1 = H_{{\mathbb S}^+} \oplus H_{{\mathbb S}^+}^\bot , \end{aligned}$$

and we introduce action–angle variables on the tangential sites by setting

$$\begin{aligned}&\eta _j := \sqrt{\frac{2}{\pi }} \, \Lambda _j^{1/2} \sqrt{\xi _j + I_j} \cos (\theta _j ) , \quad \psi _j := - \sqrt{\frac{2}{\pi }} \, \Lambda _j^{- 1/2} \sqrt{\xi _j + I_j} \, \sin (\theta _j ) \, \nonumber \ \\&\Lambda _j := \sqrt{j (1 + \kappa j^2)^{-1}},\quad \, j \in {\mathbb S}^+, \end{aligned}$$

(2.2)

where $\xi _j > 0$, $j=1, \ldots , \nu$, are positive constants, and $| I_j | \le \xi _j$. The symplectic 2-form in (1.9) then reads

$$\begin{aligned} {\mathcal {W}} := \left( \mathop \sum \limits _{j \in {\mathbb S}^+} d \theta _j \wedge d I_j \right) \oplus {\mathcal {W}}_{|H_{{\mathbb S}^+}^\bot } \end{aligned}$$

and the Hamiltonian system (2.1) transforms into the new Hamiltonian system

$$\begin{aligned} \dot{\theta }= \partial _I H_\varepsilon ( \theta , I, z ), \quad \ \dot{I} = - \partial _\theta H_\varepsilon ( \theta , I, z ), \quad z_t = J \nabla _z H_\varepsilon ( \theta , I, z ) \end{aligned}$$

(2.3)

generated by the Hamiltonian

$$\begin{aligned} H_\varepsilon := {\mathcal {H}}_\varepsilon \circ A = \varepsilon ^{-2} H \circ \varepsilon A \end{aligned}$$

(2.4)

where

$$\begin{aligned} A (\theta , I, z) := v (\theta , I) + z := \sum _{j \in {\mathbb S}^+} \sqrt{\frac{2}{\pi }} \begin{pmatrix} \ \Lambda _j^{1/2} \sqrt{\xi _j + I_j} \, \cos (\theta _j ) \\ - \Lambda _j^{- 1/2} \sqrt{\xi _j + I_j} \, \sin (\theta _j ) \\ \end{pmatrix} \cos (jx) + z. \end{aligned}$$

(2.5)

We denote by

$$\begin{aligned} X_{H_\varepsilon } := (\partial _I H_\varepsilon , - \partial _\theta H_\varepsilon , J \nabla _z H_\varepsilon ) \end{aligned}$$

the Hamiltonian vector field in the variables $( \theta , I, z ) \in \mathbb T^\nu \times \mathbb R^\nu \times H_{{\mathbb S}^+}^\bot$. The involution $\rho$ in (1.13) becomes

$$\begin{aligned} \tilde{\rho }: (\theta , I, z ) \mapsto (- \theta , I, \rho z ). \end{aligned}$$

(2.6)

By (1.8) and (2.4), the Hamiltonian $H_\varepsilon$ reads (up to a constant)

$$\begin{aligned} H_\varepsilon = {\mathcal {N}} + \varepsilon P, \quad {\mathcal {N}} := H_L \circ A = {\mathbf {\omega }} (\kappa ) \cdot I + \frac{1}{2} (z, \Omega z)_{L^2_x}, \quad P := P_\varepsilon \circ A, \end{aligned}$$

(2.7)

where ${\mathbf {\omega }} (\kappa )$ is defined in (1.23) and $\Omega$ in (1.16). We look for an embedded invariant torus

$$\begin{aligned} i : \mathbb T^\nu \rightarrow \mathbb T^\nu \times \mathbb R^\nu \times H_{{\mathbb S}^+}^\bot , \quad \varphi \mapsto i (\varphi ) := ( \theta (\varphi ), I (\varphi ), z (\varphi )) \end{aligned}$$

of the Hamiltonian vector field $X_{H_\varepsilon }$ filled by quasi-periodic solutions with Diophantine frequency $\omega \in \mathbb R^\nu$.

2.1 Nash–Moser theorem of hypothetic conjugation

The Hamiltonian $H_\varepsilon$ in (2.7) is a perturbation of the isochronous system with Hamiltonian ${\mathcal {N}}$. The expected quasi-periodic solutions of the corresponding Hamiltonian system (2.3) will have a shifted frequency—to be found—close to the linear frequencies $\omega _j (\kappa )$ in (1.18), which depend on the nonlinear term P and the amplitudes $\xi _j$.

In view of that, we introduce the family of Hamiltonians

$$\begin{aligned} H_\alpha := {\mathcal {N}}_\alpha + \varepsilon P, \quad {\mathcal {N}}_\alpha := \alpha \cdot I + \frac{1}{2} (z, \Omega z)_{L^2_x}, \quad \alpha \in \mathbb R^\nu , \end{aligned}$$

(2.8)

which depend on a constant vector $\alpha \in \mathbb R^\nu$. For the value $\alpha = \mathbf {\omega }(\kappa )$, we have $H_{\alpha } = H_\varepsilon$. Then, we look for a zero $(i, \alpha )$ of the nonlinear operator

$$\begin{aligned} {\mathcal {F}} (i, \alpha )&:= {\mathcal {F}} (i, \alpha , \omega , \kappa , \varepsilon ) := \omega \! \cdot \! \partial _\varphi i (\varphi ) - X_{H_\alpha }(i(\varphi ))\nonumber \\&= \omega \! \cdot \! \partial _\varphi i (\varphi ) - (X_{{\mathcal {N}}_\alpha } + \varepsilon X_{P}) (i(\varphi ) ) \nonumber \\&:= \left( \begin{array}{c} \omega \! \cdot \! \partial _\varphi \theta (\varphi ) - \alpha - \varepsilon \partial _I P ( i(\varphi ) ) \\ \omega \! \cdot \! \partial _\varphi I (\varphi ) + \varepsilon \partial _\theta P( i(\varphi ) ) \\ \omega \! \cdot \! \partial _\varphi z (\varphi ) - J (\Omega z + \varepsilon \nabla _z P ( i(\varphi ) )) \end{array} \right) \end{aligned}$$

(2.9)

for some Diophantine vector $\omega \in \mathbb R^\nu$. If ${\mathcal {F}} (i, \alpha ) = 0$, then $\varphi \mapsto i (\varphi )$ is an embedded torus, invariant for the Hamiltonian vector field $X_{H_\alpha }$, filled by quasi-periodic solutions with frequency $\omega$.

Since each Hamiltonian $H_\alpha$ in (2.8) is reversible, we look for reversible solutions of $\mathcal {F}(i, \alpha ) = 0$, namely satisfying ${\tilde{\varrho }} i (\varphi ) = i (- \varphi )$ (see (2.6)), i.e.

$$\begin{aligned} \theta (-\varphi ) = - \theta (\varphi ), \quad I(-\varphi ) = I(\varphi ), \quad z (- \varphi ) = ( \varrho z)(\varphi ). \end{aligned}$$

(2.10)

The Sobolev norm of the periodic component of the embedded torus

$$\begin{aligned} {\mathfrak I}(\varphi ) := i (\varphi ) - (\varphi ,0,0) := ( {\Theta } (\varphi ), I(\varphi ), z(\varphi )), \quad \Theta (\varphi ) := \theta (\varphi ) - \varphi , \end{aligned}$$

is

$$\begin{aligned} \Vert {\mathfrak I} \Vert _s := \Vert \Theta \Vert _{H^s_\varphi } + \Vert I \Vert _{H^s_\varphi } + \Vert z \Vert _s \end{aligned}$$

where $\Vert z \Vert _s := \Vert z \Vert _{H^s_{\varphi ,x}} = \Vert \eta \Vert _s + \Vert \psi \Vert _s$, see (1.20).

For the next theorem, we recall that $k_0$ is the index of non-degeneracy provided by Proposition 2.3 and it depends only on the linear unperturbed frequencies $\omega _j (\kappa )$. Therefore, it is considered as an absolute constant and we will often omit to write explicitly the dependence of the constants with respect to $k_0$. We look for quasi-periodic solutions with frequency $\omega$ belonging to a $\delta$-neighborhood (independent of $\varepsilon$)

$$\begin{aligned} {\mathtt \Omega } := \left\{ \omega \in \mathbb R^\nu \, : \, \mathrm{dist} \left( \omega , {\mathbf {\omega }}[\kappa _1, \kappa _2] \right) < \delta , \ \delta > 0 \right\} \end{aligned}$$

of the unperturbed linear frequencies ${\mathbf {\omega }}[\kappa _1, \kappa _2]$ defined in (1.23).

