Ultra Discrete Limit in Soliton Theory and Non-Archimedean Valuation

In this article, I have studied the ultra discrete limit, which is currently studied in soliton theory, from point of view of valuation theory. A quantity obtained after taking the ultra discrete limit should be regarded as non-archimedean valuation, which is related to the p-adic valuation in number theory. The ultra discrete difference-difference equations, whose domain and range are given by integers, are currently studied in soliton theory [TS, TTMS]. In this article, I will show that a quantity obtained by the ultra discrete limit in the soliton theory should be regarded as the non-archimedean valuation whose typical example is the p-adic valuation in the number theory [C,VVZ]. Thus we will start from the preliminary of p-adic number theory. Next we will review the recent development of the ultra discrete soliton theory [TTMS]. By construct a p-adic soliton equation, we will show resemblance between the p-adic valuation of the p-adic soliton equation and ultra discrete differencedifference soliton equation. The ultra discrete limit has the same structure as the p-adic valuation. Finally we will comment upon physical and mathematical meanings of the ultra-discrete limit. Let us consider p-adic field Qp for a prime number p [C, I, VVZ]. For a rational number u ∈ Q which are given by u = v w p (v and w are coprime to the prime number p and m is an integer), we will define the p-adic valuation of u as a map from Q to a set of integers Z, ordp(u) := m, for u 6= 0, and ordp(u) := ∞, for u = 0. This valuation has following properties (I); for u, v ∈ Q, (1) ordp(uv) = ordp(u) + ordp(v). (2) ordp(u+ v) ≥ min(ordp(u), ordp(v)). If ordp(u) 6= ordp(v), ordp(u+ v) = min(ordp(u), ordp(v)). This property (I-1) means that ordp is a homomorphism from the multiplicative group Q × of Q to the additive group Z. The p-adic metric is given by |v|p = p p. It is obvious that it is a metric because it has the properties (II); for u, v ∈ Q, (1) if |v|p = 0, v=0. (2) |v|p ≥ 0. (3) |vu|p = |v|p|u|p. (4) |u+ v|p ≤ max(|u|p, |v|p) ≤ |u|p + |v|p. 1 The p-adic field Qp is given as a completion of Q with respect to this metric so that properties (I) and (II) survive for Qp. Further, we note that the multiplicative group in Qp is given as Q × p ≈ Z⊕ F × p ⊕ Zp, where F × p is a multiplicative group in a finite field of order p and Zp is integer part of Qp. As the properties of p-adic metric, the convergence condition of series ∑ m xm is identified with the vanishing condition of sequence |xm|p → 0 for m → ∞ due to the relation,