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Kenji Kajiwara

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Development of Theory of Integrable Systems: New Mathematical Methodology

Kenji Kajiwara
Degree: Doctor of Engineering (the University of Tokyo)
Research Interests: Theory of Integrable Systems
Unit: Pure Mathematics

Report

Wave phenomenon is a basic form of energy propagation, and is the main research target in the fluid dynamics, which is a major field in Math-for-Industry. In particular, a nonlinear wave called a soliton, which has character of both wave and particle simultaneously, is widely recognized as a fundamental mode of nonlinear phenomena as well as chaos and fractals. The existence of solitons is made possible by an “act of miracle” both in physics and mathematics. From physical point of view, a solitary wave that stably propagates cannot be derived from linear theory, and thus, is considered to exist on a subtle balance between nonlinearity and dispersion. From mathematical point of view, although it is in general difficult to analyze the nonlinear partial differential equations describing solitons, they possess a remarkable property that they can be exactly solved. Behind these miracles lies the mathematics of “infinite-dimensional space with symmetry of infinite degrees of freedom”. A family of functional equations that share this property is called “integrable systems”. Deep understanding of the underlying mathematical structures of the integrable systems enables various applications. Below are three such examples:

1.1.Discretization and ultra-discretization: Recently, a method to discretize independent variable (discretization) and dependent variables (ultra-discretization) of integrable differential equations preserving the background mathematical structures has been developed. Namely, one can construct integrable difference equations or cellular automata by a systematic method from given integrable differential equations.

The figure to the left shows a typical interaction of solitons. In this figure, the large soliton with a higher velocity comes from the left and passes the smaller, slower soliton. Although their amplitudes, velocities, and shapes do not change through the interaction, the locations of each wave is shifted, confirming that the interaction is nonlinear. On the other hand, the figure below shows an automaton that describes solitons.

There are rows of boxes and balls. At each time, from the left to the right, each ball is moved to the empty box closest to the right once, and the time is advanced by one when all balls have been moved. This simple model describes solitons, and a sound correspondence to a partial differential equation can be established through the ultra-discretization. Discretization and ultra-discretization preserving integrability have been applied to a broad range of mathematical sciences and engineering, including numerical analysis and traffic flow analysis.

2.1.Discrete Painlevé equations and elliptic curves: A family of difference equations called the discrete Painlevé equations are formulated as an addition theorem on a moving cubic curve in the complex projective plane.
Discrete Painlevé equations are closely related to the soliton equations, and they admit various special functions as particular solutions, such as the Bessel functions and their generalizations. Thus, integrable systems are closely related to pure mathematics such as algebraic geometry. Additionally, it has been recently discovered that the discrete Painlevé equations are closely related to probability theory and combinatorics.

3.Discrete soliton equations and discrete differential geometry: Classical theory of curves and surfaces, which were developed in the 19th century, are closely related to soliton equations and their transformation theory, and soliton equations appear as equations describing various surfaces. Theory of curves and surfaces which is consistent with the discretization preserving the integrability as discussed in 1 have been developed over the last ten years, which can be considered as a theoretical foundation for visualization. Hence, this is expected to be one of the major topics in Math-for-Industry. The figures below show a surface made from solutions to a soliton equation called the Sine-Gordon equation, and a discretized surface made from solutions to the discrete Sine-Gordon equation.

Hence, the theory of integrable systems provides with a methodology for the exactly analyzable objects in a broad range of mathematical sciences, including fluid dynamics, visualization, and probability. I would like to make the best use of this strength of the theory of integrable systems to play a role as a “libero” in Math-for-Industry.

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