Nonlinear Evolution Equations and Soliton Solutions



Series: Mathematics Research Developments
BISAC: MAT017000

This book studies the methods for solving non-linear, partial differential equations that have physical meaning, and soliton theory with applications. Specific descriptions on the formation mechanism of soliton solutions of non-linear, partial differential equations are given, and some methods for solving this kind of solution such as the Inverse Scattering Transform method, Backlund Transformation method, Similarity Reduction method and several kinds of function transformation methods are introduced. Integrability of non-linear, partial differential equations is also discussed. This book is suitable for graduate students whose research fields are in applied mathematics, applied physics and non-linear science-related directions as a textbook or a research reference book. This book is also useful for non-linear science researchers and teachers as a reference book.

The characteristics of this book are:

1. The author provides clear concepts, rigorous derivation, thorough reasoning, and rigorous logic in the book. Since the research boom of non-linear, partial differential equations was rising in the 1960s, the research on non-linear, partial differential equations and soliton theory has only been several decades, which can be described as a very young discipline compared to the other branches in mathematics. Although there are a few related books, they are mostly in highly specialized interdisciplinary areas. There is no book which is suitable for cross-disciplines and for people with college level mathematics and college physics background. This book fills that gap.

2. The book is easy to be understood by readers since it provides step-by-step approaches. All results in the book have been deduced and collated by the author to make sure that they are correct and perfect.

3. The derivation from the physical models to mathematical models is emphasized in the book. In mathematical physics, we cannot just simply consider the mathematical problems without a physical image, which often plays the key role for understanding the mathematical problems.

4. Mathematical transformation methods are provided. The basic idea of various methods for solving non-linear, partial differential equations is to simplify the complex equations into simple ones through some transformations or decompositions. However, we cannot find any patterns for using such transformations or decompositions, and certain conjectures and assumptions have to be used. However, the skill and the logic of using the transformations and decompositions are very important to researchers in this field.
(Imprint: Nova)

Table of Contents

Table of Contents

(Dr. Zhijun Qiao)
pp. vii-viii

pp. ix-xiii

Chapter 1
Typical Equations and their Solitary Wave Solutions
pp. 1-70

Chapter 2
The Inverse Scattering Method and Multiple Solitary Wave Solutions
pp. 71-158

Chapter 3
Bäcklund Transformation
pp. 159-190

Chapter 4
Integrability and Painlevè Property of Nonlinear
Differential Equations
pp. 191-218

Chapter 5
Similarity Transformation and Similar Solutions of Nonlinear
Partial Differential Equations
pp. 219-280

Chapter 6
Special Transformation Methods for Solving Nonlinear Partial
Differential Equations
pp. 281-352

Chapter 7
Optical Solitons
pp. 353-400

Appendix A
Elliptic Functions and Elliptic Equations
pp. 401-410

Appendix B
The First Integral and the Solution Methods of First-Order
Partial Differential Equations
pp. 411-434

Appendix C
Some Concepts and Terminologies Associated with Fluctuations
pp. 435-452

Appendix D
Introduction of Perturbation Method
pp. 453-454

Appendix E
Hypergometric Function and Hypergometric Series
pp. 455-458

pp. 459-466

Author Contact Information
pp. 467

pp. 469-472


This book is written for graduate students and teachers whose fields are applied mathematics, applied physics, nonlinear science, and other relative areas. It can be used as textbook and reference book, and it is also available for researchers engaged in research on nonlinear science and theory of solitons as a reference book.

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