Standing at a beach, you can see wavefronts approaching the shore at different angles, forming patterns that beg for mathematical analysis. Essentially, they consist of a finite number of line segments meeting at vertices. This raises interesting questions concerning the web-like patterns that they form at each moment in time and whether one could predict the future dynamics of the waves from those patterns. These are the sorts of questions that motivate Yuji Kodama in his new book, Solitons in Two-Dimensional Shallow Water. This valuable addition to the existing literature on soliton theory straddles the line between being a graduate textbook and a research monograph.

The first three chapters of the book are not very different from what can be found in many other books on “soliton theory”. In the first chapter, nonlinear partial differential equations modeling the dynamics of surface water waves are derived from physical principles. The second chapter utilizes the amazing fact that these equations are algebraically integrable to write exact “soliton” solutions in terms of exponential functions. The third chapter involves a detailed discussion of the geometry of Grassmannian varieties.

However, because of the book’s narrow focus, many of the usual topics addressed by books on this subject are barely discussed or not mentioned at all. Although there are many integrable “soliton equations”, this book focuses almost exclusively on the KP Equation. This is understandable since the KP Equation (which has two spatial variables and one time variable) is frequently used to model water waves on the ocean surface. Most books on soliton theory say something about the famous and fascinating connection between the (quasi)-periodic solutions of these equations and the geometry of line bundles over complex projective curves, but as this is not relevant to the web-like patterns formed from a finite number of line segments that are the main subject of the book, they get no mention here. Similarly, this book says nothing about the subject’s important connections to particle physics and quantum field theories or the use of solitons of light for data transmission. And, although the infinite-dimensional Grassmannian manifold that was found by M. Sato to parametrize the solution set of the KP equation is mentioned, Kodama is able to focus only on certain finite-dimensional subsets where the non-singular, web-like solutions of interest live.

One subject that receives far more attention here than in most soliton theory books is combinatorics. Between the last section of Chapter 3 and the end of Chapter 5, combinatorial objects like Young diagrams, “pipedreams”, and Go diagrams are used to classify and categorize the wave patterns that can arise in these special solutions to the KP Equation.

The last three chapters connect all of that pure mathematics with actual water waves. Chapter 6 uses numerical methods to study the stability of these exact solutions, supporting the claim that they are idealized but still reasonable models of the ocean waves we see in the real world. Chapter 7 considers the inverse problem that would allow one to predict the dynamics of some real waves if given a snapshot of their web-like appearance at one moment in time. And, finally, Chapter 8 argues that the theory presented in this book can be used to understand the phenomenon of Mach reflection.

Much of the information in this book can be found in the existing literature, especially in recent research papers by Kodama and his co-authors. However, there is also some new material and it is very nice to have it all grouped together. In fact, the material is self-contained in that all terms, notations, and results that would not be familiar to an undergraduate math major are given within the book itself. Between that fact and the inclusion of many exercises at the end of the chapter, there is the possibility that this could serve as a good textbook for a special topics course or independent study. However, it is very dense and lacks the sort of motivational discussion that one would find in some of the more gentle introductions to soliton theory. Consequently, this book would probably work best for someone who already has prior experience with the subject.

In conclusion, this book (based on Kodama’s lectures at a week-long NSF-CBMS conference at the University of Texas-Pan American in 2013) is a nice monograph that demonstrates a connection between very abstract mathematical objects and an easily observed real-world phenomenon. It is also unusual in its emphasis on connections between soliton theory and combinatorics. This book could be used either by professional mathematicians interested in working in this area or as a textbook by graduate students, but I fear that the learning curve might be too steep for a reader who did not already know something about Sato’s Grassmannian and the KP Hierarchy.

Alex Kasman is a professor of mathematics at the College of Charleston. He has published many research papers in math, biology, and physics journals. In addition, he has two books: the AMS published his gentle introduction to soliton theory and the MAA published a book of his mathematical short stories.