Theorem 2.1

(Nash–Moser theorem of hypothetic conjugation). Fix finitely many tangential sites ${\mathbb S}^+ \subset \mathbb N^+$ and let $\nu := |{\mathbb S}^+ |$. Let $\tau \ge 1$. There exist constants $\varepsilon _0 > 0$, $a_0 := a_0 (\nu , \tau , k_0) > 0$ and $k_1 := k_1(\nu , k_0, \tau ) > 0$ such that, for all $\gamma = \varepsilon ^a$, $0< a < a_0$, $\varepsilon \in (0, \varepsilon _0)$, there exists a ${k_0}$-times differentiable function

$$\begin{aligned} \alpha _\infty : {\mathtt \Omega } \times [\kappa _1, \kappa _2] \mapsto \mathbb R^\nu , \quad \alpha _\infty (\omega , \kappa ) = \omega + r_\varepsilon (\omega , \kappa ),\quad |r_\varepsilon |^{k_0, \gamma } \le C \varepsilon \gamma ^{-(1 + k_1)},\nonumber \\ \end{aligned}$$

(2.11)

a family of embedded tori $i_\infty$ defined for all $\omega \in {\mathtt \Omega }$ and $\kappa \in [\kappa _1, \kappa _2]$ satisfying the reversibility property (2.10) and

$$\begin{aligned} \Vert i_\infty (\varphi ) - (\varphi ,0,0) \Vert _{s_0}^{k_0, \gamma } \le C \varepsilon \gamma ^{-(1 + k_1)}, \end{aligned}$$

a sequence of ${k_0}$-times differentiable functions $\mu _j^\infty : {\mathtt \Omega } \times [\kappa _1, \kappa _2] \rightarrow \mathbb R$, $j \in \mathbb N^+ \setminus {\mathbb S}^+$, of the form

$$\begin{aligned} \mu _j^\infty (\omega , \kappa ) = \mathtt m_3^\infty (\omega , \kappa ) j^\frac{1}{2} (1 + \kappa j^2)^{\frac{1}{2}} + \mathtt m_1^\infty (\omega , \kappa ) j^{\frac{1}{2}} + r_j^\infty (\omega , \kappa ) \end{aligned}$$

(2.12)

satisfying

$$\begin{aligned} | \mathtt m_3^\infty - 1|^{k_0, \gamma } + | \mathtt m_1^\infty |^{k_0, \gamma } \le C \varepsilon , \qquad \sup _{j \in {\mathbb S}^c} |r_j^\infty |^{k_0, \gamma } \le C \varepsilon \gamma ^{- k_1}, \end{aligned}$$

(2.13)

such that for all $(\omega , \kappa )$ in the Cantor-like set

$$\begin{aligned} {\mathcal {C}}_\infty ^{\gamma }:= & {} \bigg \{ (\omega , \kappa ) \in {\mathtt \Omega } \times [\kappa _1, \kappa _2] \, : \, |\omega \cdot \ell | \ge \gamma \langle \ell \rangle ^{-\tau }, \, \forall \ell \in \mathbb Z^{\nu } \setminus \{ 0 \}, \nonumber \\&|\omega \cdot \ell + \mu _j^\infty (\omega , \kappa ) | \ge 4 \gamma j^{\frac{3}{2}} \langle \ell \rangle ^{- \tau }, \, \forall \ell \in \mathbb Z^\nu , \, j \in \mathbb N^+ \setminus {\mathbb S}^+ \nonumber \\&\quad \text {(1-Melnikov conditions)} \nonumber \\&|\omega \cdot \ell + \mu _j^\infty (\omega , \kappa ) - \varsigma \mu _{j'}^\infty (\omega , \kappa ) | \ge \frac{4 \gamma | j^{\frac{3}{2}} - \varsigma j'^{\frac{3}{2}}|}{ \langle \ell \rangle ^{\tau }}, \nonumber \\&\quad \forall \ell \in \mathbb Z^\nu ,j, j' \in \mathbb N^+ \setminus {\mathbb S}^+,\varsigma = \pm 1, (2\hbox {-}{Melnikov}) \bigg \} \end{aligned}$$

(2.14)

the function $i_\infty (\varphi ) := i_\infty (\omega , \kappa , \varepsilon )(\varphi )$ is a solution of ${\mathcal {F}}( i_\infty , \alpha _\infty (\omega , \kappa ) , \omega , \kappa , \varepsilon ) = 0$. As a consequence, the embedded torus $\varphi \mapsto i_\infty (\varphi )$ is invariant for the Hamiltonian vector field $X_{H_{\alpha _\infty (\omega , \kappa )} }$ and it is filled by quasi-periodic solutions with frequency $\omega$.

In Theorem 2.1, we are not concerned about the measure of ${\mathcal {C}}_\infty ^\gamma$, in particular in investigating if it is not empty. Note that the Cantor-like set ${\mathcal {C}}_\infty ^\gamma$ in (2.14) for which a solution exists is defined only in terms of the “final” solution $i_\infty$ and the “final” normal perturbed frequencies $\mu _j^\infty$, $j \in \mathbb N^+ \setminus {\mathbb S}^+$, which are defined for all the values of $(\omega , \kappa ) \in {\mathtt \Omega } \times [\kappa _1, \kappa _2]$ by a Whitney-type extension argument.

Theorem 2.1 is called of “hypothetic conjugation” because it does not prove the existence of a quasi-periodic solution for the original Hamiltonian $H_\varepsilon$ but just for a nearby Hamiltonian $H_{\alpha _\infty (\omega , \kappa ) }$. The aim is, now, to deduce Theorem 1.1 from Theorem 2.1: we have to prove the existence of quasi-periodic solutions of the water-waves equations (1.4), and not only of the system with modified Hamiltonian $H_\alpha$ with $\alpha := \alpha _\infty (\omega , \kappa )$. Therefore, we have to prove that the curve of the unperturbed linear frequencies

$$\begin{aligned}{}[\kappa _1, \kappa _2] \ni \kappa \mapsto \mathbf {\omega }(\kappa ) := ( \sqrt{j(1+ \kappa j^2) } )_{j \in {\mathbb S}^+} \in \mathbb R^\nu \end{aligned}$$

intersects the image $\alpha _\infty ({\mathcal {C}}^\gamma _\infty )$, under the map $\alpha _\infty$ of the Cantor set ${\mathcal {C}}^\gamma _\infty$, for “most” values of $\kappa \in [\kappa _1, \kappa _2]$.

The above functional setting perspective is in the spirit of the Théoréme de conjugaison hypothétique of Herman proved by Fejoz [17] for finite-dimensional Hamiltonian systems. A relevant difference is that in [17], in addition to $\alpha$, also the normal frequencies are introduced as independent parameters, unlike in Theorem 2.1. Actually, for PDEs, it seems more convenient the present formulation: it is, indeed, a major point of the work to know the asymptotic expansion (1.29) of the Floquet exponents.

2.2 Measure estimates and degenerate KAM theory

For any $\beta \in \alpha _{\infty }({\mathcal {C}}_{\infty }^{\gamma })$, Theorem 2.1 proves the existence of an embedded invariant torus filled by quasi-periodic solutions with Diophantine frequency $\omega = \alpha _\infty ^{-1}(\beta , \kappa )$ for the Hamiltonian

$$\begin{aligned} H_\beta = \beta \cdot I + \frac{1}{2} ( z, \Omega z)_{L^2_x} + \varepsilon P. \end{aligned}$$

Theorem 2.2 now proves that for “most” values of $\kappa \in [\kappa _1, \kappa _2]$, the vector $\beta = \mathbf {\omega }(\kappa ) \in \alpha _\infty ({\mathcal {C}}^\gamma _\infty )$. Hence, for such values of $\kappa$, we have found an embedded invariant torus for the Hamiltonian $H_\varepsilon$ in (2.7), filled by quasi-periodic solutions with Diophantine frequency $\omega = \alpha _\infty ^{-1}( \mathbf {\omega }(\kappa ), \kappa )$.

Theorem 2.2

(Measure estimates). Let

$$\begin{aligned} \gamma = \varepsilon ^a, \ \ 0< a < \min \{a_0, 1 / (1 + k_0 + k_1) \}, \quad \tau > k_0 ( \nu + 4). \end{aligned}$$

(2.15)

Then, the measure of the set

$$\begin{aligned} {\mathcal {G}}_\varepsilon := \left\{ \kappa \in [\kappa _1, \kappa _2] \, : \left( \alpha _\infty ^{-1} ( {\mathbf {\omega }} (\kappa ), \kappa ), \kappa \right) \in {\mathcal {C}}^\gamma _\infty \right\} \end{aligned}$$

(2.16)

satisfies $|{\mathcal {G}}_\varepsilon | \ge (\kappa _2 - \kappa _1) - C \varepsilon ^{a/k_0}$ as $\varepsilon \rightarrow 0$.

