Why collisions always matter for a resonance even in the presence of islands

For a monochromatic wave Zakharov and Karpman (1963 Sov. Phys. JETP v.16, 351) proved Landau damping is a collisional process. They were able to do this by retaining the nonlinear term and solving in the weak collision limit for a finite amplitude plasma wave. The same procedure that works for a steady state plasma wave (Catto 2025a JPP, v.91, E80) can also be used to demonstrate that the lower hybrid current drive (LHCD) efficiency is reduced in the intense applied field limit (Catto 2025b JPP, v.91, E75) where quasilinear theory fails. Similar complications arise in a drift kinetic treatment of an optimized stellarator with a single helicity departure from quasisymmetry (QS). In this case a large enough departure from QS in the weak collision frequency limit causes the transition from the linear superbanana plateau or resonant plateau regime to a reduced transport nonlinear regime (Catto 2025c JPP, v.91, E25). Not surprisingly, in all these cases the transition from a linearized collisional evaluation to a nonlinear solution occurs when island width effects are no longer suppressed by collisional boundary layer effects.