Universality of the recurrent exchange between instability patterns near maximal-height periodic waves

Establish whether the recurrent exchange between figure-8 instability (bands crossing the imaginary axis) and figure-infinity instability (bands crossing the real axis) is universal near the limit to the periodic wave of maximal height and occurs in other wave models, including the Whitham equation.

Background

In motivating the paper, the authors discuss the spectral instability of Stokes waves and the observed transformation between figure-8 and figure-infinity instability patterns, involving exchanges between co-periodic and anti-periodic eigenvalues as dominant instabilities. They note a conjecture that this recurrent exchange is universal near maximal-height periodic waves and may manifest in other models.

The present work confirms analogous transformations for cnoidal waves in the focusing mKdV equation, reinforcing interest in the broader universality question across different dispersive wave models such as the Whitham equation.