Algebraic and Analytic Aspects of Soliton Type Equations

This is a review of two of the fundamental tools for analysis of soliton equations: i) the algebraic ones based on Kac-Moody algebras, their central extensions and their dual algebras which underlie the Hamiltonian structures of the NLEE; ii) the construction of the fundamental analytic solutions (FAS) of the Lax operator and the Riemann-Hilbert problem (RHP) which they satisfy. The fact that the inverse scattering problem for the Lax operator can be viewed as a RHP gave rise to the dressing Zakharov-Shabat, one of the most effective ones for constructing soliton solutions. These two methods when combined may allow one to prove rigorously the results obtained by the abstract algebraic methods. They also allow to derive spectral decompositions for non-self-adjoint Lax operators.