Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras
contents
1 The KdV equation and its analysis
We look for symmetries of the KdV equation taking the form of infinitesimal transformations by a nonlinear evolution equation. The KdV equation is itself a nonlinear evolution equation, but we will see how to derive it in terms of compatibility conditions between linear equations.
The best possible compass to guide us in mathematics and the natural sciences is the notion of symmetry. Following this compass, up anchor and away over the wide ocean of solitons!
2 The KdV hierarchy
The value of mathematics is its unrestrained freedom of expression, the license to introduce new concepts. You can probably still remember the amazing experience of meeting the complex numbers for the first time. In this chapter, we introduce the inverse of the differential operator \partial/\partial x. We then see the astonishing power with which this gives rise to the higher order KdV equations.
3 The Hirota equation and vertex operators
Hirota's theory of equations of bilinear type is a classic instance of freedom of expression in mathematics. In 1970s, Hirota introduced an effective method for constructing solutions of KdV equation and other solition equations, although at the time it was not clear that his methods had any connections with other areas of mathematics. However, a useful idea in mathematics does not remain in isolation for long. We will see how the Hirota equations relates to the vertex operators from elementary particle theory.
4 The calculus of Fermions
As we become more familiar with solitons and their structural properties, the algebraic laws governing the symmetry behind the equations come gradually to the fore. The scence changes for a while to this algebraic world in this chapter we explain Fermions and their calculus.
5 The Boson-Fermion correspondence
Although the construction of Bosons and Fermions in the preceding chapter proceeded along parallel lines, in character the two are remarkably different. Despite this, the main theme of this chapter is that we can actually realisze each of them in terms of the other. To pull off this kind of stunt, the essential idea is to make use of infinite sums of Bosons and Fermions. The generating functions we introduce provide a glipse of the atmosphere of quantum field theory.
6 Transformation groups and tau functions
We start by showing that the space of all quadratic expressions in Fermions has a natural structure of an infinite dimensional Lie algebra. The rest of the chapter is taken up with a treatment of the group corresponding to this Lie algebra as a transformation group taking solutions of the KP equations into other solutions. To describe what happens in geometric language, this group actions moves the vacuum vector around an orbit, each point of which is a tau function as in Chapter 2. In the huge infinite space of all functions, the orbit of this action is a submanifold, and its defining equation is nothing other than the Hirota equation.
7 The transformation group of the KdV equation
Coming down from the abstract heights of the KP hierarchy, we return to particular case of the KdV hierarchy. The world of solutions is then narrower, and the corresponding transformation group is also smaller. We give an introduction to the affine Lie algebra \hat{\mathfrak{sl_2}}, which appears as the infinitesimal transformations of the KdV hierarchy.
8 Finite dimensional Grassmannians and Plucker relations
The scene changes once more, and this chapter gives an introduction to Grassmann varieties. The link between this classical notion of projective geometry and the material of the book so far is provided by the Plucker relations. We explain the Plucker relations in the case of finite dimensional vector spaces as an introduction to the material of the following chapters.
9 Infinite dimensional Grassmannians
Chapter 6 showed how the space of all tau functions of the KP hierarchy in Fock space is the orbit of the vacuum state under a group action. In this chapter we show that this orbit is really a Grassmannian, and we consider further the equations which describe it, the bilinear identity. On the way, we touch on the Clifford group and character polynomials.
10 The bilinear identity revisited
In this chapter, we show how the bilinear identity discussed in previous chapters can be rewritten as the Plucker relations for an infinite dimensional Grassmannian. The chapter can also be seen as an exercise in applying Wick's theorem. Moreover, we derive the Hirota equation from the Plucker relations.