Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs

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