نگاهی اجمالی به نظریه سالیتون، جبر و هندسه معادلات دیفرانسیل جزئی غیرخطی، ویرایش دوم

This book challenges and intrigues from beginning to end. It would be a treat to use for a capstone course or senior seminar. --William J. Satzer, MAA Reviews on Glimpses of Soliton Theory (First Edition) 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 $\wp$-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 MathematicaR 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. 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