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Motivated by problems arising in geophysical fluid dynamics, we investigateresonant and near resonant wave interactions in nonlinear wave equations withquadratic nonlinearity, We place a special focus on interactions between slowwave modes, with zero frequency in the linear limit, and fast modes. Theseregularly occur in geophysical fluid systems with conserved potential vorticityor similar conserved quantities. A general multi-scale asymptotic expansion isused to show how the higher order nonlinear interaction coefficients arederived as a combination of the first order terms arising at the triadinteraction level. From the general nth-order interaction coefficient wepresent a proof by induction how the limiting effect for particularcombinations of slow and fast modes pushes their interactions to a slowertimescale, and we show how this is linked to the form of the conservedquantities. We compare near resonant expansions with exact resonant expansions,and show how near-resonances allow higher order expansions to be reduced tojust the most dominant contributions. These contributions then occur at oneorder higher in the expansion when compared to the analogous exact resonanceexpansion.

Quartet, higher order and near resonant interactions in nonlinear wave equations