On solutions to evolution equations defined by lattice operators

Abstract

We discuss a specific form of evolution equations defined by lattice operators. We give exact solutions for a class of those equations and evaluate the complexity of the solutions. Moreover we discuss the relationship between them and binary cellular automata, and analyze their asymptotic behavior utilizing the explicit expression of the solution.

Lattice equations and their solutions with complexity of polynomial class

“…We discuss initial value problems to the equation described by (1) where u n i is a state variable of lattice element and f is defined by a combination of lattice operations. The solution u n i is obtained by applying (1) recursively to the initial data u 0 i . We introduce the following special notations to simplify this recursive applications.…”

“…If polynomial, 'polynomial class'. The polynomial class is referred to as P k if the growth is O(n k ) at most after the reference [1].…”

“…where f is a function of m arguments defined by lattice operators and u n i is a state value u dependent on an integer site number i and an integer time n. Let us call this type of equations 'lattice equation'. Assume that the initial time is n = 0 and we consider a solution u n i (n > 0) determined by (1) and initial data {u 0 i }. General solution can be obtained by applying (1) recursively.…”

“…General solution can be obtained by applying (1) recursively. Among equations in the form of (1), the equations to which complexity of general solution is of polynomial order are reported in the reference [1]. We discuss the same class of equations and call it 'polynomial class' in this paper.…”

Max-plus equation with two conserved quantities and one monotonically decreasing quantity

“…There are various levels of discreteness, for example, difference equation, coupled map lattice, cellular automaton (CA), and so on. Among them, max-plus equation is quite novel expression and studied intensively in recent years [1][2][3][4][5][6].…”