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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."
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."
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