Determine the envelope function A(x, t) of a Gaussian wavepacket with frequency and wavenumber k propagating

Determine the envelope function A(x, t) of a Gaussian wavepacket with frequency ω and wavenumber k propagating in a one-dimensional medium (e.g., an optical pulse in a fiber optic) when ∂²ω/∂k² is significant (i.e., approximate ω(k) to second order, instead of only to first order as in (3.3.5)). What happens to the width of the wavepacket over time?

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w(k') = w(k) + Vzw|z •· (k' − k) + · · · ≈w+vg.k, 8 (3.3.5)