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!