math-ph — Mathematical Physics

Nonlinear Wave Limits of Reversible Finite-State Cellular Automata

Contributed by Zhenyu Sun

We construct reversible finite-state cellular automata whose integer lifts approximate smooth solutions of one-dimensional semilinear wave equations. The kick–drift update is bijective for every odd modulus. Separate rounding of spatial edge contributions and local nonlinear forces gives an exact one-cell support bound for the full local operator algebra. Under a strict Courant condition, a discrete energy estimate bounds the field and staggered-velocity errors by $C_T(h^2+\tau^2+\delta/\tau^2)$ on a fixed physical time interval, where $h$, $\tau$, and $\delta$ are the spatial, temporal, and field spacings. An explicit modulus bound prevents modular wrapping throughout that interval. The choices $\tau=\nu h$, $0<\nu<1$, and $\delta=O(h^4)$ yield second-order convergence. Coupled scalar fields have a common-speed, locally Lorentz-covariant classical limit. Equal bare masses and equal quartic self-couplings preserve a symmetric invariant manifold. Unequal quartic couplings introduce a cubic source into the antisymmetric field equation, and we derive the first variation of the antisymmetric field. Numerical experiments compare exact integer trajectories with a nonlinear traveling wave and an independent continuum solver, measure light and massive dispersion branches, and examine continuously interacting wave packets. The accompanying programs reproduce the reported calculations.