Clearly, Theorems 2.1 and 2.2 imply Theorem 1.1 with the Cantor-like set ${\mathcal {G}} := {\mathcal {G}}_\varepsilon$ and frequency vector $\tilde{\omega }= \alpha _\infty ^{-1}( \mathbf {\omega }(\kappa ), \kappa )$.

Let us sketch the proof of Theorem 2.2. By (2.11), for $\varepsilon$ small, for any $\kappa \in [\kappa _1, \kappa _2]$, the function $\alpha _\infty (\cdot , \kappa )$ is invertible:

$$\begin{aligned}&\beta = \alpha _\infty (\omega , \kappa ) = \omega + r_\varepsilon (\omega , \kappa ) \quad \Longleftrightarrow \omega = \alpha _\infty ^{-1}(\beta , \kappa ) = \beta + {\tilde{r}}_\varepsilon (\beta , \kappa )\\&\quad \mathrm{with } \quad |{\tilde{r}}_\varepsilon |^{k_0, \gamma } \le C \varepsilon \gamma ^{-(1 + k_1)}. \end{aligned}$$

We then consider the perturbed tangential frequency vector

$$\begin{aligned} \omega _\varepsilon (\kappa ) := \alpha _\infty ^{-1}( {\mathbf {\omega }} (\kappa ), \kappa ) = {\mathbf {\omega }} (\kappa ) + {\mathtt r}_\varepsilon ( \kappa ), \quad {\mathtt r}_\varepsilon ( \kappa ) := {\tilde{r}}_\varepsilon ({\mathbf {\omega }} (\kappa ), \kappa ), \end{aligned}$$

(2.17)

which satisfies

$$\begin{aligned} | \partial _\kappa ^k {\mathtt r}_\varepsilon (\kappa ) | \le C \varepsilon \gamma ^{-(1 + k_1 + k)}, \quad \forall 0 \le k \le k_0. \end{aligned}$$

We also denote, with a small abuse of notations, the perturbed normal Floquet exponents

$$\begin{aligned} \mu _j^\infty (\kappa ):= & {} \mu _j^\infty ( \omega _\varepsilon (\kappa ), \kappa ) := \mathtt m_3^\infty (\kappa ) j^\frac{1}{2} (1 + \kappa j^2)^{\frac{1}{2}} + \mathtt m_1^\infty (\kappa ) j^{\frac{1}{2}} + r_j^\infty (\kappa ),\nonumber \\&\quad \forall j \in \mathbb N^+ \setminus {\mathbb S}^+, \end{aligned}$$

(2.18)

where

$$\begin{aligned} \mathtt m_3^\infty (\kappa ) := \mathtt m_3^\infty (\omega _\varepsilon (\kappa ), \kappa ), \quad \mathtt m_1^\infty (\kappa ) := \mathtt m_1^\infty (\omega _\varepsilon (\kappa ), \kappa ), \quad r_j^\infty (\kappa ) := r_j^\infty (\omega _\varepsilon (\kappa ), \kappa ). \end{aligned}$$

For proving the measure estimate of Theorem 2.2, the key point is to prove the following transversality property.

Proposition 2.3

(Transversality). There exist $k_0 \in \mathbb N$, $\rho _0 > 0$ such that, for any $\kappa \in [ \kappa _1, \kappa _2]$,

$$\begin{aligned} \mathop \mathrm{max}\limits _{k \le k_0} |\partial _\kappa ^{k} \{{\mathbf {\omega }} (\kappa ) \cdot \ell \} |&\ge \rho _0 \langle \ell \rangle , \quad \forall \ell \in \mathbb Z^\nu \setminus \{ 0 \}, \\ \mathop \mathrm{max}\limits _{k \le k_0} |\partial _\kappa ^{k} \{{\mathbf {\omega }} (\kappa ) \cdot \ell + { \Omega }_j (\kappa ) \} |&\ge \rho _0 \langle \ell \rangle , \quad \forall \ell \in \mathbb Z^\nu , \, j \in \mathbb N^+ \setminus {\mathbb S}^+, \\ \mathop \mathrm{max}\limits _{k \le k_0} |\partial _\kappa ^{k} \{{\mathbf {\omega }} (\kappa ) \cdot \ell + { \Omega }_j (\kappa ) - { \Omega }_{j'}(\kappa ) \} |&\ge \rho _0 \langle \ell \rangle , \quad \forall (\ell , j, j') \ne (0, j, j), \, j, j' \in \mathbb N^+ \setminus {\mathbb S}^+, \\ \mathop \mathrm{max}\limits _{k \le k_0} |\partial _\kappa ^{k} \{{\mathbf {\omega }} (\kappa ) \cdot \ell + { \Omega }_j (\kappa ) + { \Omega }_{j'}(\kappa ) \} |&\ge \rho _0 \langle \ell \rangle , \quad \forall \ell \in \mathbb Z^{\nu }, j, j' \in \mathbb N^+ \setminus {\mathbb S}^+ . \end{aligned}$$

We call (following [34]) $\rho _0$ the “amount of non-degeneracy” and $k_0$ the “index of nondegeneracy”.

The above conditions are stable under perturbations which are small in ${\mathcal {C}}^{k_0}$-norm, and, therefore, hold for the perturbed tangential frequency $\omega _\varepsilon (\kappa )$ defined in (2.17) and the perturbed Floquet exponents $\mu _j (\kappa )$ introduced in (2.18). It follows that

Lemma 2.4

For $\varepsilon$ small enough, for all $\kappa \in [\kappa _1, \kappa _2]$,

$$\begin{aligned} \mathop \mathrm{max}\limits _{k \le k_0} |\partial _\kappa ^{k} \{\omega _\varepsilon (\kappa ) \cdot \ell \} |&\ge \rho _0 \langle \ell \rangle / 2 , \quad \forall \ell \in \mathbb Z^\nu \setminus \{ 0 \}, \, \\ \mathop \mathrm{max}\limits _{k \le k_0} |\partial _\kappa ^{k} \{\omega _\varepsilon (\kappa ) \cdot \ell + \mu _j^\infty (\kappa ) \} |&\ge \rho _0 \langle \ell \rangle / 2 , \quad \forall \ell \in \mathbb Z^\nu , \, j \in \mathbb N^+ \setminus {\mathbb S}^+, \\ \mathop \mathrm{max}\limits _{k \le k_0} |\partial _\kappa ^{k} \{\omega _\varepsilon (\kappa ) \cdot \ell + \mu _j^\infty (\kappa ) - \mu _{j'}^\infty (\kappa ) \} |&\ge \rho _0 \langle \ell \rangle / 2 , \quad \forall (\ell , j, j') \ne (0, j, j), \, j, j' \in \mathbb N^+ \setminus {\mathbb S}^+, \\ \mathop \mathrm{max}\limits _{k \le k_0} |\partial _\kappa ^{k} \{\omega _\varepsilon (\kappa ) \cdot \ell + \mu _j^\infty (\kappa ) + \mu _{j'}^\infty (\kappa ) \} |&\ge \rho _0 \langle \ell \rangle / 2 , \quad \forall \ell \in \mathbb Z^{\nu }, j, j' \in \mathbb N^+ \setminus {\mathbb S}^+. \end{aligned}$$

As a consequence, the classical Rüssmann lemma (Theorem 17.1 in [34]) implies that the measure of the $\kappa \in [\kappa _1, \kappa _2]$ where

$$\begin{aligned}&| \omega _\varepsilon (\kappa ) \cdot \ell + \mu _j^\infty (\omega _\varepsilon (\kappa ) , \kappa ) - \varsigma \mu _{j'}^\infty (\omega _\varepsilon (\kappa ) , \kappa ) | <\frac{4 \gamma | j^{\frac{3}{2}} - \varsigma j'^{\frac{3}{2}}|}{ \langle \ell \rangle ^{\tau }}, \\&\quad \ell \in \mathbb Z^\nu ,j, j' \in \mathbb N^+ \setminus {\mathbb S}^+,\varsigma = \pm 1, \end{aligned}$$

has measure $O(\gamma ^{1/k_0})$, and similarly for the other Melnikov conditions in (2.14). Together with the asymptotic of the eigenvalues, this implies that the Cantor-like set of non-resonant parameters ${\mathcal {G}}_\varepsilon$ defined in (2.16) has asymptotically full measure.

