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 specialised 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 emphasised 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.
