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When I deal with soliton problem in optics, I get a complex amplitude function in the time domain as $$E(t)=\text{sech}\left(\frac{t}{a}\right)\text{exp}[i(bt^2+c)]$$ where $a$, $b$ and $c$ are non-zero constant. The problem is how to analytically get the Fourier transformation of the function $E(t)$. $$E(\omega)=\int^\infty_{-\infty}\text{sech}\left(\frac{t}{a}\right)\text{exp}[i(bt^2+c)]\text{exp}(-i\omega t)\text{d}t$$ When I numerically calculated this Fourier transformation, I got a soomth envelop much like sech function in the frequency domain. So, I wonder if there is an analytical result of it.

Thank you in advance!

Chao Mei
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    Use the fact that $\mathrm{sech}(z) = \frac{2}{e^z+e^{-z}}$, find the singularities of the integrand and apply the residue formula. – Abezhiko Jan 14 '23 at 07:35
  • Unfortunately, I tried to use the residue theorem to solve this problem, but couldn't get the result. The main reason is that there is a square term in the cosine term, which makes its contour integral cannot be calculated. – Chao Mei Jun 18 '23 at 13:57

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