On the solvability of the discrete nonlinear Schrodinger equation with subcubic potential

In this paper, we analyze the solvability of the discrete nonlinear Schr\"odinger equation \begin{equation*} i\beta(\Delta_t+\nabla_t)\phi(t,k) +\gamma |\phi(t,k)|^2\phi(t,k) +\varepsilon \Delta_k^2\phi(t,k-1) = g(t,\phi(t,k)), \end{equation*} where $\Delta_t$ and $\Delta_k$ denote the standard forward difference operators in the variables $t$ and $k$, respectively, $\nabla_t$ denotes the standard backward difference operator in $t$, and \begin{equation*} \Delta_k^2\phi(t,k-1) = \phi(t,k+1)-2\phi(t,k)+\phi(t,k-1) \end{equation*} is the discrete Laplacian operator in the spatial variable $k$. Throughout, we will assume the parameters $\beta$ and $\varepsilon$ are positive real numbers, the parameter $\gamma$ is a nonzero real number, and the potential function $g:\mathbb{Z}\times\mathbb{C}\to \mathbb{C}$ is continuous.