Integrable systems, originating from classical mechanics, have expanded significantly over the past quarter-century, especially after the discovery of solitons. Now, integrable systems encompass a wide range of dynamical systems for which, in principle, explicit expressions for solutions and various physical quantities can be systematically derived. This scope spans systems with continuous and discrete time, finite and infinite degrees of freedom, and both classical and quantum domains. With this expansion, integrable systems research has achieved considerable development.
In addition to its autonomous progress, integrable systems research has seen recent breakthroughs. These include the discovery of new nonlinear mathematical structures that can be descred by integrable systems, the exploration of novel integrable systems, and efforts to model diverse phenomena using integrable system frameworks. Furthermore, integrable systems research has stimulated important studies in applied mathematics to validate the mathematical techniques developed within this field. Of particular interest is the intersection of integrable systems and computer science, a convergence that has gained attention since the discovery of solitons by Zabusky and Kruskal in 1965, offering new avenues for applications in mathematical sciences.
The term Applied Integrable Systems (AIS) is an undefined term to describe this new trend in integrable systems research. We anticipate that, as the field continues to yield fruitful results, AIS will become a well-established concept, representing these advances within the broader scientific community.
There have been notable recent advances in the historically rich field of integrable systems research, a tradition that dates back to Newton. The first major development is the discovery of ultradiscrete solitons, a novel nonlinear mathematical concept. While solitons were already known to exist in systems such as the KdV equation and the continuous and discrete Toda equations, it has since been found that they also exist in ultradiscrete systems, where even the dependent variables are discretized. This has led to applied research on ultradiscrete solitons, including studies on traffic flow and stochastic dynamical systems. The second advance is the development of numerical computation algorithms based on discrete integrable systems, particularly the formulation of high-precision and high-speed singular value decomposition algorithms derived from the discrete Lotka-Volterra system. Thirdly, significant progress has been made in the numerical analysis of integrable systems, including numerical integration methods that preserve solutions, conserved quantities, and canonical structures.
The deepening and broadening of integrable systems research shows no sign of slowing. However, to date, the researchers achieving these outcomes have not had a shared academic society for collaborative activities and have relied solely on seminars and workshops for academic exchange.
To foster the diverse and sustainable development of Japan’s integrable systems research, which leads the world in this field, we have now established the activity group of Applied Integrable Systems within the Japan Society for Industrial and Applied Mathematics.
Chief Organaizer Hidetomo Nagai (Tokai University) hdnagai (AT) tokai.ac.jp
Organaizer Keisuke Matsuya (Musashino University) keimatsu (AT) musashino-u.ac.jp
Organaizer Hiroshi Miki (Doshisha University) hmiki (AT) mail.doshisha.ac.jp