A posteriori KAM existence conjecture for whiskered tori in the Boussinesq equation

Let $4\mu>0$ be a parameter in the Boussinesq equation such that the center space has dimension $2\ell\geq 2$. Fix a Diophantine exponent $\nu>\ell$, a regularity exponent $m>5/2$, and a positive analyticity radius $\rho_0$. Let $\omega^0$ be the frequency vector of the small-amplitude motions, and let $X=H^{\rho,m}(\mathbb T)\times H^{\rho,m-2}(\mathbb T)$. For a positive function $a$, write $B_{a(\varepsilon)}(\omega^0)\subset\mathbb R^\ell$ for the ball of radius $a(\varepsilon)$ centered at $\omega^0$.

A posteriori KAM existence conjecture. There exist three explicit functions $a,b_d,b_a:\mathbb R^+\to\mathbb R^+$ such that

$$a(s)\to0,\qquad b_d(s),b_a(s)\to\infty\quad\text{as }s\to0,$$

and, for sufficiently small $\varepsilon$, every $\omega\in\mathcal D(b(\varepsilon),\nu)\cap B_{a(\varepsilon)}(\omega^0)$ admits an analytic embedding $K:D_{\rho_0}\to X$ solving the invariant-torus equation with frequency $\omega$. Moreover,

$$\frac{\left|\mathcal D(b(\varepsilon),\nu)\cap B_{a(\varepsilon)}(\omega^0)\right|}{\left|B_{a(\varepsilon)}(\omega^0)\right|}\longrightarrow1,$$

and the map sending $\omega$ to $K$ is Lipschitz in the topology of analytic embeddings from $D_{\rho'}$ to $X$ for every $\rho'<\rho_0$.

This is an a posteriori KAM assertion for analytic invariant tori near the small-amplitude frequency vector. The source presents it as a conjectural extension of the theorem under consideration; the supplied material does not establish its resolution.