We show a direct connection between a cellular automaton and integrable nonlinear wave equations. We also present the N-soliton formula for the cellular automaton. Finally, we propose a general method for constructing such integrable cellular automata and their N-soliton solutions.

A new soliton cellular automaton is proposed. It is defined by an array of an infinite number of boxes, a finite number of balls and a carrier of balls. Moreover, it reduces to a discrete equation obtained from discrete modified Korteweg-de Vries equation through a limit. Algebraic expression of soliton solutions is also proposed.

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Conserved quantities and symmetries of the KP equation from the point of view of the Sato theory that provides a unifying approach to soliton equations is studied. Conserved quantities are derived from the generalized Lax equations. Some reductions of the KP hierarchy such as KdV, Boussinesq, a coupled KdV, and Sawada–Kotera equation are also considered. By expansion of the squared eigenfunctions of the Lax equations in terms of the τ function, symmetries of the KP equations are obtained. The relationship of this procedure to the two-dimensional recursion operator newly found by Fokas and Santini is discussed.