On Soliton Collisions between Localizations in Complex Elementary Cellular Automata: Rules 54 and 110 and Beyond

In this paper we present a single-soliton two-component cellular automata (CA) model of waves as mobile self-localizations, also known as: particles, waves, or gliders; and its version with memory.The model is based on coding sets of strings where each chain represents a unique mobile self-localization.We will discuss briefly the original soliton models in CA proposed with filter automata, followed by solutions in elementary CA (ECA) domain with the famous universal ECA Rule 110, and reporting a number of new solitonic collisions in ECA Rule 54.A mobile self-localization in this study is equivalent a single soliton because the collisions of these mobile self-localizations studied in this paper satisfies the property of solitonic collisions.We also present a specific ECA with memory (ECAM), the ECAM Rule φR9maj:4, that displays single-soliton solutions from any initial codification (including random initial conditions) for a kind of mobile self-localization because such automaton is able to adjust any initial condition to soliton structures.