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

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.