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.