The authors consider the domain \(\mathcal{D}_{\eta}=\{(x,y)\in \mathbb{T} \times \mathbb{R}:-h<y<\eta (t,x)\}\), with \(\mathbb{T}=\mathbb{R}/2\pi \mathbb{Z}\) and where \(y=\eta (t,x)\) is the free upper surface. This domain represents a bi-dimensional ocean with finite depth whose water evolves under the action of pure gravity. The authors consider the system \[ \begin{aligned} &\Delta \Phi =0\text{ in }\mathcal{D}_{\eta},\\ &\partial_{t}\Phi +\frac{1}{2}| \nabla \Phi |^{2}+g\eta =0\text{ at }y=\eta (t,x),\\ &\partial_{y}\Phi =0\text{ at }y=-h,\\ &\partial_{t}\eta =\partial_{y}\Phi -\partial_{x}\eta \text{ at }y=\eta (t,x). \end{aligned} \] They prove the existence and the linear stability of Cantor families of small-amplitude time quasi-periodic standing water wave solutions to this problem. They observe that this problem can be written in an equivalent way as \(\partial_{t}\eta =G(\eta,h)\psi\), \(\partial_{t}\psi =-g\eta -\frac{\psi_{x}^{2}}{2}+\frac{1}{2(1+\eta_{x}^{2})} (G(\eta,h)\psi +\eta_{x}\psi_{x})^{2}\), where \(G(\eta,h)\) is the Dirichlet-Neumann operator defined as \[ G(\eta,h)\psi =-(\Phi_{y}-\eta_{x}\Phi_{x})\mid_{y=\eta (t,x)}. \] Introducing the Hamiltonian \[ H(\eta,\psi)=\frac{1}{2}\int_{\mathbb{T}}\psi G(\eta,h)\psi dx+\frac{g}{2}\int_{\mathbb{T}}\eta^{2}dx, \] the authors also observe that the last system can be written as \[ \begin{aligned} &\partial_{t}\eta =\nabla_{\psi}H(\eta,\psi),\\ &\partial_{t}\psi =-\nabla_{\eta}H(\eta,\psi),\\ &\partial_{t}u=J\nabla_{u}H(u), \end{aligned} \] with \(u=\left(\begin{smallmatrix} \eta \\ \psi \end{smallmatrix}\right)\) and \(J=\left(\begin{smallmatrix} 0 & \mathrm{Id} \\ -\mathrm{Id} & 0 \end{smallmatrix}\right)\). The authors change the scale of the depth \(h\) to \(\mathsf{h}\).
The main result of the paper proves that for every choice of the tangential sites \(\mathbb{S}^{+}\subset \mathbb{N}\setminus \{0\}\), there exists \( \overline{s}>\frac{| \mathbb{S}^{+}| +1}{2}\), \( \varepsilon_{0}\in (0,1)\) such that for every vector \(\overrightarrow{a} =(a_{j})_{j\in \mathbb{S}^{+}}\), with \(a_{j}>0\) for all \(j\in \mathbb{S}^{+}\) and \(| \overrightarrow{a}| \leq \varepsilon_{0}\), there exists a Cantor-like set \(\mathcal{G}\subset \lbrack \mathsf{h}_{1},\mathsf{h}_{2}]\) with asymptotically full measure as \(\overrightarrow{a}\rightarrow 0\), that is \(\lim_{\overrightarrow{a}\rightarrow 0}| \mathcal{G} | =\mathsf{h}_{2}-\mathsf{h}_{1}\), such that, for any \(\mathsf{h} \in \mathcal{G}\), the gravity water waves system has a time quasi-periodic solution \(u(\widetilde{\omega}t,x)=(\eta (\widetilde{\omega}t,x),\psi (\widetilde{\omega}t,x))\), with Sobolev regularity \((\eta,\psi)\in H^{\overline{s}}(\mathbb{T}^{\nu}\times \mathbb{T},\mathbb{R}^{2})\), with a Diophantine frequency vector \(\widetilde{\omega}=(\widetilde{\omega}_{j})_{j\in \mathbb{S}^{+}}\in \mathbb{R}^{\nu}\) depending on \(\mathsf{h}\) and \(\overrightarrow{a}\), of the form \[ \begin{aligned} &\eta (\widetilde{\omega} t,x)=\sum_{j\in \mathbb{S}^{+}}a_{j}\cos (\widetilde{\omega}_{j}t)\cos (jx)+r_{1}(\widetilde{\omega}_{j}t),\\ &\psi (\widetilde{\omega} t,x)=-\sum_{j\in \mathbb{S}^{+}}\frac{a_{j}}{\widetilde{\omega}_{j}(\mathsf{h})}\sin (\widetilde{\omega}_{j}t)\cos (jx)+r_{2}(\widetilde{\omega}_{j}t), \end{aligned} \] with \(\widetilde{\omega}(\mathsf{h},\overrightarrow{a})\rightarrow \widetilde{\omega}(\mathsf{h})=(\omega_{j}(\mathsf{h}))_{j\in \mathbb{S}^{+}}\) as \(\overrightarrow{a}\rightarrow 0\), and the functions \(r_{1}(\widetilde{\omega}_{j}t)\), \(r_{1}(\widetilde{\omega}_{j}t)\) are \( o(| \overrightarrow{a}|)\)-small in \(H^{\overline{s}}(\mathbb{T}^{\nu}\times \mathbb{T},\mathbb{R})\). The solution \((\eta(\widetilde{\omega}t,x),\psi(\widetilde{\omega}t,x))\) is even in \(x\), \(\eta\) is even in \(t\) and \(\psi \) is odd in \(t\). The authors then prove that these quasi-periodic solutions are linearly stable, that is for all \(s\) belonging to a suitable interval \([s_{1},s_{2}]\), for any initial datum \( y(0)\in \mathbb{R}^{\nu}\), \(w(0)\in H_{x}^{s-1/4}\times H_{x}^{s+1/4}\), the solutions \(y(t)\), \(w(t)\) of the system \[ \begin{aligned} &\dot{\phi}=K_{20}(\widetilde{\omega}t)[y]+K_{11}^{T}(\widetilde{\omega}t)[w],\\ &\dot{y}=0,\\ &\dot{w}=JK_{02}(\widetilde{\omega}t)[w]+JK_{11}^{T}(\widetilde{\omega}t)[y]\end{aligned} \] satisfy \(y(t)=y(0)\) and \[ \| w(t)\|_{H_{x}^{s-1/4}\times H_{x}^{s+1/4}}\leq C(\| w(0)\|_{H_{x}^{s-1/4}\times H_{x}^{s+1/4}}+|y(0)|) \] for every \(t\in \mathbb{R}\). Here \(K_{20}\), \(K_{11}\) and \(K_{02}\) are operators associated to the above Hamiltonian \(H\).
For the proof of the existence of time quasi-periodic solutions to the problem, the authors introduce the system linearized around equilibrium \((0,0)\): \(\partial_{t}\eta =G(0, \mathsf{h})\psi \), \(\partial_{t}\psi =-\eta \), where \(G(0,\mathsf{h})=D\tanh (\mathsf{h}D)\) is the Dirichlet-Neumann operator at the flat surface \(\eta =0\). They consider the above problem as a singular perturbation of the linearized one. The main tools of the proof are the Nash-Moser theory, a degenerate KAM theory, an analysis of the linear operators involved in the problem, and the theory of pseudodifferential operators. The long paper presents the details of the proof, and the paper ends with appendices devoted to properties of the Dirichlet-Neumann operator, of Whitney differentiable functions and to a Nash-Moser-Hörmander implicit function theorem.