The transversality Proposition 2.3 is proved by arguments of degenerate KAM theory as developed in [5]. It uses in an essential way that the unperturbed frequencies $\kappa \mapsto \omega _j (\kappa )$ are analytic, are simple (on the subspace of the even functions), grow asymptotically as $j^{3/2}$ and are non-degenerate in the following sense:

Definition 2.1

A function $f := (f_1, \ldots , f_N ) : [ \kappa _1, \kappa _2] \rightarrow \mathbb R^N$ is non-degenerate if, for any $c := (c_1, \ldots , c_N ) \in \mathbb R^N \setminus \{0\}$, the function $f \cdot c = f_1 c_1 + \cdots + f_N c_N$ is not identically zero on the whole interval $[ \kappa _1, \kappa _2]$.

From a geometric point of view, f non-degenerate means that the image of the curve $f([\kappa _1,\kappa _2]) \subset \mathbb R^N$ is not contained in any hyperplane of $\mathbb R^N$. For such reason, a curve f which satisfies the non-degeneracy property of Definition 2.1 is also referred as an essentially non-planar curve, or a curve with full torsion. Proposition 2.3 is deduced by the following non-degeneracy properties.

Lemma 2.5

The frequency vectors ${\mathbf {\omega }} (\kappa ) \in \mathbb R^\nu$, $(\sqrt{\kappa }, {\mathbf {\omega }} (\kappa )) \in \mathbb R^{\nu +1}$ and

$$\begin{aligned}&( {\mathbf {\omega }} (\kappa ), \Omega _j(\kappa )) \in \mathbb R^{\nu +1}, j \in \mathbb N^+ {\setminus } {\mathbb S}^+, \quad ( {\mathbf {\omega }} (\kappa ), { \Omega }_j(\kappa ), { \Omega }_{j'}(\kappa )) \in \mathbb R^{\nu +2} , \\&\quad \forall j, j' \in \mathbb N^+ \setminus {\mathbb S}^+ , \ j \ne j', \end{aligned}$$

are non-degenerate.

This lemma is proved by a direct verification. Setting $\lambda _0(\kappa ):= \sqrt{\kappa }$ and $\lambda _j(\kappa ) := \sqrt{j (1 + \kappa j^2)}$, $j \ge 1$, it is sufficient to show that, for any N, for any $0 \le j_1< \cdots < j_N$, the matrix

$$\begin{aligned} {\mathcal {A}}(\kappa ) := \begin{pmatrix} \lambda _{j_1}(\kappa ) &{}\quad \dots &{}\quad \lambda _{j_N}(\kappa ) \\ \partial _\kappa \lambda _{j_1}(\kappa ) &{}\quad \dots &{}\quad \partial _\kappa \lambda _{j_N}(\kappa ) \\ \vdots &{}\quad \ddots &{}\quad \vdots \\ \partial _\kappa ^{N - 1} \lambda _{j_1}(\kappa ) &{}\quad \dots &{}\quad \partial _\kappa ^{N - 1} \lambda _{j_N}(\kappa ) \end{pmatrix} \end{aligned}$$

is non-singular at some value of $\kappa \in [\kappa _1, \kappa _2]$. Actually, ${\mathcal {A}}(\kappa )$ turns out to be non-singular for all $\kappa \in [\kappa _1, \kappa _2]$. A direct computation shows that $\mathrm{det} {\mathcal {A}}(\kappa )$ is proportional to a Van der Monde determinant.

2.3 Analysis of the linearized operators

In addition to the previous bifurcation analysis, the other key step for the Nash–Moser iterative scheme is to prove that the operator $d_{\alpha , i} {\mathcal {F}}$ obtained linearizing (2.9) at any approximate solution is, for most values of the parameters $(\omega , \kappa )$, invertible, and that its inverse satisfies tame estimates in Sobolev spaces.

The linearized operator $d_{\alpha , i} {\mathcal {F}}$ is quite complicated because all the components $(\theta , I, z )$ components in the system (2.9) are coupled among them. Therefore, we first implement the procedure developed in Berti–Bolle [10], and used in [3], which consists in introducing a convenient set of symplectic variables near the approximate invariant torus such that the linearized equations become (approximately) decoupled in the action–angle components and the normal one. As a consequence, the problem reduces to “almost-approximately” invert a quasi-periodic linear operator restricted to the normal directions. Actually, since this symplectic change of variables modifies, up to a translation, only the finite-dimensional action component, this operator turns out to be just the linearized water-waves system in the original coordinates, restricted to the normal directions. More precisely,

$$\begin{aligned} \Pi _{{\mathbb S}^+}^\bot {\mathcal L}_{| H_{{\mathbb S}^+}^\bot } \qquad \mathrm{where} \qquad {\mathcal L} := \omega \! \cdot \! \partial _\varphi {\mathbb I}_2 - J \partial _u \nabla _u H (U(\varphi )) \end{aligned}$$

is obtained linearizing (1.4), (1.7) at an approximate solution $U (\varphi ) = (\eta , \psi )(\varphi , x)$, changing $\partial _t \rightsquigarrow \omega \! \cdot \! \partial _\varphi$, and denoting the $2 \times 2$-identity matrix by

$$\begin{aligned} {\mathbb I}_2 := \begin{pmatrix} \mathrm{Id} &{}\quad 0 \\ 0 &{}\quad \mathrm{Id} \\ \end{pmatrix}. \end{aligned}$$

Using formula (1.5), the linearized operator ${\mathcal L}$ is computed to be

$$\begin{aligned} {\mathcal L} = \omega \! \cdot \! \partial _\varphi {\mathbb I}_2 + \begin{pmatrix} \partial _x V + G(\eta ) B &{}\quad - G(\eta ) \\ (1 + B V_x) + B G(\eta ) B - \kappa \partial _x c \partial _x &{}\quad V \partial _x - B G(\eta ) \\ \end{pmatrix} \end{aligned}$$

(2.19)

where the functions $B:= B(\varphi , x )$, $V := V(\varphi , x )$ are defined by (1.6) and $c := c(\varphi , x) := (1 + \eta _x^2)^{-3/2}$. The operator ${\mathcal L}$ is real, even and reversible.

Notation

In (2.19) and hereafter, any function a is identified with the corresponding multiplication operators $h \mapsto ah$, and, where there is no parenthesis, composition of operators is understood. For example, $\partial _x c \partial _x$ means: $h \mapsto \partial _x (c \partial _x h)$.

The key part of the analysis consists, now, in diagonalizing (actually it is sufficient to “almost” diagonalize) the quasi-periodic linear operator ${\mathcal L}$, via linear changes of variables close to the identity, which map Sobolev spaces into itself and satisfy tame estimates. These changes of variables have two well-different tasks:

• 1. Transform ${\mathcal L}$ to an operator of the form (2.20) which has constant coefficients up to pseudo-differential remainders of order zero (actually more regularizing on the off-diagonal terms). These steps are exposed in Sects. 2.3.1–2.3.5.

• 2. Reduce quadratically the size of the perturbative terms ${\mathcal R}$, ${\mathcal Q}$, see Sect. 2.3.6.

For the search of periodic solutions, i.e. [1, 19, 20, 23, 29], there is no need to perform the task 2, because it is possible to invert the linearized operator in (2.20) simply by a Neumann-argument. Indeed, for periodic solutions, a sufficiently regularizing operator in the space variable is also regularizing in the time variable, on the characteristic Fourier indices which correspond to the small divisors. This is clearly not true for quasi-periodic solutions. That is why we will completely diagonalize the linear operator in (2.20) by a KAM scheme. For that, we need to analyze more in detail the pseudo-differential nature of the remainders after each conjugation step.

The approximate solution $U (\varphi , x)$ at which we linearize is assumed to be bounded in a low Sobolev norm (as it is satisfied by any approximate solutions along the Nash–Moser iteration). Moreover, $U(\varphi , x)$ is supposed to be ${\mathcal {C}}^\infty (\mathbb T^\nu _{\varphi } \times \mathbb T_x )$ because, along the Nash–Moser iteration, each approximate solution is actually a trigonometric polynomial in $(\varphi , x)$ (with clearly more and more harmonics). As a consequence, all the coefficients of the linearized operator ${\mathcal L }$ in (2.19) are ${\mathcal {C}}^\infty$. This allows to work in the usual framework of ${\mathcal {C}}^\infty$ pseudo-differential symbols. For the Nash–Moser convergence, we shall then perform quantitative estimates in Sobolev spaces.

