nKdV Hierarchies

I haven't had a new post in a while, so let's finish up some of the drafts.

http://math.berkeley.edu/~ilya/software2/tmp/Dickey-classical-W.pdf

Write an nth order differential operator L.



Consider the algebra generated by the u symbols and their derivatives A.

Everything in A can be differentiated. Take the quotient by the subspace of closed elements.

Also consider a ring of Laurent polynomials in the derivative symbol but the other way around. Meaning they can go down as far as they need but have to cut out at the top.
This ring splits up into the positive part and negative part.

You can also take residues in the same manner with taking the term that looks like  .

Because the top coefficient of L is 1 we can actually take it to a fractional power m/n.   which commutes with L.

But let's mess that up by a violent projection to the positive part. Now we have a n-2 order differential operator which we can call the time derivative of L which describes how the u's are changing. Since the first u that shows up is at the right place, we are OK.

If you take different fractions, you see lots of commuting times, so we get lots of conserved quantities which are written via residues of all those     for all m.

Let n=2. Then we only have one u and this becomes KdV classic.