薩摩 順吉

We present the singularity analysis of the ultradiscrete analogue of linearisable mappings of Quispel-Roberts-Thompson (QRT) type. The ultradiscretisation method used here is one which keeps track of signs and thus can be applied without the positivity restrictions of the classical ultradiscretisation approach. We show that in all cases the mappings possess confined singularities. The same is true for two non-autonomous equations, which are equally linearisable in the discrete case. We construct explicitly the solutions of the ultradiscrete mappings analysed here. © 2013 American Institute of Physics.

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.

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 Painleve II equation and those of the q-Painleve II equation which have a determinantal structure is also established.

Ultradiscrete singularity confinement test, which is an integrability detector for ultradiscrete equations with parity variables, is applied to various ultradiscrete equations. The ultradiscrete equations exhibit singularity structures analogous to those of the discrete counterparts. Exact solutions to linearisable ultradiscrete equations are also constructed to explain the singularity structures. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3682229]

Ultradiscretization with parity variables, which keeps the information of original variables' sign, is applied to the q-Painleve II equation of type A(6) (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.

We apply a method that we have already proposed for the systematic discretisation of differential systems to the construction of discrete analogues of the Painleve equations. The important point is that in the case of the Painleve equations one needs to preserve the integrability. We show that this is indeed possible using our systematic procedure applied to the equations obtained from the Hamiltonian description of the Painleve equations. (C) 2010 Elsevier B.V. All rights reserved.

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

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.

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.

We examine two integrable discrete lattice equations obtained by Levi and Yamilov. We show that the first one is a form of the lattice KdV equation already obtained by Hirota and Tsujimoto, while the second one is a discrete form of mKdV. We present the Miura transformations between the various equations involved, including the more familiar potential form of the lattice mKdV.

We present two models for an epidemic where the individuals are infective over it fixed period of time and which never becomes endemic i.e., no infective individuals remain after the epidemic has run its course. The first model is based oil it delay-difference scheme. We show that, as a function of the delay (which corresponds to the Period of infectiveness) the percentage of non-infected population varies over a wide range. We present also a variant of our model where the recovery rate follows a Poisson law and obtain it discrete version of the SIR model. We estimate the percentage of non-infected population in the two models, show that they lead to almost the same values and present an explanation of this fact. The second model is based oil the assumption that the infection is spread by carriers. Under the hypothesis that the carriers are relatively long-lived, and that the number of the infected ones is a relatively small fraction of the total population of potential carriers, we show that the model reduces to the same version of the discrete SIR obtained by our first model. (C) 2007 Elsevier Ltd. All rights reserved.

Exact solutions of the ultradiscrete Sine-Gordon equation which have oscillating structure are constructed. They are considered to be a counterpart of the breather solution of the Sine-Gordon equation. They are given by setting specific parameters in the discrete soliton solutions and ultradiscretizing the resulting solutions.

We show that a discretization of a continuous system may entail 'hidden' delays and thus introduce instabilities. In this case, while the continuous system has an attractive fixed point, the instabilities present in the equivalent discrete one may lead to the appearance of a limit cycle. We explain that it is possible, thanks to the proper staggering of the discrete variables, to eliminate the hidden delay. However, in general, other instabilities may appear in the discrete system which can even lead to chaotic behaviour.

The Backlund transformation for the discrete Korteweg-de Vries equation is introduced in the bilinear form. The superposition formula is also derived from the transformation. An ultradiscrete analogue of the transformation is presented by means of the ultradiscretization technique. This analogue gives the Backlund transformation for the box and ball system. The ultradiscrete soliton solutions for the system are also discussed with explicit examples.

We examine the transition between discrete and ultradiscrete (cellular-automaton-like) systems, the dynamics of which exhibit limit cycles. Motivated by results obtained previously for three-dimensional systems, we consider here a more manageable two-dimensional model. We show that one can follow the changes in dynamics of the system when a parameter that tunes the discrete-ultradiscrete transition is varied. In particular we explain the phenomenon of the splitting of a discrete limit cycle to a profusion of periodic orbits at the ultradiscrete limit. (C) 2007 Elsevier B.V. All rights reserved.

