Lie Algebras and Equations of Korteweg–de Vries Type

Drinfeld–Sokolov hierarchies on truncated current Lie algebras

“…It can be shown (see [9], [11], [2]) that Λ is a regular element of g, i.e., that the centralizer Λ is a regular semisimple element and therefore its isotropic subalgebra g Λ is a (Heisenberg) subalgebra H of g spanned, in the case of the Kac-Moody Lie algebra of type A…”

“…. Then it follows from [9] (proposition 3.3 page 1990) and the definition of truncated current Lie algebra that for any i > 0 the operator ad(…”

“…which can be regarded as an equation in g. It is well known ( [9] and [11]) that Λ is a regular element, g has the decomposition g = Ker(ad(Λ)) ⊕ Im(ad(Λ)) with Ker(ad(Λ)) = H. Hence, we can solve it uniquely for H 0,0 , and for T −l,0 up to an element of Ker(ad(Λ)). Now for k = 1 equation (3.6) becomes…”

“…Introduction. Surely the discovery made by Drinfeld and Sokolov in their famous and celebrated paper [9], that to any affine Kac-Moody Lie algebra and to any vertex of its (extended) Dynkin diagram corresponds a hierarchy of completely integrable nonlinear partial differential equations, is a hallmark in the theory of integrable systems. These hierarchies (usually called Drinfeld-Sokolov hierarchies) have been further intensively studied in the last twenty years by many mathematicians with the help of various techniques (see [8], [11], [1], [4], [5], [15], [3], [27], [10], [16]).…”

On Multicomponent mKDV Equations on Symmetric Spaces of DIII - Type and Their Reductions

“…While the reductions of first type can be applied both to MNLS and MMKdV equations, the reductions of second type can be applied only to MMKdV equations. Under them "half" of the members of the Hamiltonian hierarchy become degenerated [3,9]. For both classes of reductions we find examples with groups of reductions isomorphic to Z 2 , Z 3 and Z 4 .…”