An integral equation which generalizes the dispersion relation for electrostatic plasma modes with wave numbers in the presence of a large-scale magnetoacoustic (pump) oscillation is derived. This calculation is accurate to all orders in the parameter ⋅ where is the amplitude of the particle displacement induced by the pump wave. Expanding in the latter parameter, we obtain a second-order differential equation including three-wave processes, non-linear frequency shifts and nonlinear Landau damping. The electromagnetic wave may disintegrate into two Bernstein waves (three-wave interaction) if the appropriate resonance condition (between the real parts of the frequencies) can be satisfied approximately. However, a Bernstein wave can still be excited when the resonance condition is not fulfilled. In the latter case, instability arises through inverse non-linear Landau damping, and the non-linear dispersion relation can be approximated, to a high degree of accuracy, by a three-by-three matrix. In the former case, the dispersion relation reduces, to a good approximation, to the standard two-by-two matrix only if the linear decrements of both electrostatic waves are sufficiently small; otherwise, the three-wave process identifies with the non-linear Landau instability calculated in the appropriate limit (of matching between the real parts of the frequencies). The relative importance of resonant and off-resonant coupling of the pump wave to the plasma is discussed.