 Focusing nonlinear Schrodinger modulation dispersion

For $-i{A}_t+\chi A_{xx}-\mu A+\delta A|A{|}^2=0$ with positive coefficients, linearize around $A=\sqrt{\mu /\delta }$ using real amplitude and phase perturbations. Their two-by-two dispersion relation gives the displayed equation. The modulational instability band is $0<K^2<2\mu /\chi$, with maximum growth rate $\mu$ at $K^2=\mu /\chi$.