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Journal of Integrable Systems
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  This is an Open Access Journal Open Access journal
ISSN (Online) 2058-5985
Published by Oxford University Press Homepage  [396 journals]
  • Linearization of the box–ball system: an elementary approach

    • Authors: Kakei S; Nimmo J, Tsujimoto S, et al.
      Abstract: Kuniba, Okado, Takagi and Yamada have found that the time-evolution of the Takahashi–Satsuma box–ball system can be linearized by considering rigged configurations associated with states of the box–ball system. We introduce a simple way to understand the rigged configuration of $\mathfrak{sl}_2$-type, and give an elementary proof of the linearization property. Our approach can be applied to a box–ball system with finite carrier, which is related to a discrete modified KdV equation, and also to the combinatorial $R$-matrix of $A_1^{(1)}$-type. We also discuss combinatorial statistics and related fermionic formulas associated with the states of the box–ball systems. A fermionic-type formula, we obtain for the finite carrier case, seems to be new.Communicated by: Rei Inoue Yamazaki
      PubDate: Thu, 15 Feb 2018 00:00:00 GMT
      DOI: 10.1093/integr/xyy002
      Issue No: Vol. 3, No. 1 (2018)
       
  • Asymptotic analysis of non-autonomous discrete hungry integrable systems

    • Authors: Shinjo M; Nakamura Y, Iwasaki M, et al.
      Abstract: Discrete hungry integrable systems have an interesting application in the computation of eigenvalues of totally non-negative (TN) matrices. In the case where the system’s variables are restricted to be all positive, they or their combinations converge to TN matrix eigenvalues as the discrete-time variable goes to infinity. To accelerate the convergence, arbitrary parameters are introduced that act as implicit shifts of origin without loss of the positivity of the variables in matrix similarity transformations. Discrete hungry integrable systems with the arbitrary parameters depending on independent variables are called non-autonomous. In this study, we show the convergence of the solution to non-autonomous discrete hungry integrable systems to matrix eigenvalues without assuming positivity of the variables. Thus, we present convergence theorems, which can contribute to computing eigenvalues of general matrices, not just TN matrices. To this end, we show determinant solutions to two non-autonomous discrete hungry integrable systems. We also clarify a Bäcklund transformation, which relates these two non-autonomous discrete hungry integrable systems.Communicated by: Prof. Kenji Kajiwara
      PubDate: Sat, 27 Jan 2018 00:00:00 GMT
      DOI: 10.1093/integr/xyy001
      Issue No: Vol. 3, No. 1 (2018)
       
 
 
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