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Three-wave instability is a fundamental process that has importantapplications in many branches of physics. It is widely accepted that theresonant condition $\omega_{z}\approx\omega_{x}+\omega_{y}$ for participatingwaves is the criteria for the onset of the instability. We show that thiscondition is neither sufficient nor necessary, instead, the exact criteria forthe onset of the instability is that a positive-action mode resonates with anegative-action mode. This mechanism is imposed by the topology and geometry ofthe spectral space. Guided by this new theory, additional instability bandspreviously unknown are discovered.

On the physical mechanism of three-wave instabilities -- resonance between positive- and negative-action modes