arXiv subjects

Daniel Maroncelli

Publications and source records attributed to Daniel Maroncelli.

3 recordsLinked to original sources

arXiv2026

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.

math.AP↗

arXiv2018

Nonlinear scalar discrete multipoint boundary value problems at resonance

In this work we provide conditions for the existence of solutions to nonlinear boundary value problems of the form \begin{equation*} y(t+n)+a_{n-1}(t)y(t+n-1)+\cdots a_0(t)y(t)=g(t,y(t+m-1)) \end{equation*} subject to \begin{equation*} \sum_{j=1}^nb_{ij}(0)y(j-1)+\sum_{j=1}^nb_{ij}(1)y(j)+\cdots+\sum_{j=1}^nb_{ij}(N)y(j+N-1)=0 \end{equation*} for $i=1,\cdots, n$. The existence of solutions will be proved under a mild growth condition on the nonlinearity, $g$, which must hold only on a bounded subset of $\{0,\cdots, N\}\times\mathbb{R}$.

math.DS↗

arXiv2015

Periodic behaviour of nonlinear second order discrete dynamical systems

In this work we provide conditions for the existence of periodic solutions to nonlinear, second-order difference equations of the form \begin{equation*} y(t+2)+by(t+1)+cy(t)=g(t,y(t)) \end{equation*} where $c\neq 0$, and $g:\mathbb{Z}^+\times\mathbb{R}\to \mathbb{R}$ is continuous and periodic in $t$. Our analysis uses the Lyapunov-Schmidt reduction in combination with fixed point methods and topological degree theory.

math.CA↗