= Solution
Put $g=\gamma P_0$. For a temporally uniform envelope, $A_z=igA$ and $A_{TT}=0$, so the propagation equation reduces to $-gA+\gamma P_0A=0$. Thus \b[the <power> stays $P_0$ and the accumulated nonlinear <wave phase> is $g z$]. The <optical Kerr effect> gives an intensity-dependent <refractive index>, and hence <self-phase modulation>. For the usual <power> normalization $\gamma\simeq n_2\omega_0/(cA_{\rm eff})$, with effective guided-mode area $A_{\rm eff}$. Losslessness preserves the background <power>; it does not remove the <self-phase modulation>. The carrier <wave phase> has already been factored out.
Write $A=(\sqrt{P_0}+\epsilon)e^{igz}$. To first order,
$$
|\sqrt{P_0}+\epsilon|^2(\sqrt{P_0}+\epsilon)
=P_0^{3/2}+P_0(2\epsilon+\epsilon^*)+O(|\epsilon|^2).
$$
Cancelling the background and its phase derivative leaves
$$
i\epsilon_z-\frac{\beta_2}{2}\epsilon_{TT}+g(\epsilon+\epsilon^*)=0.
$$
The last term couples a <perturbation> to its <complex conjugate>: the pump can transfer <energy> into a pair of sidebands. It is not an independent <wave phase> of each Fourier component. This <sideband coupling> is the linear origin of <fibre modulational instability>.
For real $K,\Omega,c_1,c_2$, put $d=\beta_2\Omega^2/2$. Matching the two exponential coefficients gives
$$
\begin{pmatrix}d+g-K&g\\g&d+g+K\end{pmatrix}
\begin{pmatrix}c_1\\c_2\end{pmatrix}=0.
$$
A nonzero pair requires a zero <determinant>, hence
$$
\boxed{K^2=d(d+2g)=\frac{\beta_2^2\Omega^2}{4}
\left(\Omega^2+\frac{4g}{\beta_2}\right).}
$$
For unstable complex $K$, the real-coefficient assumption is too restrictive. The consistent pair is $\epsilon=u e^{iKz-i\Omega T}+v^*e^{-iK^*z+i\Omega T}$, with complex $u,v$. Its equations give the same <determinant> and allow the growing and decaying branches. Merely inserting imaginary $K$ while treating the two original coefficients as real would not respect <complex conjugation>.
For $P_0>0$ and nonzero $\beta_2,\gamma$, exponential growth requires \b[$\beta_2\gamma<0$] and
$$
0<\Omega^2<-\frac{4\gamma P_0}{\beta_2}.
$$
For ordinary positive Kerr nonlinearity this means anomalous <group velocity dispersion>, $\beta_2<0$. Define $\Omega_c=\sqrt{4g/|\beta_2|}$. The growing branch is $K=-iG$, with <amplitude> growth rate
$$
G(\Omega)=\frac{|\beta_2||\Omega|}{2}\sqrt{\Omega_c^2-\Omega^2},\qquad
\boxed{|\Omega|_{\max}=\Omega_c/\sqrt2,\quad G_{\max}=g.}
$$
Sideband <power> grows at rate $2G$, rather than $G$. The cutoff and zero-frequency modes have no exponential growth; in a degenerate zero-frequency sector a secular phase response can remain. Positive dispersion with positive $\gamma$ gives real $K$ and oscillatory <perturbations>.
Initially small ripples grow into pronounced <intensity> peaks and deplete the uniform background. Dispersion and <self-phase modulation> can then support localized bright pulses. A <fundamental temporal bright soliton> obeys $\gamma P_sT_0^2=|\beta_2|$, so its width is comparable to $1/\Omega_c$ when its peak <power> is comparable to the original <power>. The dominant ripple spacing is $2\pi/|\Omega|_{\max}$, not simply $1/\Omega_c$. Propagation distance $z$ is the evolution variable; $T$ measures time within the envelope. An ideal conservative fibre need not settle permanently into a unique pulse train: nonlinear compression, interactions and recurrence depend on the seed and boundary conditions.
\Image[/past-exam-of-the-nonlinear-optics-and-quantum-states-of-light-course-of-the-university-of-cambridge/2014/paper-1-modulational-instability.png]
{title=Original instability gain and calculated nonlinear growth of a weak periodic modulation in a focusing fibre model}
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