Numerical validity of the island hypotheses in dark-gray parameter regions

Consider the repulsive one-dimensional Gross–Pitaevskii equation with cosine potential, parameters $\omega$ and $A$, and its associated Poincaré map $T$. Let $\Delta_n^\pm$ be the sets defined in the paper, and let $\mu(S)$ denote the area of a set $S$. The phase-space islands $D_i$, their $v$- and $h$-curves and strips, and the sets $\Delta_n^\pm$ are those used in Hypotheses 1–3. Numerical validity conjecture. If the parameters $\omega$ and $A$ belong to the dark-gray areas (Regions 1 and 2) of the parameter plane $(\omega,A)$ in Figure CodZones, then Hypotheses 1–3 hold: the Poincaré map has the stated $v$- and $h$-strip properties, and

$$\lim_{n\to\infty}\mu(\Delta_n^\pm)=0.$$

The claim is based on a systematic numerical investigation and concerns the parameter regimes in which the proposed coding construction for gap solitons is expected to apply; the source provides no proof or resolution beyond this numerical support.