  In this paper, we introduce a novel first-order derivative for functions on a
lattice graph, and establish its weak (1, 1) estimate as well as strong (p, p)
estimate for p > 1 in weighted spaces. This derivative is designed to
reconstruct the discrete Laplacian, enabling an extension of the theory of
nonlinear wave equations, including quasilinear wave equations, to lattice
graphs. We prove the local well-posedness of generalized quasilinear wave
equations and the long-time well-posedness of these equations for small initial
data. Furthermore, we prove the global well-posedness of defocusing semilinear
wave equations for large initial data.
