Volume 7, Issue 4
Integrable Properties of a Variant of the Discrete Hungry Toda Equations and Their Relationship to Eigenpairs of Band Matrices

East Asian J. Appl. Math., 7 (2017), pp. 785-798.

Published online: 2018-02

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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 dhToda$_Ⅰ$ equation and dhToda$_Ⅱ$ 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$_Ⅰ$ equation has been investigated with respect to the properties of integrable systems, but the dhToda$_Ⅱ$ 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 dhToda$_Ⅱ$ equation by focusing on an infinite sequence. We show that the resulting determinant solution firmly covers the general solution to the dhToda$_Ⅱ$ equation, and provide an asymptotic analysis of the general solution as discrete-time variable goes to infinity.

• Keywords

Discrete hungry Toda equation, determinant solution, Lax pair, asymptotic behavior.

15A15, 15A18, 37K10, 37K40, 39A12, 40A05, 45M05, 65F15

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@Article{EAJAM-7-785, author = {}, title = {Integrable Properties of a Variant of the Discrete Hungry Toda Equations and Their Relationship to Eigenpairs of Band Matrices }, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {4}, pages = {785--798}, 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 dhToda$_Ⅰ$ equation and dhToda$_Ⅱ$ 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$_Ⅰ$ equation has been investigated with respect to the properties of integrable systems, but the dhToda$_Ⅱ$ 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 dhToda$_Ⅱ$ equation by focusing on an infinite sequence. We show that the resulting determinant solution firmly covers the general solution to the dhToda$_Ⅱ$ equation, and provide an asymptotic analysis of the general solution as discrete-time variable goes to infinity.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.300716.300517a}, url = {http://global-sci.org/intro/article_detail/eajam/10720.html} }
TY - JOUR T1 - Integrable Properties of a Variant of the Discrete Hungry Toda Equations and Their Relationship to Eigenpairs of Band Matrices JO - East Asian Journal on Applied Mathematics VL - 4 SP - 785 EP - 798 PY - 2018 DA - 2018/02 SN - 7 DO - http://doi.org/10.4208/eajam.300716.300517a UR - https://global-sci.org/intro/article_detail/eajam/10720.html KW - Discrete hungry Toda equation, determinant solution, Lax pair, asymptotic behavior. AB -

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 dhToda$_Ⅰ$ equation and dhToda$_Ⅱ$ 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$_Ⅰ$ equation has been investigated with respect to the properties of integrable systems, but the dhToda$_Ⅱ$ 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 dhToda$_Ⅱ$ equation by focusing on an infinite sequence. We show that the resulting determinant solution firmly covers the general solution to the dhToda$_Ⅱ$ equation, and provide an asymptotic analysis of the general solution as discrete-time variable goes to infinity.

Yusuke Nishiyama, Masato Shinjo, Koichi Kondo & Masashi Iwasaki. (2020). Integrable Properties of a Variant of the Discrete Hungry Toda Equations and Their Relationship to Eigenpairs of Band Matrices . East Asian Journal on Applied Mathematics. 7 (4). 785-798. doi:10.4208/eajam.300716.300517a
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