Combinatorial geometry of the 2D Toda lattice and Davey Stewartson equation

The KP equation is a prototypical $(2+1)$-dimensional integrable PDE.

Its soliton solutions are famously parametrized by the Sato Grassmannian.

In seminal work, Kodama and Williams made the surprising discovery that the combinatorics of soliton solutions are intimately related to the combinatorics of the totally positive Grassmannian as pioneered by Postnikov.

They introduced novel algorithmic methods inspired by polyhedral structures arising from tropical geometry.

Soliton solutions to the 2D Toda lattice and the Davey--Stewartson equation, two closely related integrable systems with soliton solutions, are also classified by the Sato Grassmannian.

Kodama suggested that the methods of his work with Williams could generalize to these two integrable equations.

In this work, we show that this is indeed the case.

We derive algorithms to produce contour plots from elements in the totally nonnegative Grassmannian in both cases.

In the asymptotic setting, we recover and refine previous work of Biondini and Wang; as well as Biondini, Kireyev and Maruno.