Rogue wave patterns associated with Okamoto polynomial hierarchies

We show that new types of rogue wave patterns exist in integrable systems, and these are described by root structures Okamoto polynomial hierarchies. These arise when the τ functions solutions determinants Schur polynomials with index jumps three, an internal free parameter waves gets large. demonstrate Manakov system three-wave resonant interaction system. For each system, we derive asymptotic predictions its under a large through Unlike previously reported associated Yablonskii–Vorob'ev hierarchy, feature present is mapping from structure Okamoto-hierarchy to shape pattern linear only leading order, but becomes nonlinear next order. As consequence, current often deformed, sometimes strongly structures, unless underlying very Our analytical compared true solutions, excellent agreement observed, even deformed structures.