Customer Reviews

Solitons: An Introduction (Cambridge Texts in Applied Mathematics)

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The short text by Drazin and Johnson on Solitons discusses a method called the "inverse scattering transform", developed predominately in the 1950s and 1960s, to solve certainly "slightly" nonlinear, dispersive equations such as the Korteweg de Vries equation (KdV). Although the book is short, it adequately presents the method, through informal discussion and examples, such as applying it to the KdV, so that the method can be understood as a practical computational approach. In fact, there are many explicit computational examples in the text to illustrate the method, both in simple applications to the KdV and in more general cases, such as matrix equations.

The inverse scattering method is, in itself, rather beautiful, and transforms the nonlinear equation under consideration into exact linear integral equations, whose formal solutions can be discussed in terms of spectral decompositions. I liken the approach to solving the quadratic equation by "completing the square".

The authors also place the method in context of broader studies of nonlinear equations. For example, a brief presentation indicating the physical significance of the KdV is provided that shows that this equation is expected to have some fairly general applicability, and that solitons can be expected as somewhat general features to observe in nature. Other discussions consider the mathematics of solitons in more abstract settings, and there is a short discussion of numerical methods and results at the end of the book.

Many problems are provided at the end of each chapter. The problems take one through the previous chapter and force re-examination, more careful consideration and overall review, point by point sequentially of what was discussed in the chapter. Some of the problems are also intended to supplement the text, and give consideration to points that were of interest but could not be covered in the main text.

I personally found that I did not have adequate time to cover all of the material in the text. I feel, however, that the care and clarity with which the inverse scattering transform was presented did allow me to gain an appreciation of the method. I also had time to work only a small fraction of the problems.

For a short, "informal", presentation, I found the overall presentation to be very clear. This is, despite the shortness, and summary nature of some of the text, a deep book, requiring very dedicated effort to master (more effort than I could expend). I do consider it to be a very nice introduction to the study of solitons, and it has motivated me to further study. With the vast improvements we have seen in computation over the past fifty years, solitions have become an important area of applied mathematics.

I recommend this book as a good introduction to solitons, and a very good discussion of the inverse transform method. The mathematical pre-requisites are kept pretty minimal by the focus on "informal" mathematics. However, this, the clarity of presentation, and the shortness of the book should not deceive the reader: the book is very deep and to fully appreciate requires considerable dedication.

I am still in the process of learning solitons from this book, hence it is not very easy for me to offer an opinion of the merits of the book. But I assume that there is a whole batch of copies out there with awful printing quality between pages 117 and 147. The ink from the one side of each page has penetrated almost all the way to the other side making the reading very tedious. I returned my first copy, only to have it replaced with one having exactly the same problem