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B. Grammaticos - One of the best experts on this subject based on the ideXlab platform.

  • The road to the Discrete Analogue of the Painlevé property: Nevanlinna meets singularity confinement
    Computers & Mathematics with Applications, 2003
    Co-Authors: Alfred Ramani, B. Grammaticos, T Tamizhmani, K. M. Tamizhmani
    Abstract:

    The question of integrability of Discrete systems is analysed in the light of the recent findings of Ablowitz et al., who have conjectured that a fast growth of the solutions of a differ- ence equation is an indication of nonintegrability. The study of the behaviour of the solutions of a mapping is based on the theory of Nevanlinna. In this paper, we show how this approach can be implemented in the csse of second-order mappings which include the Discrete Painlevk equations. Since the Nevanlinna approach does offer only a necessary condition which is not restrictive enough, we complement it by the singularity confinement requirement, first in an autonomous setting and then for deautonomisation. We believe that this three-tiered approach is the closest one can get to a Discrete Analogue of the Painlevb property. @ 2003 Elsevier Science Ltd. All rights reserved.

  • what is the Discrete Analogue of the painleve property
    Anziam Journal, 2002
    Co-Authors: A. Ramani, B. Grammaticos
    Abstract:

    We analyse the various integrability criteria which have been proposed for Discrete systems, focusing on the singularity confinement method. We present the exact procedure used for the derivation of Discrete Painleve equations based on the deautonomisation of integrable autonomous mappings. This procedure is then examined in the light of more recent criteria based on the notion of the complexity of the mapping. We show that the low-growth requirements lead, in the case of the Discrete Painleve equations, to exactly the same results as singularity confinement. The analysis of linearisable mappings shows that they have special growth properties which can be used in order to identify them. A working strategy for the study of Discrete integrability based on singularity confinement and low-growth considerations is also proposed.

  • What is the Discrete Analogue of the Painlevé property
    The ANZIAM Journal, 2002
    Co-Authors: A. Ramani, B. Grammaticos
    Abstract:

    AbstractWe analyse the various integrability criteria which have been proposed for Discrete systems, focusing on the singularity confinement method. We present the exact procedure used for the derivation of Discrete Painlevé equations based on the deautonomisation of integrable autonomous mappings. This procedure is then examined in the light of more recent criteria based on the notion of the complexity of the mapping. We show that the low-growth requirements lead, in the case of the Discrete Painlevé equations, to exactly the same results as singularity confinement. The analysis of linearisable mappings shows that they have special growth properties which can be used in order to identify them. A working strategy for the study of Discrete integrability based on singularity confinement and low-growth considerations is also proposed.

  • on a novel q Discrete Analogue of the painleve vi equation
    Physics Letters A, 1999
    Co-Authors: B. Grammaticos, A. Ramani
    Abstract:

    Abstract We present the Discrete, q -, form of the Painleve VI equation written as a three-point mapping and analyse the structure of its singularities. This Discrete equation goes over to P VI at the continuous limit and degenerates towards the Discrete q -P V through coalescence. It possesses special solutions in terms of the q -hypergeometric function. It can bilinearised and, under the appropriate assumptions, ultradiscretised. A new Discrete form for P V is also obtained which is of difference type, in contrast with the `standard' form of the Discrete P V . Finally, we present the `asymmetric' form of q -P VI as a system of two first-order mappings involving seven arbitrary parameters.

  • INTEGRABILITY OF Discrete-TIME SYSTEMS
    NATO ASI Series, 1992
    Co-Authors: B. Grammaticos, George Karra'a, V.g. Papageorgiou, Alfred Ramani
    Abstract:

    A new integrability criterion for Discrete-time systems, based on the notion of the confinement of the singularities that may appear in rational mappings is presented. Discrete Analogues of the Painleve equations are derived as second-order, nonautonomous mappings, with the help of this integrability detector. Moreover a parallel between continuous and Discrete systems is established by showing that to each kind of “continuous” integrability there exists a “Discrete” Analogue.

B.g. Pachpatte - One of the best experts on this subject based on the ideXlab platform.

Yasuhiro Ohta - One of the best experts on this subject based on the ideXlab platform.

  • A two-component generalization of the reduced Ostrovsky equation and its integrable semi-Discrete Analogue
    Journal of Physics A: Mathematical and Theoretical, 2017
    Co-Authors: Bao-feng Feng, Ken-ichi Maruno, Yasuhiro Ohta
    Abstract:

    In the present paper, we propose a two-component generalization of the reduced Ostrovsky equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis-Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and $N$-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Backlund transformation of the above extended BKP hierarchy, an integrable semi-Discrete Analogue of two-component reduced Ostrovsky equation is constructed by defining an appropriate Discrete hodograph transform and dependent variable transformations. Especially, the backward difference form of above semi-Discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their $N$-soliton solutions in terms of pffafians are also provided.

