Resonant three-wave triad (${\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3=0$)

Three Fourier modes form a spatial resonance when their wavevectors sum to zero. Products of the conjugate second and third modes then force the first mode, with cyclic counterparts. Such terms couple both amplitudes and phases, unlike self-saturation terms proportional only to squared magnitudes. For three equal wavenumbers, the vectors are separated by $120$ degrees and support hexagonal pattern interactions.