We construct special solutions of the Hirota-Miwa equation for which the tau-function is a polynomial in the independent variables. Three different methods are presented: direct construction ( obtained also as a limit of the soliton solutions), and the derivation of the solutions in two different determinant forms, namely Grammian and Casorati. Introducing the appropriate ansatz, we write the Hirota-Miwa equation in a nonlinear form for a single variable. In terms of the latter, the solutions obtained are rational and are reminiscent of the lump solutions for the continuous analogue of the Hirota-Miwa equation, namely the KP equation.

An ultradiscrete analogue of the Miura transformation is constructed through the bilinear form of the discrete KdV and modified KdV equations. This transformation maps solutions of the 'box and ball system with a carrier' to those of the 'box and ball system'. Explicit examples of solutions are also discussed. (c) 2006 Elsevier B.V. All rights reserved.

The discrete modified Korteweg-de Vries equation admits exact solutions with nondefinite sign, which describe interaction among solitons with positive and negative amplitude. In this Letter a transformation of hyperbolic sine type is proposed in order to ultradiscretize this equation and solutions. (c) 2006 Elsevier B.V. All rights reserved.

We present a three-component model which could represent the reaction of the organism to pathogen invasion. A continuous-time (differential) model is constructed first. Its discrete analogue is then derived and is used for numerical simulations which show a great variety of behaviours. We also construct a cellular automaton for the model using the ultra-discretisation procedure. 2005 Elsevier B.V. All rights reserved.

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.

We present the discrete systems which result from the discrete Painlev equations q-P-VI and d-P-V associated to the affine Weyl group E-7((1)). Two different procedures ("limits" and "degeneracies") Lire used, giving rise to a host of new discrete Painleve equations but also to some equations which are integrable through linearisation. (C) 2004 Elsevier B.V. All rights reserved.

We present a novel method for the reduction of integrable two-dimensional discrete systems to one-dimensional mappings. The procedure allows for the derivation of nonautonomous systems, which are typically discrete (difference or q) Painleve equations, or of autonomous ones. In the latter case we produce the discrete analogue of an integrable subcase of the Henon-Heiles system.

An ultradiscrete system corresponding to the sine-Gordon equation is proposed. A new dependent variable for the discrete sine-Gordon equation is introduced in order to apply the procedure of ultradiscretization. The ultradiscrete system possesses exact solutions which are directly related to soliton solutions of the discrete equation. (C) 2004 Elsevier B.V. All rights reserved.

We investigate possible extensions of the susceptible-infective-removed (SIR) epidemic model. We show that there exists a large class of functions representing interaction between the susceptible and infective populations for which the model has a realistic behaviour and preserves the essential features of the classical SIR model. We also present a new discretisation of the SIR model which has the advantage of possessing a conserved quantity, thus making possible the estimation of the non-infected population at the end of the epidemic. A cellular automaton SIR is also constructed on the basis of the discrete-time system. (C) 2003 Elsevier B.V. All rights reserved.

More than 20 years ago, it was discovered that the solutions of the Kadomtsev-Petviashvili (KP) hierarchy constitute an infinite-dimensional Grassmann manifold and that the Plucker relations for this Grassmannian take the form of Hirota bilinear identities. As is explained in this contribution, the resulting unified approach to integrability, commonly known as Sato theory, offers a deep algebraic and geometric understanding of integrable systems with infinitely many degrees of freedom. Starting with an elementary introduction to Sato theory, followed by an expose of its interpretation in terms of infinite-dimensional Clifford algebras and their representations, the scope of the theory is gradually extended to include multi-component systems, integrable lattice equations and fully discrete systems. Special emphasis is placed on the symmetries of the integrable equations described by the theory and especially on the Darboux transformations and elementary Backlund transformations for these equations. Finally, reductions to lower dimensional systems and eventually to integrable ordinary differential equations are discussed. As an example, the origins of the fourth Painleve, equation and of its Backlund transformations in the KP hierarchy are explained in detail.

