KdV hierarchies and their relation to Kac-Moody algebras (2024)
The Korteweg-de Vries (KdV) equation equation was one of the first integrable systems which caught the attention of many researchers during the 19th century due to its several distinguishing characteristics. On the one hand, it admitted a special kind of solution called a soliton: a solitary wave capable of traveling without losing its form and that whose interactions are elastical; on the other hand, the KdV equation exhibited a large quantity of independent conserved quantities regardless of its high nonlinearity. One of the major breakthroughs in the study of the KdV equation came from Drinfeld and Sokolov in 1985, who showed there existed an algebraic structure underlying the KdV system associated to the recently famous Kac-Moody algebras. Moreover, beginning from an arbitrary affine Kac-Moody algebra, Drinfeld and Sokolov gave a method of constructing a hierarchy of evolution equations that could be solved under the same scheme under which the KdV equation could. The purpose of this work is to provide a fundamental background in which Drinfeld and Sokolov’s construction can be understood.