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Integrable Properties of a Variant of the Discrete Hungry Toda Equations and Their Relationship to Eigenpairs of Band Matrices

Part of: Convergence and divergence of infinite limiting processes Qualitative behavior Basic linear algebra Difference equations Infinite-dimensional Hamiltonian systems Difference and functional equations Numerical linear algebra

Published online by Cambridge University Press:  31 January 2018

Yusuke Nishiyama ,
Masato Shinjo ,
Koichi Kondo  and
Masashi Iwasaki
Yusuke Nishiyama*
Affiliation:
Graduate School of Science and Engineering, Doshisha University, 1-3, Tatara-miyakodani, Kyotanabe, Kyoto, 610-0394, Japan
Masato Shinjo*
Affiliation:
Graduate School of Informatics, Kyoto University, Yoshida-Hommachi, Sakyo-ku, Kyoto, 606-8501, Japan
Koichi Kondo*
Affiliation:
Graduate School of Science and Engineering, Doshisha University, 1-3, Tatara-miyakodani, Kyotanabe, Kyoto, 610-0394, Japan
Masashi Iwasaki*
Affiliation:
Faculty of Life and Environmental Science, Kyoto Prefectural University, 1-5, Nakaragi-cho, Shimogamo, Sakyo-ku, Kyoto, 606-8522, Japan
*
*Corresponding author. Email addresses:dup0344@mail4.doshisha.ac.jp (Y. Nishiyama), mshinjo@amp.i.kyoto-u.ac.jp (M. Shinjo), kokondo@mail.doshisha.ac.jp (K. Kondo), imasa@kpu.ac.jp (M. Iwasaki)
*Corresponding author. Email addresses:dup0344@mail4.doshisha.ac.jp (Y. Nishiyama), mshinjo@amp.i.kyoto-u.ac.jp (M. Shinjo), kokondo@mail.doshisha.ac.jp (K. Kondo), imasa@kpu.ac.jp (M. Iwasaki)
*Corresponding author. Email addresses:dup0344@mail4.doshisha.ac.jp (Y. Nishiyama), mshinjo@amp.i.kyoto-u.ac.jp (M. Shinjo), kokondo@mail.doshisha.ac.jp (K. Kondo), imasa@kpu.ac.jp (M. Iwasaki)
*Corresponding author. Email addresses:dup0344@mail4.doshisha.ac.jp (Y. Nishiyama), mshinjo@amp.i.kyoto-u.ac.jp (M. Shinjo), kokondo@mail.doshisha.ac.jp (K. Kondo), imasa@kpu.ac.jp (M. Iwasaki)
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Abstract

The Toda equation and its variants are studied in the filed of integrable systems. One particularly generalized time discretisation of the Toda equation is known as the discrete hungry Toda (dhToda) equation, which has two main variants referred to as the dhTodaI equation and dhTodaII equation. The dhToda equations have both been shown to be applicable to the computation of eigenvalues of totally nonnegative (TN) matrices, which are matrices without negative minors. The dhTodaI equation has been investigated with respect to the properties of integrable systems, but the dhTodaII equation has not. Explicit solutions using determinants and matrix representations called Lax pairs are often considered as symbolic properties of discrete integrable systems. In this paper, we clarify the determinant solution and Lax pair of the dhTodaII equation by focusing on an infinite sequence. We show that the resulting determinant solution firmly covers the general solution to the dhTodaII equation, and provide an asymptotic analysis of the general solution as discrete-time variable goes to infinity.

Keywords

Discrete hungry Toda equationdeterminant solutionLax pairasymptotic behavior

MSC classification

Secondary: 15A15: Determinants, permanents, other special matrix functions 15A18: Eigenvalues, singular values, and eigenvectors 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) 37K40: Soliton theory, asymptotic behavior of solutions 39A12: Discrete version of topics in analysis 40A05: Convergence and divergence of series and sequences 45M05: Asymptotics 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 4 , November 2017 , pp. 785 - 798
Copyright
Copyright © Global-Science Press 2017 

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