Asymptotic behavior of discrete Schr\"{o}dinger equations on the hexagonal triangulation

TL;DR

This paper establishes decay estimates and Strichartz inequalities for the discrete Schrödinger equation on a hexagonal triangulation, showing faster decay than on standard lattices and applying results to nonlinear variants.

Contribution

It provides the first decay rate analysis for DS on hexagonal triangulation and extends to nonlinear equations, using advanced oscillatory integral techniques.

Findings

Decay rate of ^t^{-rac{3}{4}} on hexagonal triangulation

Faster decay than on ^2 lattice (^{-rac{2}{3}})

Application to discrete nonlinear Schrf6dinger equation

Abstract

In this article, we prove the decay estimate for the discrete Schr\"odinger equation (DS) on the hexagonal triangulation. The $l^1\rightarrow l^\infty$ dispersive decay rate is $\left\langle t\right\rangle^{-\frac{3}{4}}$, which is faster than the decay rate of DS on the 2-dimensional lattice $\mathbb{Z}^2$, which is $\left\langle t\right\rangle^{-\frac{2}{3}}$, 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 Schr\"odinger equation (DNLS) on the hexagonal triangulation.