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Starting from a gaussian waveform propagating in a dispersive medium, is it possible to obtain an expression for the waveform at a generic time [itex]t[/itex], when the dispersion is not negligible?

I know that a generic gaussian pulse (considered as an envelope of a carrier at frequency [itex]k_c[/itex]) can be expressed for [itex]t = 0[/itex] as

[itex]g(x) = e^{ - \frac{1}{2} \left( \frac{x - x_0}{\sigma_x} \right)^2} e^{i k_c (x - x_0)}[/itex]

and its Fourier transform is

[itex]G(k) = \displaystyle \frac{\sigma_x}{\sqrt{2 \pi}} e^{- \frac{(k - k_c)^2 \sigma_x^2}{2}} e^{-i (k - k_c)x_0}[/itex]

Then, the pulse at a generic time [itex]t[/itex] can be obtained as

[itex]G(x,t) = \displaystyle \int_{-\infty}^{+\infty} G(k) e^{i(kx - \omega(k) t)} dk[/itex]

with [itex]\omega(k) = \omega(k_c) + (k - k_c) \displaystyle \left. \frac{d \omega}{dk} \right|_{k = k_c} + (k - k_c)^2 \left. \frac{d^2 \omega}{dk^2} \right|_{k = k_c}[/itex].

The first derivative is known as the group velocity. Note that the second derivative is considered too.

I don't know how to proceed in order to obtain an explicit form for [itex]G(x,t)[/itex] which can show its evolution during time, showing the velocity of propagation of the pulse and its possible broadening. With a simple substitution of [itex]\omega(k)[/itex] and solution of the integral I did not get such an explicit form. Is there anyone who can help, or anyone who does know a site/document dealing with this topic?

Thank you anyway,

Emily