Asymptotic behavior of discrete Schrödinger equations on the hexagonal triangulation

In this article, we prove the decay estimate for the discrete Schrödinger equation (DS) on the hexagonal triangulation. The $l^1\to l^{\mathrm{\infty }}$ dispersive decay rate is ${⟨t⟩}^{-\frac{3}{4}}$, which is faster than the decay rate of DS on the 2-dimensional lattice ${\mathbb{Z}}^2$, which is ${⟨t⟩}^{-\frac{2}{3}}$, see [32]. The proof relies on the detailed analysis of singularities of the corresponding phase function and the theory of uniform estimates on oscillatory integrals developed by Karpushkin [15]. Moreover, we prove the Strichartz estimate and give an application to the discrete nonlinear Schrödinger equation (DNLS) on the hexagonal triangulation.