In this paper, we analyze the solvability of the discrete nonlinear Schrödinger equation
$ \begin{equation*} i\beta(\Delta_t+\nabla_t) \varphi(t,k) +\gamma | \varphi(t,k)|^2 \varphi(t,k) + \varepsilon \Delta_k^2 \varphi(t,k-1) = g(t, \varphi(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 \varphi(t,k-1) = \varphi(t,k+1)-2 \varphi(t,k)+ \varphi(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.
Citation: Daniel Maroncelli. On the solvability of the discrete nonlinear Schrödinger equation with subcubic potential[J]. Electronic Research Archive, 2026, 34(7): 4803-4811. doi: 10.3934/era.2026212
In this paper, we analyze the solvability of the discrete nonlinear Schrödinger equation
$ \begin{equation*} i\beta(\Delta_t+\nabla_t) \varphi(t,k) +\gamma | \varphi(t,k)|^2 \varphi(t,k) + \varepsilon \Delta_k^2 \varphi(t,k-1) = g(t, \varphi(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 \varphi(t,k-1) = \varphi(t,k+1)-2 \varphi(t,k)+ \varphi(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.
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Daniel Maroncelli. On the solvability of the discrete nonlinear Schrödinger equation with subcubic potential[J]. Electronic Research Archive, 2026, 34(7): 4803-4811. doi: 10.3934/era.2026212
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