The continuum (cubic) NLS has a standing wave solution, for $\omega < 0$ ,
$$u_{\omega}(x,t) = e^{-i \omega t} \psi_{|\omega|}(x),$$ where $\psi_{|\omega|}(x)$ is the unique solution to the nonlinear elliptic problem \begin{align}-\Delta_{x}\psi_{|\omega|} - |\psi_{|\omega|}|^{2}\psi_{|\omega|} &= \omega \psi_{|\omega|}, \tag{1}\end{align} which is real-valued, radially symmetric, and decreasing to zero at spatial infinity.
In [1], they consider solving the initial value problem for discrete nonlinear Schrödinger equation (DNLS)
\begin{align} i \partial_{t}u_{n}(t) &= -h^{-2} \left ( \delta^{2} u\right )_n - |u_n|^{2}u_n, \; n\in\mathbb{Z}^{d} \tag{2} \end{align}
in spatial dimension $d=1$ with the initial condition
\begin{align} u_{n}(0) &= e^{i \nu \cdot x_{n}}\psi_{|\omega|}(x_{n}),\; x_{n}=nh,\; n\in \mathbb{Z} \tag{3} \end{align}
How can they use $\psi_{|\omega|}$ in their initial condition if we do not know what it actually is? In (1.3) they say there exist a unique solution, but what is it?