Ultradiscrete Soliton Equations Research Articles

Discrete and Ultradiscrete Periodic Phase Soliton Equations

We propose new type of discrete and ultradiscrete soliton equations, which admit extended soliton solution called periodic phase soliton solution. The discrete equation is derived from the discrete DKP equation and the ultradiscrete one is obtained by applying the ultradiscrete limit. The soliton solutions have internal freedom and change their shape periodically during propagation. In particular, the ultradiscrete solution reduces into the solution to the ultradiscrete hungry Lotka-Volterra equation in a special case.

Solutions to the ultradiscrete KP hierarchy and its reductions

We propose a recursive representation of solutions to an ultradiscrete analogue of the discrete KP hierarchy, which is the master equation of discrete soliton equations. We also propose a class of solutions which can be used to start the recursion. Finally, as an application, we discuss the compatibility condition of ultradiscrete soliton equations.

Ultradiscrete soliton equations derived from ultradiscrete permanent formulae

We propose formulae of ultradiscrete permanent. Utilizing the formulae, ultradiscrete soliton equations and their multi-soliton solutions are obtained by a simple process. Changing variables and parameters of the formulae, we can derive the ultradiscrete Toda, KdV and hungry Lotka–Volterra equations. An extended version of the ultradiscrete hungry Lotka–Volterra equation is also proposed.

Solutions to the ultradiscrete Toda molecule equation expressed as minimum weight flows of planar graphs

We define a function by means of the minimum weight flow on a planar graph and prove that this function satisfies the ultradiscrete Toda molecule equation, its Bäcklund transformation and the two-dimensional Toda molecule equation. The method we employ in the proof can be considered as fundamental to the integrability of ultradiscrete soliton equations.

Bilinear equations and Bäcklund transformation for a generalized ultradiscrete soliton solution

Ultradiscrete soliton equations and Bäcklund transformation for a generalized soliton solution are presented. The equations include the ultradiscrete Korteweg–de Vries (KdV) equation or the ultradiscrete Toda equation in a special case. We also express the solution by the ultradiscrete permanent, which is defined by ultradiscretizing the signature-free determinant, that is, the permanent. Moreover, we discuss a relation between Bäcklund transformations for discrete and ultradiscrete KdV equations.

New Solutions to the Ultradiscrete Soliton Equations

New solutions to the ultradiscrete soliton equations, such as the Box–Ball system, the Toda equation, etc. are obtained. One of the new solutions which we call a “negative‐soliton” satisfies the ultradiscrete KdV equation (Box–Ball system) but there is not a corresponding traveling wave solution for the discrete KdV equation. The other one which we call a “static‐soliton” satisfies the ultradiscrete Toda equation but there is not a corresponding traveling wave solution for the discrete Toda equation. A collision of a soliton with a negative‐soliton generates many balls in a box over the capacity of the box in the Box–Ball system, while a collision of a soliton with the static‐soliton describes, in the ultradiscrete limit, transmission of a soliton through junctions of a “nonuniform Toda equation.” We have obtained exact solutions describing these phenomena.

Soliton-like dynamics of filtrons of cyclic automata

Iteration of certain automata mappings performed over a stringof symbols reveals some soliton-like coherent entities. These discrete solitons are called filtrons of automata. They were first discovered (and called particles) in binarystrings for parity rule filter cellular automata (CAs) by Park,Steiglitz and Thurston. Since then, further models based on filterCAs capable of supporting the filtrons have been proposed.There are soliton CAs, fast rules, ultradiscrete soliton equations, integrable CAs, filter transducers, crystal modelsand box-ball systems among them. In this paper we apply aunified automaton description for such discretesoliton-supporting systems. We show a wide class of cyclic automata, so-called filter automata (FAs), that support thefiltrons. FAs generalize some of the models mentioned above.For FAs we present a technique called ring computation, a kindof inverse method. This computation applies inverse FAs and anedge description of a filtron. Our technique is aimed atpredicting the existence of a filtron that could be supportedby a given FA. We present various filtrons and quasi-filtrons supported by automata, especially the family of automata that isequivalent to box-ball system with a carrier.