Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras

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 worldl 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