SpringerBriefs in Mathematical Physics

KP Solitons and the Grassmannians

Combinatorics and Geometry of Two-Dimensional Wave Patterns

Authors: Kodama, Yuji

  • 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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eBook $39.99
price for USA (gross)
  • ISBN 978-981-10-4094-8
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA
  • ISBN 978-981-10-4093-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this book

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)

  • Introduction to KP Theory and KP Solitons

    Kodama, Yuji

    Pages 1-24

  • Lax-Sato Formulation of the KP Hierarchy

    Kodama, Yuji

    Pages 25-40

  • Two-Dimensional Solitons

    Kodama, Yuji

    Pages 41-51

  • Introduction to the Real Grassmannian

    Kodama, Yuji

    Pages 52-63

  • The Deodhar Decomposition for the Grassmannian and the Positivity

    Kodama, Yuji

    Pages 64-89

Buy this book

eBook $39.99
price for USA (gross)
  • ISBN 978-981-10-4094-8
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA
  • ISBN 978-981-10-4093-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
KP Solitons and the Grassmannians
Book Subtitle
Combinatorics and Geometry of Two-Dimensional Wave Patterns
Authors
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 and Tables
19 b/w illustrations
Topics