The structures underlying soliton solutions in integrable hierarchies

We point out that a common feature of integrable hierarchies presenting soliton solutions is the existence of some special “vacuum solutions” such that the Lax operators evaluated on them, lie in some abelian subalgebra of the associated Kac-Moody algebra. The soliton solutions are constructed out of those “vacuum solitons” by the dressing transformation procedure. This talk is concerned with the structures responsible for the appearance of soliton solutions for a large class of non linear differential equations. In spite of the great variety of types of equations presenting soliton solutions, some basic features seem to be common to all of them. Pratically all such theories have a representation in terms of a zero curvature condition [1], and the corresponding Lax operators lie in some infinite dimensional Lie algebra, in general a Kac-Moody algebra Ĝ. We argue that one of the basic ingredients for the appearence of soliton solutions in such theories is the existence of “vacuum solutions” corresponding to Lax operators lying in some abelian (up to central term) subalgebra of Ĝ. Using the dressing transformation procedure [2] we construct the solutions in the orbit of those vacuum solutions, and conjecture that the soliton solutions correspond to some special points in those orbits. The talk is based on results obtained in collaboration with J.L. Miramontes and J. Sanchez Guillén and reported in ref. [3]. We consider non-linear integrable hierarchies of equations which can be formulated in terms of a system of first order differential LNΨ = 0 , LN ≡ ∂ ∂tN − AN (1) 1) Talk given at the I Latin American Symposium on High Energy Physics, I SILAFAE, Merida, Mexico, November/96 2) Partially supported by a CNPq research grant where the variables tN are the various “times” of the hierarchies, and their number may be finite or infinite. The equations of the hierarchies are then equivalent to the integrability or zero-curvature conditions of (1) [LN , LM ] = 0 . (2) Therefore, the Lax operators are “flat connections” AN = ∂Ψ ∂tN Ψ . (3) The type of integrable hierarchy considered here is based on a Kac-Moody algebra Ĝ, furnished with an integral gradation Ĝ = ⊕ i∈ZZ Ĝi and [Ĝi, Ĝj] ⊆ Ĝi+j . (4) The connections are of the form