Celebratio Mathematica — Chern

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article S.-S. Chern and C. K. Peng: “Lie groups and KdV equations,” Manuscr. Math. 28 : 1–3 (1979), pp. 207–217. MR 535702 Zbl 0408.35074

In recent years there have been extensive studies of evolution equations with soliton solutions, among which the most important ones are the Korteweg-deVries and sine-Gordon equations. We will show that the algebraic basis of these mathematical phenomena lies in Lie groups and their structure equations; their explicit solutions with special properties often give the evolution equations. The process is thus similar to the introdution of a “potential”. In fact, from \( \textrm{SL}(2;\mathbf{R}) \), the special linear group of all \( (2\times 2) \)-real unimodular matrices, one is led naturally to the KdV and MKdV (= modified Korteweg-deVries) equations of higher order. A Miura transformation exists between them. Following H. H. Chen [1976], this leads to the Bäcklund transformations of the KdV.

article S. S. Chern and C. K. Peng: “On the Bäcklund transformations of KdV equations and modified KdV equations,” J. China Univ. Sci. Tech. 11 : 3 (1981), pp. 1–6. MR 701775

incollection T. E. Cecil and S. S. Chern: “Dupin submanifolds in Lie sphere geometry,” pp. 1–48 in Differential geometry and topology (Tianjin, China, 1986–1987). Edited by B. Jiang, C.-K. Peng, and Z. Hou. Lecture Notes in Mathematics 1369. Springer (Berlin), 1989. Proceedings of the special year at Nankai Institute of Mathematics. MR 1001178 Zbl 0678.53003

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