2.3.1 Reduction of ${\mathcal L}$ to constant coefficients in decreasing symbols

The goal of the first steps is to reduce ${\mathcal L}$ to a quasi-periodic linear operator of the form

$$\begin{aligned} (h, \bar{h}) \mapsto \left( \omega \! \cdot \! \partial _\varphi + {\mathrm i} \mathtt m_3 T(D) + {\mathrm i} {\mathtt m}_1 |D|^{\frac{1}{2}} \right) h + {\mathcal R} h + {\mathcal Q} \bar{h}, \quad h \in \mathbb C, \end{aligned}$$

(2.20)

where $\mathtt m_3, {\mathtt m}_1 \in \mathbb R$ are constants coefficients, satisfying $\mathtt m_3 \approx 1$, ${\mathtt m}_1 \approx 0$, the principal symbol operator

$$\begin{aligned} T(D) = |D|^{1/2} (1 - \kappa \partial _{xx})^{1/2}, \end{aligned}$$

and the remainders ${\mathcal R} := {\mathcal R} (\varphi )$, ${\mathcal Q} := {\mathcal Q} (\varphi )$ are small bounded operators acting in the Sobolev spaces $H^s$, which satisfy tame estimates. More precisely, in view of a KAM reducibility scheme that will completely diagonalize the operator (2.20) (Sect. 2.3.6), we need that all the derivatives

$$\begin{aligned} \partial _\varphi ^{\beta } \partial _{\omega , \kappa }^k {\mathcal R}, \quad \partial _\varphi ^{\beta } \partial _{\omega , \kappa }^k {\mathcal Q}, \quad |\beta | \le \beta _0, \ |k| \le k_0, \end{aligned}$$

(2.21)

for $\beta _0$ large enough (depending on the Diophantine exponent $\tau$), satisfy tame estimates.

2.3.2 Symmetrization and space–time reduction of ${\mathcal L}$ at the highest order

The first part of the analysis is similar to Alazard–Baldi [1]. We first conjugate the linear operator ${\mathcal L}$ in (2.19) by the change of variable

$$\begin{aligned} \mathcal {Z}:= \begin{pmatrix} 1 &{}\quad 0 \\ B &{}\quad 1 \end{pmatrix}, \qquad \mathcal {Z}^{-1} = \begin{pmatrix} 1 &{}\quad 0 \\ - B &{}\quad 1 \end{pmatrix} \end{aligned}$$

obtaining

$$\begin{aligned} {\mathcal L}_0 := \mathcal {Z}^{-1} \mathcal {L}\mathcal {Z}= \omega \! \cdot \! \partial _\varphi {\mathbb I}_2 + \begin{pmatrix} \partial _x V \ &{}\quad \ - G(\eta ) \\ a - \kappa \partial _x c \partial _x \ &{}\quad V \partial _x \end{pmatrix} \end{aligned}$$

(2.22)

for some function $a(\varphi , x)$. This step amounts to introduce (a linearized version of) the “good unknown of Alinhac”.

As a second step, we conjugate $\mathcal {L}_0$ with an operator of the form ${\mathcal S} {\mathcal Q}{\mathcal B}$, where ${\mathcal B }$ is a change of variable

$$\begin{aligned} (\mathcal {B}h)(\varphi ,x) := h(\varphi , x+\beta (\varphi ,x)) \end{aligned}$$

(2.23)

induced by a $\varphi$-dependent family of diffeomorphisms of the torus

$$\begin{aligned} y = x + \beta (\varphi ,x) \qquad \Leftrightarrow \qquad x = y + \tilde{\beta }(\varphi ,y), \end{aligned}$$

(2.24)

${\mathcal Q}$ is a matrix-valued multiplication operator

$$\begin{aligned} {\mathcal Q} := \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad q \end{pmatrix}, \qquad {\mathcal Q}^{-1} := \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad q^{-1} \end{pmatrix}, \end{aligned}$$

for a function $q (\varphi , x )$ close to 1, and $\mathcal {S}$ is the vector-valued Fourier multiplier

$$\begin{aligned} \mathcal {S}= \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad G \end{pmatrix}, \quad G := |D|^{- \frac{1}{2}} (1 + \kappa D^2)^{\frac{1}{2}} \in OPS^{1/2}. \end{aligned}$$

Choosing properly the small periodic functions $\beta (\varphi ,x)$ and $q(\varphi , x ) - 1$, one gets

$$\begin{aligned}&\mathcal {L}_1 = \mathcal {S}^{-1} {\mathcal Q}^{-1} \mathcal {B}^{-1} \mathcal {L}_0 \mathcal {B}{\mathcal Q} \mathcal {S}\nonumber \\&\quad = \omega \! \cdot \! \partial _\varphi {\mathbb I}_2 + \begin{pmatrix} a_{1} \partial _x + a_{2} &{} \!\! \!\! - m_3(\varphi ) T(D) + \sqrt{\kappa } \, a_{3} \mathcal {H}|D|^{\frac{1}{2}} + {\mathcal R}_{1,B} \\ m_3(\varphi ) T(D) - \frac{a_{4}}{\sqrt{\kappa }} \, |D|^{\frac{1}{2}} \mathcal {H}+ m_3(\varphi ) \pi _0 + {\mathcal R}_{1,C} &{} \!\! \!\! a_{1} \partial _x + {\mathcal R}_{1,D} \end{pmatrix}\nonumber \\ \end{aligned}$$

(2.25)

for suitable functions $a_1 , a_2$, $a_3 , a_4$ and pseudo-differential operators remainders ${\mathcal R}_{1,B}$, ${\mathcal R}_{1,C}$, ${\mathcal R}_{1,D} \in OPS^0$ which are $O(\varepsilon )$ small, in low Sobolev norm. All the coefficients and the operators depend in a tame way, i.e. at most linearly, in the high Sobolev norm of the approximate solution $\Vert u \Vert _{s + \sigma }$ with a possible fixed loss of derivatives $\sigma$. Note that the coefficient $m_3 (\varphi )$ of the highest order operator ${\mathcal L}_1$ in (2.25) is independent of the space variable. The operator $\pi _0$ is the $L^2$ projector of the constants, that, for simplicity of exposition, we neglect in the sequel.

We then write ${\mathcal L}_1$ as an operator acting on the complex variables

$$\begin{aligned} h := \eta + {\mathrm i} \psi , \quad \bar{h} := \eta - {\mathrm i} \psi , \end{aligned}$$

obtaining

$$\begin{aligned} {\mathcal L}_1= & {} \omega \! \cdot \! \partial _\varphi {\mathbb I}_2 + {\mathrm i} m_3(\varphi ) \mathbf{T}(D) + \mathbf{A}_1(\varphi , x) \partial _x + {\mathrm i} (\mathbf{A}_0^{(I)}(\varphi , x) \nonumber \\&+ \mathbf{A}_0^{(II)}(\varphi , x)) \mathcal {H}|D|^{\frac{1}{2}}+ \mathbf{R}_1^{(I)} + \mathbf{R}_1^{(II)} \end{aligned}$$

(2.26)

where

$$\begin{aligned} \mathbf{T}(D) := \begin{pmatrix} T(D) &{}\quad 0 \\ 0 &{}\quad - T(D) \end{pmatrix}, \quad \mathbf{A}_1(\varphi , x) := \begin{pmatrix} a_1(\varphi , x) &{}\quad 0 \\ 0 &{}\quad a_1(\varphi , x) \end{pmatrix}, \end{aligned}$$

$$\begin{aligned} \mathbf{A}_0^{(I)} (\varphi , x):= & {} \begin{pmatrix} a_5(\varphi , x) &{}\quad 0 \\ 0 &{}\quad - a_5 (\varphi , x) \end{pmatrix}, \\ \mathbf{A}_0^{(II)}(\varphi , x):= & {} \begin{pmatrix} 0 &{}\quad a_{6}(\varphi , x) \\ - a_{6} (\varphi , x) &{}\quad 0 \end{pmatrix} , \end{aligned}$$

and

$$\begin{aligned} \mathbf{R}_{1}^{(I)}:= & {} \begin{pmatrix} r_1^{(I)}(x, D) &{}\quad 0 \\ 0 &{}\quad \overline{r_1^{(I)} (x,D)} \end{pmatrix} , \nonumber \\ \mathbf{R}_1^{(II)}:= & {} \begin{pmatrix} 0 &{}\quad r_1^{(II)}(x, D) \\ \overline{r_1^{(II)} (x, D)} &{}\quad 0 \end{pmatrix}\in OPS^0 \end{aligned}$$

(2.27)

are $O(\varepsilon )$-pseudo-differential operators. Note that ${\mathcal L}_1$ in (2.26) is block-diagonal (in ($u, \bar{u}$)) up to order $|D|^{1/2}$.

