Graduate School of Science and Engineering Electrical and Electronic Engineering

Laboratory of Applied Mathematics

Website of Prof.Kondo 【In Japanese】

Aiming to build a bridge between mathematics and engineering

Staff

Research Topics

• Nonlinear waves and soliton theory
• Development of integrable numerical algorithms
• Nonlinear integrable systems based on system of orthogonal functions

Research Contents

In the Laboratory of Applied Mathematics, we are conducting research on applied mathematics based on nonlinear integrable systems (continuous systems, discrete systems, and ultra-discrete systems), computational mathematics (numerical analysis and computer algebra system), geometry, and other fields for better understanding the mathematical structures at work deep inside physics and engineering.

1. Generally, it is difficult problem whether nonlinear differential equations have solutions. However, nonlinear equations called soliton equations can be shown to be integrable, and moreover they are known to have many types of solutions. Also, one characteristic of nonlinear integrable systems is the ability to configure discrete integrable systems with a discretized spatial axis or time axis while keeping their solution structure. In this laboratory, we are researching nonlinear integrable systems.


2. Numerical algorithms are the most important tools in conducting numerical analysis and computer simulations. When the recurrence formula for numerical algorithms is treated as a discrete dynamical system, there are many algorithms with desirable properties, and it has been shown that these are equivalent to discrete soliton equations. Soliton equations were originally proposed as a physics model for nonlinear waves, and soliton theory serves as a bridge to physics, mathematical engineering, and computer science. In this laboratory, we are striving to develop new, powerful numerical algorithms based on the ideas of nonlinear integrable systems.


3. Nonlinear integrable systems are known to have a wide variety of characteristics. One of the most important is Hirota bilinear forms. The method of finding multiple soliton solutions using Hirota bilinear forms is called "Hirota's direct method." The direct method is an extremely powerful tool in the theory of nonlinear integrable systems. In recent years, the relationship between system of orthogonal functions and nonlinear integrable systems has been important problem. In this laboratory, we research integrable systems based on system of orthogonal functions.

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