Cantor families of periodic solutions near nonresonant spherical capillary frequencies

Let $(n_0,j_0)$ be a pair of natural numbers with $n_0\geq2$, $j_0\geq1$, and set

$$\omega_0=\sqrt{\Lambda(n_0)}/j_0.$$

Suppose that the only natural number solution of the Diophantine equation

$$\omega_0^2j_0=\Lambda(n_0)^2=n_0(n_0-1)(n_0+1)$$

is $(j_0,n_0)$. Periodic-solution conjecture. There is a Cantor set with positive measure of parameters $\omega$, clustered near $\omega_0$, such that the spherical capillary water waves equation admits small amplitude periodic solution with frequency $\omega$. The conjecture refines the preceding existence proposal by imposing a uniqueness condition on the relevant natural-number solution of the Diophantine equation. The corresponding result for gravity-capillary standing water waves was proved by Alazard and Baldi using a Nash–Moser theorem, whereas the spherical capillary case remains unresolved.