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We develop a rigorous asymptotic derivation of two mathematical models of water waves that capture the full nonlinearity of the Euler equations up to quadratic and cubic interactions, respectively. Specifically, letting $$ \epsilon $$ ϵ denote an asymptotic parameter denoting the steepness of the water wave, we use a Stokes expansion in $$ \epsilon $$ ϵ to derive a set of linear recursion...