The Korteweg-deVries dynamical system

Abstract

As with the Neumann dynamical system, our purpose now is to introduce a dynamical system interesting in its own right, and then to show that it can, in some cases, be integrated explicitly by the theory of hyperelliptic Jacobians. More precisely, we can, following the ideas in the previous section, define an embedding of Jac C in an infinite dimensional space: On R₁, we consider a simple class of vector fields X: those which assign to f a tangent vector in $X_f \in T_{R_1 ,f} \cong R_1$ given by

$$ X_f = P\left( {f,\dot f, \cdots ,f^{\left( n \right)} } \right),{\text{ P a polynomial}} $$

. Integrating this vector field means finding an analytic function f(x,y) s.t.

$$ \frac{{\partial f}} {{\partial y}} = P\left( {f, \frac{{\partial f}} {{\partial x}},{\text{ }} \cdots {\text{, }}\frac{{\partial ^n f}} {{\partial x^n }}} \right). $$

By the Cauchy-Kowalevski Theorem, for all f(x,0) analytic in |x| < ε, there exists f(x,y) analytic in |x|,|y| < η solving this. What we want to do is to set up a sequence X₁,X₂,...of such vector fields called the Kortweg-de Vries hierarchy which a) commute [X_i,X_j] =0 — we must define this carefully — and b) are Hamiltonian in a certain formal sense, such that c) for all g, and for all hyperelliptic curves C of genus g:

$$ \operatorname{Im} \left( {Jac - \Theta } \right) = \left[ {\begin{array}{*{20}c} {orbit of all flows {\rm X}_n ,} \\ {i.e., all {\rm X}_n are tangent to image} \\ {and a codimension g subspace of} \\ {\sum {c_n {\rm X}_n } are even 0 on Image} \\ \end{array} } \right] $$