This is the first book to treat combinatorial and geometric aspects of two-dimensional
solitons. Based on recent research by the author and his collaborators the book presents new
developments focused on an interplay between the theory of solitons and the combinatorics of
finite-dimensional Grassmannians in particular the totally nonnegative (TNN) parts of the
Grassmannians. The book begins with a brief introduction to the theory of the
Kadomtsev-Petviashvili (KP) equation and its soliton solutions called the KP solitons. Owing
to the nonlinearity in the KP equation the KP solitons form very complex but interesting
web-like patterns in two dimensions. These patterns are referred to as soliton graphs. The main
aim of the book is to investigate the detailed structure of the soliton graphs and to classify
these graphs. It turns out that the problem has an intimate connection with the study of the
TNN part of the Grassmannians. The book also provides an elementary introduction to the recent
development of the combinatorial aspect of the TNN Grassmannians and their parameterizations
which will be useful for solving the classification problem. This work appeals to readers
interested in real algebraic geometry combinatorics and soliton theory of integrable systems.
It can serve as a valuable reference for an expert a textbook for a special topics graduate
course or a source for independent study projects for advanced upper-level undergraduates
specializing in physics and mathematics.
