manuscrlpta math. 28, 207 - 217 (1979) manuscripta mathemati ca t~) by Springer.Vcrlag 1979 LIE GROUPS AND KdV EQUATIONS Shiing-shen Chern and Chia-kuei Peng Dedicated to Hans Lewy and Charles B. Morrey, Jr. i. Introduction In recent years there have been extensive studies of evolution equations with soliton solutions, among which the most important ones are the Korteweg-deVrles and sine-Gordon equations. We will show that the alge- braic basis of these mathematical phenomena lies in Lie groups and their structure equations; their explicit solutions with special properties often give the evolu- tion equations, The process is thus similar to the in- troduction of a "potential". In fact, from SL(2;R), the special linear group of all (2 X 2)-real unimodular matrices, one is led naturally to the KdV and MKdV (= modified Korteweg-deYries) equations of higher order. A Miura transformation exists between them. Following H.H. Chen, [2] this leads to the B~cklund transforma- tions of the KdV equation. 2. KdV equations Let (I) SL(2;R) = = ad-bc = 1 Work done under partial support of NSF grant MCS77-23579. 0025-2611179/0028/0207/$02.20 207