KP Solitons and the Grassmannians
Combinatorics and Geometry of Two-Dimensional Wave Patterns
Authors: Kodama, Yuji
Free Preview- Is the first book to present a classification theory of two-dimensional patterns generated by the KP solitons
- Provides an introduction to totally non-negative Grassmannians and introduces combinatorial tools to study the manifolds
- Explains the combinatorial and geometric structure of the KP solitons which leads to a surprising connection among several areas of pure mathematics
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- About this book
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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.
- Table of contents (8 chapters)
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Introduction to KP Theory and KP Solitons
Pages 1-24
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Lax-Sato Formulation of the KP Hierarchy
Pages 25-40
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Two-Dimensional Solitons
Pages 41-51
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Introduction to the Real Grassmannian
Pages 52-63
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The Deodhar Decomposition for the Grassmannian and the Positivity
Pages 64-89
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Table of contents (8 chapters)
- Download Preface 1 PDF (111.9 KB)
- Download Sample pages 2 PDF (202.2 KB)
- Download Table of contents PDF (126.7 KB)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- KP Solitons and the Grassmannians
- Book Subtitle
- Combinatorics and Geometry of Two-Dimensional Wave Patterns
- Authors
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- Yuji Kodama
- Series Title
- SpringerBriefs in Mathematical Physics
- Series Volume
- 22
- Copyright
- 2017
- Publisher
- Springer Singapore
- Copyright Holder
- The Author(s)
- eBook ISBN
- 978-981-10-4094-8
- DOI
- 10.1007/978-981-10-4094-8
- Softcover ISBN
- 978-981-10-4093-1
- Series ISSN
- 2197-1757
- Edition Number
- 1
- Number of Pages
- XII, 138
- Number of Illustrations
- 19 b/w illustrations
- Topics
