Continuum limit of fourth-order Schrödinger equations on the lattice

TL;DR

This work analyzes the discrete fourth-order Schrödinger equation on the lattice $h\mathbb{Z}^2$ and establishes uniform-in-$h$ dispersive and Strichartz estimates via frequency-localized oscillatory integrals and stationary-phase analysis. By decomposing the Fourier side with Paley–Littlewood projections, the authors obtain uniform bounds for the discrete kernel and derive a sharp $L^2$-convergence rate $h^{2/3}$ in the continuum limit for the associated semilinear problem, linking discrete dynamics to the continuous model on $\mathbb{R}^2$. The study combines detailed kernel estimates, lattice harmonic analysis, and discretization/interpolation machinery to bridge discrete and continuous dispersive PDEs, with implications for numerical analysis of nonlinear dispersive flows. The results rely on uniform Strichartz bounds and nonlinear stability to quantify the continuum limit and its temporal growth.

Abstract

In this paper, we consider the discrete fourth-order Schrödinger equation on the lattice $h\mathbb{Z}^2$. Uniform Strichartz estimates are established by analyzing frequency localized oscillatory integrals with the method of stationary phase and applying Littlewood-Paley inequalities. As an application, we obtain the precise rate of $L^2$ convergence from the solutions of discrete semilinear equations to those of the corresponding equations on the Euclidean plane $\mathbb{R}^2$ in the contimuum limit $h \rightarrow 0$.

Theorems & Definitions (18)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Lemma 2.1: Paley-Littlewood inequality
• Lemma 2.2
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm-main']} assuming Theorem \ref{['thm-main 2']}