The fourth-order Schrödinger equation on lattices

In this paper, we study the fourth-order Schrödinger equation \begin{equation*} i \partial_t u + Δ^2 u - γΔu = \pm |u|^{s-1}u \end{equation*} on the lattice ${\mathbb{Z}}^d$ with dimensions $d=1,2$ and parameter $\gamma \in \mathbb{R}$. In order to establish sharp dispersive estimates, we consider the fundamental solution as an oscillatory integral and analyze the Newton polyhedron of its phase function. Furthermore, we prove Strichartz estimates which yield the existence of global solutions to nonlinear equations with small data.