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.



Longtime behaviour of the periodic KdV equation
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.


