Convergence of long-period breathers to a rogue wave

Establish whether a subsequence of the least-energy kT-periodic breather solutions converges, as k tends to infinity, to a least-energy rogue wave solution of Theorem 1.1.

Background

The paper proves the existence of least-energy rogue waves on the real time axis and, under time-periodic assumptions, least-energy breather solutions with every period kT for positive integers k. The authors observe that increasingly long temporal periods might link these two classes of solutions: as the period grows, a periodic breather could plausibly approach a temporally localized rogue wave.

The proposed convergence is not established. The authors identify it as an open problem because proving it would require controlling the variationally constructed breather solutions as their periods tend to infinity, and they note that an analogous argument has been carried out for the nonlinear Schrödinger equation.