Governing equations of envelopes created by nearly bichromatic waves and relation to the nonlinear Schrödinger equation

In this paper, we derive the governing equations of the envelope created by nearly bichromatic waves defined as the waves with wavenumber spectrum has two closely approaching peaks and spreads over around two peaks. The obtained governing equation has higher-order terms related to the distinctive quantities of nearly bichromatic waves. We show a nonlinear Schrödinger type equation governs the envelope created by nearly bichromatic waves, up to third order, under some conditions.