Towards a unified research method for ultradiscrete integrable systems

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540204
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Global analysis
• Research Institution: The University of Tokyo
• Principal Investigator: WILLOX Ralph 東京大学, 数理(科)学研究科(研究院), 教授 (20361610)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 可積分系 / 離散可積分系 / 超離散可積分系 / セルオートマトン

Outline of Final Research Achievements

Studying so-called "ultradiscrete integrable systems" - which are special discrete dynamical systems in which all the variables, including the dependent ones, only take discrete values - I discovered an entirely new method for solving such systems.
In particular, this enabled me to solve the so-called "ultradiscrete sine-Gordon equation" for which no explicit solutions are known, over the set of the integers. It also proved possible to attach a combinatorial interpretation to data that appear in the initial value problem for the ultradiscrete version of the famous KdV equation.
Furthermore, concerning integrable systems in the stage before they are turned into ultradiscrete systems, I developed a novel method for constructing higher-dimensional discrete integrable systems and I proposed a novel criterion for deciding on the integrability of lower dimensional discrete dynamical systems.

Free Research Field

数物科学・数学