Standard nearest-neighbour discretizations of Klein-Gordon models cannot preserve both energy and linear momentum

We consider nonlinear Klein-Gordon wave equations and illustrate that standard discretizations there of (involving nearest neighbours) may preserve either standardly defined linear momentum or standardly defined total energy but not both. This has a variety of intriguing implications for the 'nonpotential' discretizations that preserve only the linear momentum, such as the self-accelerating or self-decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices.