Combinatorial expressions of the solutions to initial value problems of the discrete and ultradiscrete Toda molecules
type: Item
instance of: scholarly article
title: Combinatorial expressions of the solutions to initial value problems of the discrete and ultradiscrete Toda molecules
zbMATH Open document ID: 1287.37046
DOI: 10.1088/1751-8113/46/35/355203
author: Shuhei Kamioka
author: Tomoaki Takagaki
published in: Journal of Physics A: Mathematical and Theoretical
publication date: 2013-10-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1303.4173
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37K15
Mathematics Subject Classification ID: 37E15
Mathematics Subject Classification ID: 34A33
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 6213901
zbMATH Keywords: discrete Toda molecule
zbMATH Keywords: ultradiscrete Toda molecule
zbMATH Keywords: initial value problems
zbMATH Keywords: continued fractions
zbMATH Keywords: Hankel determinants
MaRDI profile type: Publication
OpenAlex ID: W1974356912
arXiv ID: 1303.4173
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review text: The discrete Toda molecule is a discrete analogue of the Toda molecule [\textit{R. Hirota}, in: Algebraic analysis, Vol. 1, 203--216 (1989; Zbl 0705.35121)] derived using the bilinear formalism; it is a discrete integrable system which possesses a Hankel determinant solution analogous to the Toda molecule. The authors examine the initial value problems of the discrete and ultradiscrete Toda molecules with given initial value. Although it is obvious that the exact value of each dependent variable can be calculated from the initial value in finitely many arithmetic and minimizing operations, it is nontrivial how to formulate the solutions since the equations are nonlinear. Utilizing combinatorial objects, the authors derive an exact expression of the solutions to the initial value problems purely in terms of the initial value.NEWLINENEWLINEFor the discrete Toda molecule, a subtraction-free expression of the solution is derived in terms of non-intersecting paths, for which Flajolet's interpretation of continued fractions and Gessel-Viennot's lemma on determinants are applied. By ultradiscretizing the subtraction-free expression, the solution to the ultradiscrete Toda molecule is obtained. Further combinatorial observations lead to a simpler expression of the solution in terms of shortest paths on a specific graph. Finally, the authors analyze the behavior of the solution obtained in terms of shortest paths in comparison with the box-ball system.
reviewed by: Johanna Michor
description: scientific article; zbMATH DE number 6213901