Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that one can use the Wronskian method to construct a soliton solution to the KP equation from each point of the real Grassmannian. In my talk I'll describe the beautiful combinatorics (permutations, triangulations, etc) that arises when one studies the regular soliton solutions that come from the Grassmannian. We'll also see how the theory of total positivity and cluster algebras provide a natural framework for studying KP solitons.
Regular KP soliton solutions provide a good model for shallow water waves (like beach waves), and I'll end the talk with some pictures. This is joint work with Yuji Kodama.