A real-variable unidirectional reduction of deep-water gravity waves

A unidirectional reduction of the deep-water surface gravity wave problem is derived in physical space using real variables. By employing a near-identity canonical transformation, cubic interactions are eliminated from the Hamiltonian, with an exact elimination of second- and third-order bound waves. A projection operator is then constructed to isolate the unidirectional, rightward-propagating dynamics at the next asymptotic order, yielding a single nonlocal evolution equation. The model admits the third-order Stokes wave as an exact monochromatic solution, and a multiple-scales analysis recovers the Dysthe envelope equation, including the nonlocal mean-flow coupling, without requiring an auxiliary boundary value problem. Dropping four sub-leading nonlinear terms that vanish on the resonant manifold yields a more compact variant suitable for analytical study. Numerical validations demonstrate that both formulations faithfully reproduce the full Euler dynamics through modulational-instability recurrence and broadband focusing up to moderate wave steepness.