TY - JOUR
T1 - Integrable Properties of a Variant of the Discrete Hungry Toda Equations and Their Relationship to Eigenpairs of Band Matrices 
AU - Yusuke Nishiyama, Masato Shinjo, Koichi Kondo & Masashi Iwasaki
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 - <p style="text-align: justify;">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<span style="text-align: justify;">$_Ⅰ$</span> equation has been
investigated with respect to the properties of integrable systems, but the dhToda<span style="text-align: justify;">$_Ⅱ$</span> 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<span style="text-align: justify;">$_Ⅱ$</span> equation
by focusing on an infinite sequence. We show that the resulting determinant solution
firmly covers the general solution to the dhToda<span style="text-align: justify;">$_Ⅱ$</span> equation, and provide an asymptotic
analysis of the general solution as discrete-time variable goes to infinity.</p>