  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 the lattices. First, we establish the well-posedness of
generalized quasilinear Schr\"odinger equations and give a new proof of the
global well-posedness of semilinear Schr\"odinger equations. Then we provide
explicit expressions of higher order derivatives of the solution map and prove
the analytic dependence between solution and initial data. At the end, we show
the existence of wave operator and give a sufficient condition for the
asymptotic completeness.