The next step is to remove the dependence on $\varphi$ from the highest order term ${\mathrm i} m_3(\varphi )\mathbf{T}(D)$, by applying a quasi-periodic time reparametrization

$$\begin{aligned} {\mathcal P} \, \mathbb I_2 = \begin{pmatrix} {\mathcal P} &{}\quad 0 \\ 0 &{}\quad {\mathcal P} \end{pmatrix} , \quad ({\mathcal P}h)(\varphi , x) := h(\varphi + \omega p(\varphi ), x) , \end{aligned}$$

induced by the diffeomorphism

$$\begin{aligned} \vartheta := \varphi + \omega p(\varphi ) \quad \Leftrightarrow \quad \varphi = \vartheta + \omega \tilde{p}(\vartheta ) \end{aligned}$$

where $p(\varphi )$ is a small periodic function. Choosing properly p and assuming $\omega$ to be Diophantine, we get

$$\begin{aligned} \mathcal {L}_2:= & {} \omega \! \cdot \! \partial _\varphi {\mathbb I}_2 +{\mathrm i} \mathtt m_3 \mathbf{T}(D) + \mathbf{B}_1(\varphi , x) \partial _x \nonumber \\&+ {\mathrm i} \left( \mathbf{B}_0^{(I)}(\varphi , x) + \mathbf{B}_0^{(II)}(\varphi , x) \right) \mathcal {H}|D|^{\frac{1}{2}}+ \mathbf{R}_2^{(I)} + \mathbf{R}_2^{(II)} \end{aligned}$$

(2.28)

where

$$\begin{aligned} \mathbf{B}_1&= \begin{pmatrix} a_{7} (\varphi ,x) &{}\quad 0 \\ 0 &{}\quad a_{7} (\varphi ,x) \end{pmatrix}, \quad \mathbf{B}_0^{(I)} = \begin{pmatrix} a_{8} (\varphi ,x) &{}\quad 0 \\ 0 &{}\quad - a_{8} (\varphi ,x) \end{pmatrix}, \\ \mathbf{B}_0^{(II)}&= \begin{pmatrix} 0 &{}\quad a_9 (\varphi ,x) \\ a_9 (\varphi ,x) &{}\quad 0 \end{pmatrix} \end{aligned}$$

and $\mathbf{R}_2^{(I)}$ $\mathbf{R}_2^{(II)}$ are $O(\varepsilon )$ pseudo-differential operators $\mathbf{R}_2^{(I)}$ $\mathbf{R}_2^{(II)}$. All the previous transformations are real, even, and reversibility-preserving, so that $\mathcal {L}_2$ is a real, even and reversible operator.

From this point, we have to proceed quite differently with respect to [1].

2.3.3 Block-decoupling

The next step is to conjugate the operator ${\mathcal L}_2$ in (2.28) to an operator of the form

$$\begin{aligned} {\mathcal L}_M:= & {} \mathbf{\Phi }_M^{- 1} {\mathcal L}_2 \mathbf{\Phi }_M = \omega \! \cdot \! \partial _\varphi {\mathbb I}_2 \nonumber \\&+ {\mathrm i} \mathtt m_3 \mathbf{T}(D) + \mathbf{B}_1(\varphi , x) \partial _x + {\mathrm i} \mathbf{B}_0^{(I)}(\varphi , x) \mathcal {H}|D|^{\frac{1}{2}}+ \mathbf{R}_M^{(I)}+ \mathbf{R}_M^{(II)}\qquad \qquad \end{aligned}$$

(2.29)

where the remainders

$$\begin{aligned} \mathbf{R}_M^{(I)}:= & {} \begin{pmatrix} r_M^{(I)}(\varphi , x, D) &{}\quad 0 \\ 0 &{}\quad \overline{r_M^{(I)}(\varphi , x, D)} \end{pmatrix} \in OPS^0 , \\ \mathbf{R}_M^{(II)}:= & {} \begin{pmatrix} 0 &{}\quad {\mathcal R}_M^{(II)} \\ \overline{\mathcal R}_M^{(II)} &{}\quad 0 \end{pmatrix} \in OPS^{\frac{1}{2} - M} \end{aligned}$$

are $\varepsilon$ small. This is achieved by applying iteratively M-times a conjugation map which transforms the off-diagonal block operators into 1-smoother ones. Notice that the operator ${\mathcal L}_M$ in (2.29) is block-diagonal up to the smoothing remainder $\mathbf{R}_M^{(II)} \in OPS^{\frac{1}{2} - M}$. The coefficients of $\mathbf{R}_M^{(II)}$ depend on O(M) -derivatives of the approximate solution. In any case, the number of regularizing steps M will be fixed (independently on s, depending just on the Diophantine exponent $\tau$), determined by the KAM reducibility scheme.

2.3.4 Egorov analysis. Space reduction of the order $\partial _x$

The goal is now to eliminate the first-order vector field $\mathbf{B}_1(\varphi , x) \partial _x$ from ${\mathcal L}_M$. We conjugate ${\mathcal L}_M$ by the flow

$$\begin{aligned} \mathbf{\Phi }(\varphi , t) := \begin{pmatrix} \Phi (\varphi , t) &{}\quad 0 \\ 0 &{}\quad \overline{\Phi }(\varphi , t) \end{pmatrix} \end{aligned}$$

generated by the system

$$\begin{aligned} \partial _t \begin{pmatrix} u\\ \overline{u} \end{pmatrix} = {\mathrm i} \begin{pmatrix} a(\varphi , x) &{}\quad 0 \\ 0 &{}\quad - a(\varphi , x) \end{pmatrix} | D |^{\frac{1}{2}} \begin{pmatrix} u \\ \overline{u} \end{pmatrix} \end{aligned}$$

(2.30)

where $a(\varphi , x )$ is a small real valued function to be determined. Thus, $\Phi (\varphi , t)$ is the flow of the scalar linear pseudo-PDE

$$\begin{aligned} \partial _t u = {\mathrm i} a(\varphi , x) |D|^{\frac{1}{2}} u. \end{aligned}$$

(2.31)

Conjugating the operator ${\mathcal L}_M$ in (2.29) by the time one flow operator $\mathbf{\Phi }(\varphi ) := \mathbf{\Phi }(\varphi , 1)$, we get

$$\begin{aligned} {\mathcal L}_M^{(1)}= & {} \mathbf{\Phi } {\mathcal L}_M \mathbf{\Phi }^{- 1} = \omega \! \cdot \! \partial _\varphi {\mathbb I}_2 + \mathbf{\Phi }(\varphi ) \mathbf{P}_0(\varphi , x, D) \mathbf{\Phi }(\varphi )^{- 1} \\&+ \mathbf{\Phi }(\varphi ) \omega \! \cdot \! \partial _\varphi \{\mathbf{\Phi }(\varphi )^{- 1}\} + \mathbf{\Phi } \mathbf{R}_M^{(II)} \mathbf{\Phi }^{- 1} \end{aligned}$$

where we have denoted

$$\begin{aligned} \mathbf{P}_0(\varphi , x, D) = {\mathrm i} \mathtt m_3 \mathbf{T}(D) + \mathbf{B}_1(\varphi , x) \partial _x + {\mathrm i} \mathbf{B}_0^{(I)}(\varphi , x) \mathcal {H}|D|^{\frac{1}{2}} + \mathbf{R}_M^{(I)} \end{aligned}$$

the diagonal part of ${\mathcal L}_M$. Note that the terms $\mathbf{\Phi }(\varphi ) \mathbf{P}_0(\varphi , x, D) \mathbf{\Phi }(\varphi )^{- 1}$ and $\mathbf{\Phi }(\varphi ) \, \omega \! \cdot \! \partial _\varphi \{\mathbf{\Phi }(\varphi )^{- 1}\}$ are block-diagonal. They are classical pseudo-differential operators and can be analyzed by an Egorov-type argument. On the other hand, the off-diagonal term $\mathbf{\Phi } \mathbf{R}_M^{(II)} \mathbf{\Phi }^{- 1}$ is very regularizing and satisfies tame estimates. Let us see how evolves the operator

$$\begin{aligned} \mathbf{P} (\varphi , t)= & {} \mathbf{\Phi }(\varphi , t) \mathbf{P}_0 \mathbf{\Phi }(\varphi , t)^{- 1} = \begin{pmatrix} P(\varphi , t) &{}\quad 0 \\ 0 &{}\quad \overline{P}(\varphi , t) \end{pmatrix}, \nonumber \\ P(\varphi , t):= & {} \Phi (\varphi , t) p_0(\varphi , x, D) \Phi ^{- 1}(\varphi , t). \end{aligned}$$

(2.32)

under the flow of (2.30). The operator $P(\varphi , t)$ solves the usual Heisenberg equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t P (\varphi , t) = {\mathrm i} [ A(\varphi ) , P (\varphi , t)] \\ P (\varphi , 0) = P_0 := p_0(\varphi , x, D) \end{array}\right. } \qquad \mathrm{where} \qquad A (\varphi ) = a(\varphi , x) |D|^{\frac{1}{2}} \, . \end{aligned}$$

(2.33)

The Eq. (2.33) can be solved in decreasing symbols using the fact that the order of the commutator $[ A(\varphi ), Q(\varphi ) ]$ with a classical pseudo-differential operator Q is strictly less than the order of Q. More precisely, (2.33) has an approximate solution $Q(\varphi , t) := q(t, \varphi , x, D)$ expanded in decreasing orders

$$\begin{aligned} q(t, \varphi , x, \xi ) = \sum _{n = 0}^M q_n (t, \varphi , x, \xi ) , \quad q_n (t,\varphi , x, \xi ) \in S^{\frac{1}{2} (3 - n)} , \quad \forall n = 0, \ldots , M \end{aligned}$$

(2.34)

where $q_0 = p_0$ and the other lower order symbols $q_n$ are recursively computed. This shows that the diagonal term $P(\varphi , t)$ remains pseudo-differential along the conjugation. One can analyze $\mathbf{\Phi }(\varphi ) \omega \! \cdot \! \partial _\varphi \{ \mathbf{\Phi }(\varphi )^{- 1} \}$ in the same way.

