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 in0 mathematics