Fourier transform of a nonlinear Schrödinger soliton The nonlinear Schrödinger equation
$$
i\frac{\partial\psi}{\partial t} = -\frac{1}{2}\frac{\partial^2\psi}{\partial x^2}- |\psi|^2\psi
$$
has the single-soliton solution
$$
\psi(x,t)=A \frac{e^{iv(x-x_0)+i\frac{(A^2-v^2)}{2}t}}{\cosh(A(x-x_0 -vt))}.
$$
The simplest case of the soliton is with $x_0=0$ and at the initial time $t=0$, so
$$
\psi(x,0)=A \frac{e^{ivx}}{\cosh(Ax)}.
$$
Is there a closed-form expression for the spatial Fourier transform $\hat{\psi}(k,t)$ of this soliton solution ?
I have tried but I can't think of how to do the integral (and anyway Mathematica has been churning away at it for half an hour so if Mathematica can't do it then I certainly can't do it!), and google only comes up with results for "nonlinear Fourier transform" (aka scattering transform).
If there is no closed-form solution, then does anyone know any asymptotic expression for large $k$ ?
 A: We need to evaluate
$$
I(k)=\int\limits_{-\infty}^\infty dx \ \frac{e^{ikx}}{2\cosh(x)}=\int\limits_{-\infty}^\infty dx \ \frac{e^{ikx}}{e^x+e^{-x}}
$$
From which your Fourier transform follows with rescaling $Ax \to x$ and the replacement $k\to k+v/A$. We can use contour integrals to write $I$ as a sum which we can evaluate. The zeros of the denominator are at
$$
x_n=i\pi(n+1/2) \qquad, \qquad n=0,\pm1,\pm2,\dots
$$
The residue at each pole of the integrand is
$$
R_n=-(-1)^n\frac{i}{2}e^{-(n+1/2)\pi k}
$$
When $k>0$ we close the CCW integration contour in the upper half plane. The contribution due to the semicircular arc vanishes for large radius. The poles with $n\geq 0$ are enclosed. Then the residue theorem says
$$
I(k)=2\pi i\sum\limits_{n=0}^\infty R_n=\pi\sum\limits_{n=0}^\infty(-1)^ne^{-(n+1/2)\pi k}
$$
On the right is a geometric sum which yields
$$
I(k)=\frac{\pi}{2}\operatorname{sech}\left(\frac{\pi k}{2}\right)
$$
When $k<0$ a similar analysis leads to the identical expression.