As an outcome, choosing properly the function $a ( \varphi , x )$, and using the fact that the operator ${\mathcal L }_2$ is even, one can eliminate the order $\partial _x$, getting an operator of the form

$$\begin{aligned} {\mathcal L}_M^{(1)} = \omega \cdot \partial _\varphi {\mathbb I}_2 + {\mathrm i} \mathtt m_3 \mathbf{T}(D) + {\mathrm i} \left( \mathbf{C}_1(\varphi , x) + \mathbf{C}_0(\varphi , x) \mathcal {H}\right) |D|^{\frac{1}{2}} + \mathbf{R}_M^{(1)} + \mathbf{Q}_M^{(1)} \end{aligned}$$

(2.35)

where

$$\begin{aligned} \mathbf{C}_1(\varphi , x) := \begin{pmatrix} a_{10} &{}\quad 0 \\ 0 &{}\quad - a_{10} \end{pmatrix}, \quad \mathbf{C}_0(\varphi , x) := \begin{pmatrix} a_{11} &{}\quad 0 \\ 0 &{}\quad - a_{11} \end{pmatrix}\, . \end{aligned}$$

Remark 2.6

Alazard–Baldi [1] uses a semi-Fourier integral operator like $\mathrm{Op} ( e^{{\mathrm i} a(\varphi , x) \sqrt{|\xi |}} ) \in OPS_{\frac{1}{2}, \frac{1}{2}}^{0}$. The use of the flow $\Phi (\varphi )$ of (2.31) is simpler because the proof that $\Phi$, as well as its inverse $\Phi ^{-1}$, is a bounded operator on Sobolev spaces $H^s$ and satisfies tame estimates, follows by simple energy estimates (the vector field ${\mathrm i} a(\varphi , x) |D|^{1/2}$ is skew-adjoint at the highest order, see Appendix of [7]).

The fact that the diagonal terms of the conjugated operator (2.35) are still pseudo-differential is a relevant information. Indeed, the flow $\Phi (\varphi ) \sim \mathrm{Op} ( e^{{\mathrm i} a(\varphi , x) \sqrt{|\xi |}} )$ maps Sobolev spaces in itself, but each derivative

$$\begin{aligned} \partial _\varphi \Phi (\varphi ) \sim \mathrm{Op} \left( e^{{\mathrm i} a(\varphi , x) \sqrt{|\xi |}} \, {\mathrm i} \partial _\varphi a(\varphi , x) \sqrt{|\xi |} \right) \end{aligned}$$

is an unbounded operator which loses $|D|^{1/2}$ derivatives. Actually, $\partial _{\omega ,\kappa }^k \partial _\varphi ^\beta \Phi (\varphi )$ loses $| D|^{\frac{|\beta |+|k|}{2}}$ derivatives. Since the conjugated operator

$$\begin{aligned} P_+ (\varphi ) := \Phi (\varphi ) P_0 \Phi (\varphi )^{-1} = \mathrm{Op} ( c(\varphi , x, \xi )) , \quad c(\varphi , x, \xi ) \in S^m , \end{aligned}$$

(2.36)

is a classical pseudo-differential operator, the differentiated operator

$$\begin{aligned} \partial _\varphi P_+ (\varphi ) = \mathrm{Op} ( \partial _\varphi c(\varphi , x, \xi )) \in OPS^{m} \end{aligned}$$

is still a pseudo-differential operator of the same order of $P_0$ with just a symbol $\partial _\varphi c$ less regular in $\varphi$. The loss of regularity for $\partial _\varphi c$ may be compensated by the usual Nash–Moser smoothing procedure in $\varphi$. This is the reason why we require that the diagonal remainder ${\mathcal R} \in OPS^0$ is just of order zero.

On the other hand, the off-diagonal term ${\mathcal Q}_M \in OPS^{-M}$ evolves, under the flow $\mathbf{\Phi } (\varphi , t)$, according to the “skew-Heisenberg” equation obtained replacing in (2.33) the commutator with the skew-commutator. As a consequence, the symbol of ${\mathcal Q}_M^+ := \Phi (\varphi ) {\mathcal Q}_M \Phi (\varphi )^{-1}$ assumes the form $e^{{\mathrm i} a(\varphi , x) \sqrt{|\xi |}} q (\varphi , x, \xi )$, where $q (\varphi , x, \xi ) \in S^{-M}$ is a classical symbol (actually we do not prove it explicitly because it is not needed). Thus, the action of each $\partial _\varphi$ on ${\mathcal Q}_M^+$ produces an operator which loses $| D |^{\frac{1}{2}}$ derivatives in space more than ${\mathcal Q}_M$. This is why we have performed previously a large number M of regularizing steps for the off-diagonal components ${\mathcal Q}$.

2.3.5 Space reduction of the order $|D|^{1/2}$

Finally, we eliminate the x-dependence of the coefficient in front of $|D|^{\frac{1}{2}}$ in the operator ${\mathcal L}_M^{(1)}$ in (2.35), conjugating ${\mathcal L}_M^{(1)}$ by a matrix-valued multiplication operator of the form

$$\begin{aligned} \mathbf{V} := \begin{pmatrix} {\mathcal V} &{}\quad 0 \\ 0 &{}\quad \overline{\mathcal V} \end{pmatrix}, \qquad {\mathcal V} := \mathrm{Op}(v), \qquad v := v(\varphi ,x, \xi ) \in S^0. \end{aligned}$$

Choosing properly the function $v(\varphi , x )$, one finally gets

$$\begin{aligned} {\mathcal L}_M^{(2)} := \omega \! \cdot \! \partial _\varphi {\mathbb I}_2 + {\mathrm i} \mathtt m_3 \mathbf{T}(D) + {\mathrm i} \mathtt m_{1} \mathbf{\Sigma } |D|^{\frac{1}{2}} + \mathbf{R}_M^{(2)} + \mathbf{Q}_M^{(2)} , \quad \mathrm{where} \quad \mathbf{\Sigma } := \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad - 1 \end{pmatrix} \end{aligned}$$

(2.37)

which has the form stated in (2.20). We remark that all the previous transformations are real, even, and reversibility-preserving, so that ${\mathcal L}_M^{(2)}$ is real, even and reversible.

2.3.6 KAM-reducibility scheme

We are now in position to apply an iterative quadratic scheme to reduce the size of the terms $\mathbf{R}_M^{(2)}$ and $\mathbf{Q}_M^{(2)}$ (if possible). Let us explain the main idea. Consider a linear real, even, and reversible operator acting on $H_{{\mathbb S}_0}^\bot$,

$$\begin{aligned} \mathbf{L} = \omega \! \cdot \! \partial _\varphi {\mathbb I}_2^\bot + \mathbf{D} + \varepsilon \mathbf{P} , \end{aligned}$$

(2.38)

with diagonal part (with respect to the exponential basis)

$$\begin{aligned} \mathbf{D}= & {} \begin{pmatrix} {\mathrm i} \Lambda &{}\quad 0 \\ 0 &{}\quad - {\mathrm i} \Lambda \end{pmatrix} , \quad \Lambda = \mathrm{diag}_j ( \mu _j ) , \quad \mu _j = {\mathtt m}_3 \sqrt{j (1 + \kappa j^2)} + {\mathtt m}_1 j^{\frac{1}{2}} + r_j , \\&\sup _{j } |r_j| = O( \varepsilon ) \end{aligned}$$

with $r_j \in \mathbb R$ (at the first step, $\mu _j = {\mathtt m}_3 \sqrt{j (1 + \kappa j^2)} + {\mathtt m}_1 j^{\frac{1}{2}}$ by (2.37)), and a bounded perturbation

$$\begin{aligned} \mathbf{P} = \begin{pmatrix} P_1 &{}\quad P_2 \\ {\overline{P}}_2 &{}\quad {\overline{P}}_1 \end{pmatrix}. \end{aligned}$$

