A direct approach to the ultradiscrete KdV equation with negative and non-integer site values

Abstract

A generalisation of the ultra-discrete KdV equation is investigated using a direct approach. We show that evolution through one time step serves to reveal the entire solitonic content of the system.

Nonautonomous ultradiscrete hungry Toda lattice and a generalized box-ball system

“…In addition, there are studies on the relation between the BBS and the ultradiscrete Toda lattice for the case of a periodic boundary condition [6], and for the case in which the number of balls in each box can take any real value [4].…”

Darboux dressing and undressing for the ultradiscrete KdV equation

“…Notice also that for i → ±∞, φ t i ψ t+1 i ∼ φ t i ψ t i−1 → 0. Taking the limit as i → ±∞ in the second equation in (13) we see that…”

“…It is well known that the general solution of the box and ball system (udKdV with U t i ∈ {0, 1} and finite support) consists of interacting solitons made up of strings of k consecutive 1s, propagating at speed k. The most general solution in the case U t i ∈ R is more complicated but can still be described completely: it consists of interacting solitons parametrised by positive parameters ω, plus a "background" that moves with speed 1, which is the minimal speed in the system [12,13] (see also Remark 9.1).…”

“…In the last few years there has also been some interest in more general versions of the ultradiscrete KdV equation. These have the same update rule as the BBS but U t i takes arbitrary integer [9,10,11] or arbitrary real values [12,13]. In this non-binary case, the most general solution describes the interaction of pair-wise interacting solitons of arbitrary positive mass, similar to the BBS solitons, overlayed on a simply evolving background.…”

Linearization of the box–ball system: an elementary approach

“…The box-ball system (BBS for short) was introduced in 1990 as a cellular automaton model that exhibits solitonic behaviour [TS]. Since then it has been studied from various perspectives such as ultra-discretisation of discrete soliton equations [TTMS,TM,TH,KNW1,KNW2,GNN], representation theory of quantum groups [HHIKTT,IKT,Ta2], and combinatorics [TTS, A, F, FOY]. In particular, it is known to be related to the ultra-discrete limit of the discrete KdV equation [TTMS,TH,KNW1], a link which allowed for the obtention of its N -soliton solution in [TTMS,MIT2] and for the solution of its general initial value problem by means of IST techniques, similar to those for the continuous KdV equation, in [WNSRG, WRSG].…”