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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 codingsets of strings where each chain represents a unique mobile self-localization.We will discuss briefly the original soliton models in CA proposed with {\itfilter automata}, followed by solutions in elementary CA (ECA) domain with thefamous universal ECA {\it Rule 110}, and reporting a number of new solitoniccollisions in ECA {\it Rule 54}. A mobile self-localization in this study isequivalent a single soliton because the collisions of these mobileself-localizations studied in this paper satisfies the property of solitoniccollisions. We also present a specific ECA with memory (ECAM), the ECAM Rule$\phi_{R9maj:4}$, that displays single-soliton solutions from any initialcodification (including random initial conditions) for a kind of mobileself-localization because such automaton is able to adjust any initialcondition to soliton structures.

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