Transform $\mathbf{L}$ under the flow $\mathbf{\Phi } (\varphi , \tau )$ generated by a linear system

$$\begin{aligned} \partial _\tau \begin{pmatrix} u \\ \bar{u} \end{pmatrix} = \varepsilon \mathbf{W} (\varphi ) \begin{pmatrix} u \\ \bar{u} \end{pmatrix} \qquad \mathrm{where} \qquad \mathbf{W} ( \varphi ) = \begin{pmatrix} W_1 &{}\quad W_2 \\ {\overline{W}}_2 &{}\quad {\overline{W}}_1 \end{pmatrix} \end{aligned}$$

is a bounded map, to be determined. The conjugated operator $\mathbf{L} ( \tau ) = \mathbf{\Phi } (\varphi , \tau ) \mathbf{L} \mathbf{\Phi } (\varphi , \tau )^{-1}$ solves as usual the vector Heisenberg equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _\tau \mathbf{L} ( \tau ) = [ \varepsilon \mathbf{W}(\varphi ), \mathbf{L} ( \tau ) ] \\ \mathbf{L} ( 0 ) = \mathbf{L} = \omega \cdot \partial _\varphi + \mathbf{D} + \varepsilon \mathbf{P}. \end{array}\right. } \end{aligned}$$

This time, we expand $\mathbf{L}(1)$ in power of $\varepsilon$ (Lie series)

$$\begin{aligned} \mathbf{L} ( 1 )&= \mathbf{L} ( 0 ) + [ \varepsilon \mathbf{W}(\varphi ) , \mathbf{L} ( 0 ) ] + O(\varepsilon ^2 ) \\&= \omega \cdot \partial _\varphi + \mathbf{D} + { \varepsilon \left\{ \mathbf{P} + {\omega \cdot \partial _\varphi \mathbf{W}} (\varphi ) + [ \mathbf{W}, \mathbf{D} ] \right\} } + O(\varepsilon ^2 ). \end{aligned}$$

The goal is to eliminate the $\varepsilon$-term ${\omega \cdot \partial _\varphi \mathbf{W}} (\varphi ) + [ \mathbf{W}, \mathbf{D} ] + \mathbf{P}$ (if possible). This amounts to solve

$$\begin{aligned}&{\omega \cdot \partial _\varphi W_1} (\varphi ) + {\mathrm i} ( W_1 (\varphi ) \Lambda - \Lambda W_1 (\varphi )) + P_1 (\varphi ) = [ \! [ P_1 (\varphi ) ] \! ] \\&{\omega \cdot \partial _\varphi W_2} (\varphi ) + {\mathrm i} ( W_2 (\varphi ) \Lambda + \Lambda W_2 (\varphi ) ) + P_2 (\varphi ) = 0 , \end{aligned}$$

where $[ \! [ P_1 (\varphi ) ] \! ] := \mathrm{diag}([P_1 (\varphi ) ]_j^j)$. Expanding in Fourier series ${ W_i} (\varphi ) = \sum _{\ell \in \mathbb Z^\nu } {W_i} (\ell ) e^{{\mathrm i} \ell \cdot \varphi }$ and $P_i (\varphi ) = \sum _{\ell \in \mathbb Z^\nu } P_i(\ell ) e^{{\mathrm i} \ell \cdot \varphi }, i=1,2$, we are led to the following equations:

$$\begin{aligned}&{\mathrm i} \omega \cdot \ell W_1 (\ell ) + {\mathrm i} ( W_1 (\ell ) \Lambda - \Lambda W_1 (\ell ) ) + P_1 (\ell ) = [ \! [ P_1 (\ell ) ] \! ] \\&{\mathrm i} \omega \cdot \ell W_2 (\ell ) + {\mathrm i} ( W_2 (\ell ) \Lambda + \Lambda W_1 (\ell ) ) + P_2 (\ell ) = 0 . \end{aligned}$$

Representing ${ W_i} (\ell ) = ( [{ W_i} (\ell )]^j_k )_{j, k \in \mathbb Z}$ as a matrix, we get the infinitely many scalar equations

$$\begin{aligned}&{\mathrm i} \omega \cdot \ell [W_1 (\ell )]^j_k + {\mathrm i} [W_1 (\ell )]^j_k ( \mu _j - \mu _k ) + [P_1 (\ell )]^j_k = [P_1 (\ell ) ]_j^k \\&{\mathrm i} \omega \cdot \ell [W_2 (\ell )]^j_k + {\mathrm i} [W_2 (\ell )]^j_k ( \mu _j + \mu _k ) + [P_2 (\ell )]^j_k = 0 . \end{aligned}$$

These equations admit the solutions

$$\begin{aligned}&[W_1 (\ell )]^j_k = \frac{ [P_1 (\ell )]^j_k }{ {\mathrm i} ( \omega \cdot \ell + \mu _j - \mu _k )} \\&\forall (\ell , j, k) \ne (0, j, j) , \quad [W_2 (\ell )]^j_k = \frac{ [P_2 (\ell )]^j_k }{ {\mathrm i} ( \omega \cdot \ell + \mu _j + \mu _k )} , \quad \forall (\ell , j, k) \end{aligned}$$

if the corresponding denominators do not vanish. Note that since we look for solutions even in x and the eigenvalues $\mu _j$ are simple, $\mu _j - \mu _k \ne 0$ for $j \ne k$ (we could expand on the $\cos (jx)$ basis so that $j, k \in \mathbb N^+$). We actually require a quantitative lower bound for the denominators, as

$$\begin{aligned} |\omega \cdot \ell + \mu _j - \mu _{k} | \ge \frac{| j^{3/2} - {k}^{3/2}|}{ \gamma \langle \ell \rangle ^{\tau }} , \quad |\omega \cdot \ell + \mu _j + \mu _{k} | \ge \frac{| j^{3/2} + {k}^{3/2}|}{\gamma \langle \ell \rangle ^{\tau }} . \end{aligned}$$

These conditions are called the second-order Melnikov non-resonance conditions and appear in the Cantor set (2.14). After this conjugation step, we have obtained a linear operator of the same form (2.38), but with a smaller $O(\varepsilon ^2 )$ perturbation and a new diagonal part corrected by the matrix $[ \! [ P_1 (0) ] \! ] = \mathrm{diag} [P_1 (0) ]_j^j$. Since $\mathbf{P}$ is reversible, then $[P_1 (0) ]_j^j$ are purely imaginary. To apply the above classical KAM reducibility scheme to the operator ${\mathcal L}_M^{(2)}$ in (2.37), a difficulty is that the remainders $\mathbf{R}_M^{(2)}$, $\mathbf{Q}_M^{(2)}$ just satisfy tame estimates. For technical details of the proof, we refer to [7]. Here, we just mention that for the convergence, we need the tame conditions (2.21). In conclusion, the operator ${\mathcal L}_M^{(2)}$ defined in (2.37) may be conjugated to a diagonal operator of the form

$$\begin{aligned} \omega \! \cdot \! \partial _\varphi {\mathbb I}_2^\bot + {\mathrm i} \mathbf{D}_\infty , \quad \mathbf{D}_\infty = \begin{pmatrix} {\mathrm i} \Lambda _\infty &{}\quad 0 \\ 0 &{}\quad - {\mathrm i} \Lambda _\infty \end{pmatrix} , \end{aligned}$$

with

$$\begin{aligned} \Lambda _\infty = \mathrm{diag}_j ( \mu _j^{\infty } ) , \ \ \mu _j^{\infty } = {\mathtt m}_3 \sqrt{j (1 + \kappa j^2)} + {\mathtt m}_1 j^{\frac{1}{2}} + r_j^\infty , \ \ \sup _{j } r_j^{\infty } = O( \varepsilon ). \end{aligned}$$

It is then sufficient to require the first-order Melnikov conditions

$$\begin{aligned} |\omega \cdot \ell + \mu _j^\infty | \ge 2\gamma j^{\frac{3}{2}} \langle \ell \rangle ^{- \tau }, \quad \forall \ell \in \mathbb Z^\nu , \,\, j \in \mathbb N\setminus {\mathbb S}^+ \end{aligned}$$

to prove the invertibility of the diagonal operators $\omega \! \cdot \! \partial _\varphi {\mathbb I}_2^\bot + {\mathrm i} \mathbf{D}_\infty$. These conditions appear in (2.14).

Since all the transformations that we have performed above are bounded map between Sobolev spaces, we get the required tame estimates for the inverse.

After the above analysis of the linearized operator, a differentiable Nash–Moser iterative scheme constructs a sequence of better and better approximate solutions of (2.9), finally proving Theorem 2.1.