Integrable Models is a very broad topic that can be divided into two
major sub-topics:
Classical Integrable Models, the starting
point of which is nonlinear partial differential equations that are
completely integrable, and
Quantum Integrable Models, the
starting point of which is exactly solvable models in statistical
mechanics and in quantum field theory.
These two sub-topics are related, but in this course, we discuss
only Classical Integrable Models. For an introduction
to
Quantum Integrable Models (from a statistical mechanical
point of view), I refer you to Paul Pearce's honours course on
Phase
Transitions and Critical Phenomena.
The sub-topic of
Classical Integrable Models is by itself
a very broad subject and there are many complementary approaches
to it. All of these approaches are naturally related and it's one
of the challenges of the subject to explain these relationships.
In this course we concentrate on an algebraic approach that is
also known as
Sato's Theory (after the great Japanese
mathematician Mikio Sato). The reason why we choose this approach
is that (aside from the fact that time is too limited to cover more
material) it plays a central role in modern mathematical physics.
Roughly speaking, the course consists of 3 (very related) parts:
1.
Lax formulation: Integrable
nonlinear PDE's are
understood as consistency conditions of systems of
linear
PDE's (we will see what this means exactly in due course).
2.
Fermionic formulation: Solutions of integrable nonlinear
PDE's are re-written in terms of expectation values (to be defined)
of fermion (Clifford) operators (also to be defined).
3.
Geometric formulation: Solutions of integrable nonlinear
PDE's are points on a Grassmannian (to be defined) embedded in
a suitable projective space (also to be defined).
One of the challenges of the course (and a main attraction of the
entire subject) is to be able to see the connections between the
above 3 parts/formulations.
The following contains more detailed information about the contents
of the course. It is basically the table of contents of the textbook.
General Information 2009
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