The well-posedness and scattering theory of nonlinear Schr\"odinger equations on lattice graphs

TL;DR

This paper develops a new derivative concept for functions on lattice graphs, establishing well-posedness, analytic dependence, and scattering theory for nonlinear Schrödinger equations in this discrete setting.

Contribution

It introduces a novel first-order derivative on lattice graphs and proves well-posedness, analytic dependence, and scattering results for discrete nonlinear Schrödinger equations.

Findings

Established well-posedness of generalized discrete quasilinear Schrödinger equations

Proved global well-posedness of discrete semilinear Schrödinger equations

Demonstrated existence of wave operators and asymptotic completeness for small data

Abstract

In this paper, we introduce a novel first-order derivative for functions on a lattice graph, which extends the discrete Laplacian and generalizes the theory of discrete PDEs on lattices. First, we establish the well-posedness of generalized discrete quasilinear Schr\"odinger equations and give a new proof of the global well-posedness of discrete semilinear Schr\"odinger equations. Then we provide explicit expressions of higher-order derivatives of the solution map and prove the analytic dependence between the solution and the initial data. At the end, we show the existence of the wave operator and prove the asymptotic completeness of the solutions with the small data.