Longtime behaviour of the periodic KdV equation

Question

I was wondering if anyone could give a heuristic (i.e. preferably non-technical) explanation of what is the expected longtime behavior of the periodic KdV equation.

Recall the standard KdV equation is the PDE defined (after suitable normalization) on $\mathbb{R}_t\times \mathbb{R}_x$ by $$ u_t= u_{xxx}+6 u u_x $$ where we assume $u$ has certain decay properties at spatial $\infty$.

If my understanding is correct, in this setting we expect that as $t\to \infty$ the solution $u$ should decompose into a sum of solitons along with a small "radiation" term that disappears in the limit.

My question is: what happens as $t\to \infty$ when the solutions are spatial periodic instead of decaying at spatial $\infty$? That is when we consider the equation on $\mathbb{R}_t\times \mathbb{S}_x^1.$

There doesn't seem to be "room" for the solution to split into solitons -- yet the equation is completely integrable so presumably there is some structure.

For context: I don't work in dispersive equations. However, the KdV equation has arisen quite naturally in some work I have been doing on a very unrelated problem and am currently trying to understand the extant to which I can exploit this. For better or worse, the literature on the KdV is vast which is making it hard to know where to begin.

Answer 1

Smooth solutions to the periodic KdV equation evolve almost periodically in time in any reasonable topology, due to the existence of action-angle variables that make any given KdV flow equivalent to a linear flow on an infinite-dimensional torus (which, when starting from smooth data, is compact in any reasonable topology). I believe this result was first established by McKean and Trubowitz ( http://www.ams.org/mathscinet-getitem?mr=427731 ), with closely related results also established by Lax ( http://www.ams.org/mathscinet-getitem?mr=404889 ) and by Flaschka-MacLaughlin ( http://www.ams.org/mathscinet-getitem?mr=403368 ).

Note that the Hill operator $-\partial_{xx} + u$ has completely different spectral properties in the periodic and non-periodic case, which explains the different asymptotics. On ${\bf R}$ (and for reasonable choices of $u$), this operator has absolutely continuous spectrum on the positive real axis (corresponding to the radiative component of non-periodic KdV) plus a small number of negative eigenvalues (corresponding to solitons, except in the case of repeated eigenvalues in which case one gets more exotic behaviour). In contrast, the spectrum on ${\bf T}$ consists of a bunch of pairs of periodic eigenvalues going to positive infinity separated by increasingly small gaps, together with a Dirichlet eigenvalue oscillating within each gap. The periodic KdV flow preserves the periodic eigenvalues but moves each of the Dirichlet eigenvalue around in something like a sinusoidal motion, and this is essentially the action-angle perspective mentioned earlier.

I believe the book of Kappeler and Poschel ( http://www.ams.org/mathscinet-getitem?mr=1997070 ) covers all of this material.