= Solution
Write $r=\sqrt{\mu/\delta}$. At first order the amplitude-phase <perturbation> is $A=r+\varepsilon(A_1+ir\theta)+O(\varepsilon^2)$. Expanding the cubic term in the <focusing nonlinear Schrodinger equation> gives $A|A|^2=r^3+\varepsilon(3r^2A_1+ir^3\theta)+O(\varepsilon^2)$. The constant terms cancel because $\delta r^2=\mu$. Separating real and <imaginary parts> yields
$$
\boxed{(A_1)_t=\chi r\theta_{xx},\qquad r\theta_t+\chi(A_1)_{xx}+2\mu A_1=0.}
$$
For the Fourier <normal modes>, these equations give $-i\Omega\widehat A_1=-\chi rK^2\widehat\theta$ and $-ir\Omega\widehat\theta+(2\mu-\chi K^2)\widehat A_1=0$. Their <determinant> vanishes exactly when
$$
\boxed{\Omega^2=\chi K^2(\chi K^2-2\mu).}
$$
The <focusing nonlinear Schrodinger modulation dispersion> is negative for $0<K^2<2\mu/\chi$. One of the two <frequencies> then has positive imaginary part and grows exponentially: the constant state has <modulational instability>. The <growth rate> is $\sqrt{\chi K^2(2\mu-\chi K^2)}$, maximized at $K^2=\mu/\chi$ with value $\mu$. Modes beyond the unstable band have real <frequencies>. The zero-<wavenumber> phase mode is neutral, with possible linear phase drift from a uniform amplitude <perturbation>.
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