In this Letter, we present a new hierarchy which includes the intermediate long wave (ILW) equation at the lowest order. This hierarchy is thought to be a novel reduction of the 1st modified KP type hierarchy. The framework of our investigation is Sato theory. (C) 2003 Elsevier Science B.V. All rights reserved.

We apply the algebraic-geometric techniques developed for the study of mappings which have the singularity confinement property to mappings which are integrable through linearization. The main difference with respect to the previous studies is that the linearizable mappings have generically unconfined singularities. Despite this fact we are able to provide a complete description of the dynamics of these mappings and derive rigorously their growth properties.

The bilinear method introduced by Hirota to obtain exact solutions for nonlinear evolution equations is discussed. Firstly, several examples including the Korteweg-deVries, nonlinear Schrodinger and Toda equations are given to show how solutions are derived. Then after considering multi-dimensional systems such as the Kadomtsev-Petviashvili, two dimensional Toda and Hirota-Miwa equations, the algebraic structure of such nonlinear evolution systems is explained. Finally, extensions of the method including q-analogue, ultra-discrete systems and trilinear forms are also presented.

The coupled KP (cKP) equation possesses N-soliton solutions with many more degrees of freedom than the solitons for the usual KP equation have. In comparison, cKP solutions can therefore be expected to model far more complex interactions than their KP counterparts. In this paper we present some typical solutions for the cKP system (and for some of its reductions: a complex coupled KdV equation and a coupled Boussinesq equation) and we analyse their interaction and asymptotic properties.

Conserved quantities of box and ball system(BBS) are presented from the hungry Toda molecule equation, an inverse ultra-discrete limit of the BBS.

We construct a difference equation which preserves any time evolution pattern of the rule 90 elementary cellular automaton. We also demonstrate that such difference equations can be obtained for any elementary cellular automata.

It is explained how a description of integrable systems using fermion operators leads in a natural way to the notions of Darboux and binary Darboux transformations. This approach is illustrated for a nonautonomous version of the discrete KP (dKP) equation. As an application of this technique, solutions - some of which are expressed in terms of special functions - are discussed. The ultra-discrete limit is then performed to obtain a nonautonomous ultra-discrete system with particularly interesting soliton behaviour. (C) 1999 Elsevier Science Ltd. All rights reserved.

A soliton cellular automaton, which represents movement of a finite number of balls in an array of boxes, is investigated. Its dynamics is described by an ultra-discrete equation obtained from an extended Toda molecule equation. The rules for soliton interactions and factorization property of the scattering matrices (Yang-Baxter relation) are proved by means of inverse ultra-discretization. The conserved quantities are also presented and used for another proof of the solitonical nature.

The (charged) free fermion formulation of the N-component KP hierarchies is used for the description of quadrilateral lattices in terms of multicomponent KP eigenfunction potentials. Tan functions for these hierarchies are explicitly calculated, giving rise to Grammian and Casorati determinant solutions to the (discretized) Darboux or Laplace equations describing the lattices. An example of localized tau functions and eigenfunction potentials is given. (C) 1999 Published by Elsevier Science B.V.

An ultra-discrete version of the Toda molecule equation is presented. The solution and conserved quantities for the equation are also discussed. (C) 1998 Elsevier Science B.V.

Starting from the free fermion description of the one-component KP hierarchy, we establish a connection between this approach and the theory of Darboux and binary Darboux transformations. Certain difference identities-allowing for the treatment of both continuous as well as discrete evolution equations-turn out to be crucial: first to show that any solution of the associated (adjoint) linear problems can always be expressed as a superposition of KP (adjoint) wavefunctions and then to interpret Darboux (and binary Darboux) transformations as Backlund transformations in the fermion language.