Solitons are nonlinear waves which behave like interacting particles. When first proposed in the 19th century, leading mathematical physicists denied that such a thing could exist. Now they are regularly observed in nature, shedding light on phenomena like rogue waves and DNA transcription. Solitons of light are even used by engineers for data transmission and optical switches. Furthermore, unlike most nonlinear partial differential equations, soliton equations have the remarkable property of being exactly solvable. Explicit solutions to those equations provide a rare window into what is possible in the realm of nonlinearity.
Glimpses of Soliton Theory reveals the hidden connections discovered over the last half-century that explain the existence of these mysterious mathematical objects. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebro-geometric structure of soliton equations provides an elegant explanation of something seemingly miraculous.
Assuming only multivariable calculus and linear algebra, the book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass ℘-functions, the algebra of differential operators, Lax Pairs and their use in discovering other soliton equations, wedge products and decomposability, the KP Hierarchy, and Sato's theory relating the Bilinear KP Equation to the geometry of Grassmannians.
Notable features of the book include: careful selection of topics and detailed explanations to make the subject accessible to undergraduates, numerous worked examples and thought-provoking exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of Mathematica® to facilitate computation and animate solutions.
The second edition refines the exposition in every chapter, adds more homework exercises and projects, updates references, and includes new examples involving non-commutative integrable systems. Moreover, the chapter on KdV multisolitons has been greatly expanded with new theorems providing a thorough analysis of their behavior and decomposition.
Author(s): Alex Kasman
Series: Student Mathematical Library 100
Edition: 2
Publisher: American Mathematical Society
Year: 2023
Language: English
Pages: 347
Tags: Nonlinear PDEs, Solitons, KdV Equation, Elliptic Curves, Differential Algebra
Cover
1/2 title
Title page
Copyright
Contents
Preface
Differential Equations
Classification of Differential Equations
Can we write solutions explicitly?
Differential equations as models of reality
Named equations
When are two equations equivalent?
Evolution in time
Problems
Suggested Reading
Developing PDE Intuition
The Structure of Linear Equations
Examples of Linear Equations
Examples of Nonlinear Equations
Problems
Suggested Reading
The Story of Solitons
The Observation
Terminology and Backyard Study
A Less-than-enthusiastic Response
The Great Eastern
The KdV Equation
Early 20th Century
Numerical Discovery of Solitons
Hints of Nonlinearity
Explicit Formulas for n-soliton Solutions
Soliton Theory and Applications
Epilogue
Problems
Suggested Reading
Elliptic Curves and KdV Traveling Waves
Algebraic Geometry
Elliptic Curves and Weierstrass -functions
Traveling Wave KdV Solutions
Problems
Suggested Reading
KdV n-Solitons and -Functions
Introductory Remarks on KdV n-solitons
KdV -Functions
KdV 1-solitons and their -functions
KdV 2-solitons and their -functions
The 2-soliton Phase Shift
KdV n-Solitons and their -functions
Predicting the Appearance of an n-soliton
Proofs of the Main Claims in This Chapter
There's Something about KdV
Problems
Suggested Reading
Multiplying and Factoring Differential Operators
Differential Algebra
Factoring Differential Operators
Almost Division
Application to Solving Differential Equations
Producing an ODO with a Specified Kernel
Problems
Suggested Reading
Eigenfunctions and Isospectrality
Isospectral Matrices
Eigenfunctions and Differential Operators
Dressing for Differential Operators
Problems
Suggested Reading
Lax Form for KdV and Other Soliton Equations
KdV in Lax Form
Finding Other KdV-like Soliton Equations
The Non-commutative KdV Equation
Scalar Equations with Matrix Lax Operators
Connection to Algebraic Geometry
Problems
Suggested Reading
The KP Equation and Bilinear KP Equation
The KP Equation
The Bilinear KP Equation
Problems
Suggested Reading
2,4 and the Bilinear KP Equation
Wedge Products
Decomposability and the Plücker Relation
The Grassmann Cone 2,4 as a Geometric Object
Bilinear KP as a Plücker Relation
Geometric Objects and Nonlinear PDEs
Problems
Suggested Reading
Pseudo-Differential Operators and the KP Hierarchy
Motivation
The Algebra of Pseudo-Differential Operators
DOs are Not Really Operators
Application to Soliton Theory
Problems
Suggested Reading
k,n and the Bilinear KP Hierarchy
Higher Order Wedge Products
The Bilinear KP Hierarchy
Problems
Suggested Reading
Concluding Remarks
Soliton Solutions and their Applications
Algebro-Geometric Structure of Soliton Equations
Mathematica Guide
Basic Input
Some Notation
Graphics
Matrices and Vectors
Trouble Shooting: Common Problems and Errors
Complex Numbers
Algebra with Complex Numbers
Geometry with Complex Numbers
Functions and Complex Numbers
Problems
Ideas for Independent Projects
References
Glossary of Symbols
Index
Back cover
