# KP Solitons and the Grassmannians

## Combinatorics and Geometry of Two-Dimensional Wave Patterns

• Yuji Kodama
Book

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 22)

1. Front Matter
Pages i-xii
2. Yuji Kodama
Pages 1-24
3. Yuji Kodama
Pages 25-40
4. Yuji Kodama
Pages 41-51
5. Yuji Kodama
Pages 52-63
6. Yuji Kodama
Pages 64-89
7. Yuji Kodama
Pages 90-102
8. Yuji Kodama
Pages 103-119
9. Yuji Kodama
Pages 120-132
10. Back Matter
Pages 133-138

### Introduction

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.

### Keywords

KP equation soliton solutions in two-dimension totally non-negative Grassmannian Schubert decomposition and its refinements soliton graph

#### Authors and affiliations

• Yuji Kodama
• 1
1. 1.The Ohio State University ColumbusUSA