What is it about?
This article is devoted to a construction of solutions to the Korteweg-De Vries (KdV) equation describing the wave motion of shallow water. The phenomena of solitary wave in shallow water was found about 200 years ago, and in 1965 a computer simulation discovered that these solitary waves behaved something like particles, which means the waves preserve their shapes after collision. Therefore this wave is called as soliton.
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Why is it important?
The KdV equation is a typical equation representing soliton phenomena. Physicists say this phenomena is universal in the real world. However the non-linearity of the equation caused some difficulties in its rigorous investigation. In 1967 an epoch-making discovery was made by 4 mathematicians, and it turned out that many non-linear equations can be solved by this method. They have opened a new area of mathematics, namely infinite dimensional completely integrable system.
Perspectives
The 4-mathematicians related the KdV equation with 1-D Schroedinger operators, but to obtain various solutions their investigation was insufficient. In 1980 Sato created a quite different approach to this equation, and the present work is supposed to be a continuation of Sato theory. It contains a large class of initial data including almost periodic or more complicated oscillating ones.
Shinichi Kotani
Read the Original
This page is a summary of: Construction of KdV Flow: A Unified Approach, Peking Mathematical Journal, May 2023, Tsinghua University Press,
DOI: 10.1007/s42543-022-00058-w.
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