In between cleaning, packing, and freaking out about my upcoming sabbatical, I’ve been thinking more about the conservation laws problem and specifically how they connect back to the KdV equation.

Warning: the following is completely unedited rambling!

There are several routes to pursue here, but underlying them all is how to properly connect the asymptotic scales.  That is, in the asymptotics, we would like to preserve as much of the “structure” as possible.

It just so happens that a little over a year ago, Peter Olver made an off-hand comment about one of his papers during “Harvey-Fest” (a conference celebrating the 75th birthday of Harvey Segur).  Peter mentioned a separate form of the KdV equation he derived in “Hamiltonian Perturbation Theory and Water Waves“.  This presents, what I ultimately believe, is the proper way to pursue the connection.  That is, enforce the structure of the bracket through the perturbations.

For me to purse this path, I’d need to brush up a lot on my geometry.  Alternatively, I could try to pursue a different method of generating the hierarchy.  However, these seem cumbersome.  So, while I have little time to really think, I’ve been spending it thinking about how the Gardner transformation helps generate the infinite hierarchy of conservation laws.

The Gardner Transform

While I’m still trying what insight led to the Gardner transform, here’s my understanding of the process thus far.

Beginning with the KdV equation

\displaystyle u_t - 6 u u_x + u_{xxx} = 0

do the following (note: there may be sign errors below):

  1. Write u = w + \epsilon w_x + \epsilon^2w^2 (seeking intuition for this choice).
  2. Plug the above into the KdV equation.
  3. Realize that once u in terms of w is plugged in to the KdV equation, the resulting operator can be factored (magic!)
  4. Collapse the factored equation into a conservation law for w in the form (w)_t  + (~~\cdots~~)_x = 0.
  5. Let w = \sum\epsilon^n w_n.  Thus, since w is a conservation law, so is w_n.
  6. Relate w_n back to u.  For example, at leading order, u = w_0 followed by w_1 + w_{0,x} = 0.  This implies that  -u_{x} is conserved.  At the next order, we find that w_2 + w_{1,x} + w_0^2 = 0.  Thus, u_{xx} - u_0^2 is a conserved quantity as well.

One question would be, is such a connection possible with the water-wave problem?  Sure there are specific challenges (finding the GT, factoring the operator, etc), but if it’s possible, this may be a faster way to get at a connection between the two conservation laws, and then ultimately seeing how to generate more conservation laws for the water-wave problem.

I’ve made some progress on this front, but it’s not quite ready yet.

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