I am studying the work of M. Boiti and F. Pempinelli, "Nonlinear Schrödinger Equation, Bäcklund Transformations and Painlevé Transcendent" (1980).
In the nonlinear Schrödinger equation \begin{align} i\partial_t q+\partial_x^2 q+\alpha q\left|q\right|^2=0, \end{align} where $q=q(x,t)$ and $\alpha^2=1$, they define the Bäcklund relations \begin{align} \partial_t\left(q-\tilde{q}\right)&=\left(\zeta+\zeta^*\right)\partial_x\left(q-\tilde{q}\right)-\frac{i}{2}\partial_x\left(q+\tilde{q}\right)\sqrt{-4\left(\zeta-\zeta^*\right)^2-2\alpha\left|q-\tilde{q}\right|^2}+i\frac{\alpha}{2}\left(q-\tilde{q}\right)\left(\left|q\right|^2+\left|\tilde{q}\right|^2\right),\\ \partial_x\left(q-\tilde{q}\right)&=-i\left(\zeta+\zeta^*\right)\left(q-\tilde{q}\right)-\frac{1}{2}\left(q+\tilde{q}\right)\sqrt{-4\left(\zeta-\zeta^*\right)^2-2\alpha\left|q-\tilde{q}\right|^2}, \end{align} where $\zeta=\xi+i\sigma$. My question is pure algebra: When we consider $q(x,t)=u(x,t)e^{i\theta(x,t)}$, $\tilde{q}(x,t)=\tilde{u}(x,t)e^{i\tilde{\theta}(x,t)}$, or $q(x,t)-\tilde{q}(x,t)=\psi(x,t)e^{i\left(\theta(x,t)-\omega(x,t)\right)}$, the first Bäcklund relation can be equated in real and imaginary part \begin{align} \dot{\psi}&=\left[2\xi\left(\psi-u\cos\omega\right)+u'\sin\omega+u\theta'\cos\omega\right]\sqrt{16\sigma^2-2\alpha\psi^2}+u\sin\omega\left(8\sigma^2-\alpha\psi^2\right),\\ \left(\dot{\theta}-\dot{\omega}\right)\psi&=\left[\left(\theta'-2\xi\right)u\sin\omega-u'\cos\omega\right]\sqrt{16\sigma^2-2\alpha\psi^2}+\left(4\sigma^2-4\xi^2+\alpha u^2\right)\psi-8\sigma^2u\cos\omega, \end{align} respectively, with $\dot{\psi}=\partial_t\psi$ and $\psi'=\partial_x \psi$. I am unable to obtain these equations (equation (2.7) and (2.9) of the paper, which can be seen below), I only obtain some terms.
