KP line solitons and Tamari lattices

The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "critical" events where a transition to a different rooted binary tree takes place. We prove that these correspond to maximal chains in Tamari lattices (which are poset structures on associahedra). We further derive results that allow to compute details of the evolution, including the critical events. Moreover, we present some insights into the structure of the more general line soliton solutions. All this yields a characterization of possible evolutions of line soliton patterns on a shallow fluid surface (provided that the KP-II approximation applies).