A Stability Criterion for Envelope Equations

The statistical initial value problem for the envelope equation \[ \frac{{\partial w}}{{\partial t}} - \gamma \frac{{\partial ^2 w}}{{\partial x^2 }} = \chi w - \beta w^2 w^ * ,\quad \gamma ,\beta \,{\text{complex}} \], is discussed. It is shown that there exists an instability (stability) criterion \[ \beta _r \gamma _r + \beta _i \gamma _i \gtreqless 0,\quad \beta _r > 0 \], which determines whether the system underlying the above equation achieves or does not achieve a monochromatic state.