J Satsuma

Singularity confinement test for ultradiscrete equations with parity variables

We present a new ultradiscretization method which does not require that the solutions of the discrete equation have a fixed sign. We construct an ultradiscrete analogue of the singularity confinement test using this method and thereby propose an integrability test for ultradiscrete equations.

Ultradiscretization without positivity

We present a new ultradiscretization approach which can be applied to discrete systems, the solutions of which are not positive definite. This was made possible, thanks to an ansatz involving the hyperbolic-sine function. We apply this new procedure to simple mappings. For the linear and homographic mappings, we obtain ultradiscrete forms and explicitly construct their solutions. Two discrete Painleve II equations are also analysed and ultradiscretized. We show how to construct the ultradiscrete analogues of their rational and Airy-type solutions.

Linearizable QRT mappings

We derive the members of the QRT family of mappings which are integrable through linearization. We show how these mappings can be integrated and link them to known families of linearizable mappings. We study their growth properties and comment on the relation of the latter to the singularity structure of the mappings. The case of integrable linearizable mappings involving free functions of the independent variable is also discussed.

Exact solutions for discrete and ultradiscrete modified KdV equations and their relation to box-ball systems

A new class of solutions is proposed for discrete and ultradiscrete modified KdV equations. These are directly related to solutions of the box and ball system with a carrier. Moreover, an extended box and ball system and its exact solutions are discussed.

Solving the ultradiscrete KdV equation

We show that a generalized cellular automaton, exhibiting solitonic interactions, can be explicitly solved by means of techniques first introduced in the context of the scattering problem for the KdV equation. We apply this method to calculate the phase-shifts caused by interactions between the solitonic and non-solitonic parts into which arbitrary initial states separate in time.

Ultradiscrete Painlevé II equation and a special function solution

Ultradiscretization with parity variables, which keeps the information of original variables sign, is applied to the q-Painleve II equation of type A6 (q-PII). A special solution of the resulting ultradiscrete system, which corresponds to the special function solution of q-PII, is constructed. Ultradiscrete analogues of the q-Airy equation and its special solutions are also discussed in the process.

How to discretize differential systems in a systematic way

We present a systematic approach to the construction of discrete analogues for differential systems. Our method is tailored to first-order differential equations and relies on a formal linearization, followed by a Pade-like rational approximation of an exponential evolution operator. We apply our method to a host of systems for which there exist discretization results obtained by what we call the intuitive method and compare the discretizations obtained. A discussion of our method as compared to one of the Mickens is also presented. Finally we apply our method to a system of coupled Riccati equations with emphasis on the preservation of the integrable character of the differential system.

A Class of Special Solutions for the Ultradiscrete Painlevé II Equation

A class of special solutions are constructed in an intuitive way for the ultradis- crete analog of q-Painleve II (q-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter.

On the explicit integration of a class of linearizable mappings

We examine a class of second-order mappings which can be integrated by reduction to a linear equation. These mappings have been identified in our previous works where we have precisely shown how to obtain their linearization. The mappings belonging to this class are referred to as linearizable mappings of the third kind. We construct their explicit solution and obtain, for all of them, an invariant of QRT aspect but which is non-autonomous. We show that some of these third-kind mappings are related to another class of linearizable mappings, known as mappings of Gambier type, from which they are obtained through a (discrete) derivative with respect to a parameter.

Direct ultradiscretization of Ai and Bi functions and special solutions for the Painlevé II equation

Ultradiscrete Ai and Bi functions are directly derived through the ultradiscrete limit from q-difference analogues of the Ai and Bi functions, respectively. An infinite number of identities among the number of restricted partitions are obtained as by-products. A direct relationship between a class of special solutions for the ultradiscrete Painlev? II equation and those of the q-Painlev? II equation which have a determinantal structure is also established.