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Nonlinear Wave Limits of Reversible Finite-State Cellular Automata

Primarily human-written textHuman understanding: all partsmath-ph — Mathematical Physicsmath.AP — Analysis of PDEsmath.NA — Numerical Analysiscs.FL — Formal Languages and Automata Theory

Contributed by Zhenyu Sun ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

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 CT(h2+τ2+δ/τ2)C_T(h^2+\tau^2+\delta/\tau^2) on a fixed physical time interval, where hh, τ\tau, and δ\delta are the spatial, temporal, and field spacings. An explicit modulus bound prevents modular wrapping throughout that interval. The choices τ=νh\tau=\nu h, 0<ν<10<\nu<1, and δ=O(h4)\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.

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Version history

  1. v1Submitted by Zhenyu SunInitial depositCurrentOct 25, 2026