  • a two component generalization of the reduced ostrovsky equation and its integrable semi Discrete Analogue
    Journal of Physics A, 2017
    Co-Authors: Bao-feng Feng, Ken-ichi Maruno, Yasuhiro Ohta
    Abstract:

    In the present paper, we propose a two-component generalization of the reduced Ostrovsky (Vakhnenko) equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis–Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that the two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and N-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Backlund transformation of the above extended BKP hierarchy, an integrable semi-Discrete Analogue of the two-component reduced Ostrovsky equation is constructed by defining an appropriate Discrete hodograph transform and dependent variable transformations. In particular, the backward difference form of above semi-Discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their N-soliton solutions in terms of pffafians are also provided.

Ken-ichi Maruno - One of the best experts on this subject based on the ideXlab platform.

  • A two-component generalization of the reduced Ostrovsky equation and its integrable semi-Discrete Analogue
    Journal of Physics A: Mathematical and Theoretical, 2017
    Co-Authors: Bao-feng Feng, Ken-ichi Maruno, Yasuhiro Ohta
    Abstract:

    In the present paper, we propose a two-component generalization of the reduced Ostrovsky equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis-Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and $N$-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Backlund transformation of the above extended BKP hierarchy, an integrable semi-Discrete Analogue of two-component reduced Ostrovsky equation is constructed by defining an appropriate Discrete hodograph transform and dependent variable transformations. Especially, the backward difference form of above semi-Discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their $N$-soliton solutions in terms of pffafians are also provided.

  • a two component generalization of the reduced ostrovsky equation and its integrable semi Discrete Analogue
    Journal of Physics A, 2017
    Co-Authors: Bao-feng Feng, Ken-ichi Maruno, Yasuhiro Ohta
    Abstract:

    In the present paper, we propose a two-component generalization of the reduced Ostrovsky (Vakhnenko) equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis–Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that the two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and N-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Backlund transformation of the above extended BKP hierarchy, an integrable semi-Discrete Analogue of the two-component reduced Ostrovsky equation is constructed by defining an appropriate Discrete hodograph transform and dependent variable transformations. In particular, the backward difference form of above semi-Discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their N-soliton solutions in terms of pffafians are also provided.

  • Resonance and web structure in Discrete soliton systems: the two-dimensional Toda lattice and its fully Discrete and ultra-Discrete Analogues
    Journal of Physics A: Mathematical and General, 2004
    Co-Authors: Ken-ichi Maruno, Gino Biondini
    Abstract:

    We present a class of solutions of the two-dimensional Toda lattice equation, its fully Discrete Analogue and its ultra-Discrete limit. These solutions demonstrate the existence of soliton resonance and web-like structure in Discrete integrable systems such as differential-difference equations, difference equations and cellular automata (ultra-Discrete equations).

Iain Findlay - One of the best experts on this subject based on the ideXlab platform.

  • Space & time discontinuities in Liouville theory and its Discrete Analogue
    arXiv: Mathematical Physics, 2016
    Co-Authors: Anastasia Doikou, Iain Findlay
    Abstract:

    We consider the deformed harmonic oscillator as a Discrete version of the Liouville theory and study this model in the presence of local integrable defects. From this, the time evolution of the defect degrees of freedom are determined, found in the form of the local equations of motion. We also revisit the continuous Liouville theory, deriving its local integrals of motion and comparing these with previous results from the sine-Gordon point of view.Finally, the generic Backlund type relations are presented, corresponding to the implementation of time-like and space-like impurities in the continuum model.

  • space time discontinuities in liouville theory and its Discrete Analogue
    arXiv: Mathematical Physics, 2016
    Co-Authors: Anastasia Doikou, Iain Findlay
    Abstract:

    We consider the deformed harmonic oscillator as a Discrete version of the Liouville theory and study this model in the presence of local integrable defects. From this, the time evolution of the defect degrees of freedom are determined, found in the form of the local equations of motion. We also revisit the continuous Liouville theory, deriving its local integrals of motion and comparing these with previous results from the sine-Gordon point of view.Finally, the generic Backlund type relations are presented, corresponding to the implementation of time-like and space-like impurities in the continuum model.