Integrable Hamiltonian Hierarchies : Spectral and Geometric Methods

In the past decades now a famous class of evolution equations has been discovered and intensively studied, a class including the nowadays celebrated Korteweg-de Vries equation, sine-Gordon equation, nonlinear Schr. odinger equation, etc. The equations from this class are known also as the soliton equations or equations solvable by the so- called Inverse Scattering Tra- form Method. They possess a number of interesting properties, probably the most interesting from the geometric point of view of being that most of them are Liouville integrable Hamiltonian systems. Because of the importance of the soliton equations, a dozen monographs have been devoted to them. H- ever, the great variety of approaches to the soliton equations has led to the paradoxical situation that specialists in the same ?eld sometimes understand eachotherwithdi?culties. Wediscovereditourselvesseveralyearsagoduring a number of discussions the three of us had. Even though by friendship binds us, we could not collaborate as well as we wanted to, since our individual approach to the ?eld of integrable systems (?nite and in?nite dimensional) is quite di?erent. We have become aware that things natural in one approach are di?cult to understand for people using other approaches, though the - jects are the same, in our case - the Recursion (generating) Operators and theirapplicationsto?niteandin?nitedimensional(notnecessarilyintegrable) Hamiltonian systems.