Elimination of variables in simple laser equations

The usual way of obtaining rate equations (RE) and a single equation for the field amplitude (EFA) from the semiclassical laser equations (Lorenz-Haken model) is reexamined by undertaking a systematic elimination procedure developed in synergetics. The RE and EFA are justified in the case 1 (γ_⊥≫γ_∥, κ) and case 2 (γ_⊥, γ_∥≫κ), respectively. We show that, because the eliminated variable happens to contain a considerable contribution from an unstable mode, the usual elimination technique in the case 3 (γ_⊥, κγ_∥) leads to an inconsistency. As important by-products we obtain the RE and EFA for arbitrary cavity relaxation constant (κ). Some remarks are given on the direct elimination technique in the non-diagonal representation in the study of instabilities.