On the solvability of the discrete nonlinear Schrodinger equation with subcubic potential
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866917542465896448 |
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| author | Maroncelli, Daniel |
| author_facet | Maroncelli, Daniel |
| contents | In this paper, we analyze the solvability of the discrete nonlinear Schrödinger equation \begin{equation*} iβ(Δ_t+\nabla_t)ϕ(t,k) +γ|ϕ(t,k)|^2ϕ(t,k) +\varepsilon Δ_k^2ϕ(t,k-1) = g(t,ϕ(t,k)), \end{equation*} where $Δ_t$ and $Δ_k$ 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*} Δ_k^2ϕ(t,k-1) = ϕ(t,k+1)-2ϕ(t,k)+ϕ(t,k-1) \end{equation*} is the discrete Laplacian operator in the spatial variable $k$. Throughout, we will assume the parameters $β$ and $\varepsilon$ are positive real numbers, the parameter $γ$ is a nonzero real number, and the potential function $g:\mathbb{Z}\times\mathbb{C}\to \mathbb{C}$ is continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29145 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the solvability of the discrete nonlinear Schrodinger equation with subcubic potential Maroncelli, Daniel Analysis of PDEs Mathematical Physics In this paper, we analyze the solvability of the discrete nonlinear Schrödinger equation \begin{equation*} iβ(Δ_t+\nabla_t)ϕ(t,k) +γ|ϕ(t,k)|^2ϕ(t,k) +\varepsilon Δ_k^2ϕ(t,k-1) = g(t,ϕ(t,k)), \end{equation*} where $Δ_t$ and $Δ_k$ 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*} Δ_k^2ϕ(t,k-1) = ϕ(t,k+1)-2ϕ(t,k)+ϕ(t,k-1) \end{equation*} is the discrete Laplacian operator in the spatial variable $k$. Throughout, we will assume the parameters $β$ and $\varepsilon$ are positive real numbers, the parameter $γ$ is a nonzero real number, and the potential function $g:\mathbb{Z}\times\mathbb{C}\to \mathbb{C}$ is continuous. |
| title | On the solvability of the discrete nonlinear Schrodinger equation with subcubic potential |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2605.29145 |