Yuji Kodama – författare

Solitons are a fascinating topic for study and a major source of interest for potential application in optical communication. Possibly the first observation of a soliton occurred in 1838 and was made by a clerical gentleman riding a horse along a canal towpath. When a barge under tow came to a stop alongside him the bow wave did not stop, but continued to travel on its own for several miles with no change in shape. At the time this unusual phenomenon was not understood and remained unexplained. Interest was revived with the development of optical fibres and the realisation that at the high intensities possible in their very small cores the onset of non-linear effects could modify the propagation characteristics in a significant way. In a seminal paper in 1973 Hasegawa and Tappert solved a non-linear Schrodinger equation for fibre propagation and found solutions for solitary waves, i.e. solitons. Since then advances have been very rapid resulting in a much better understanding of a wide variety of soliton effects, and, crucially, the realisation that soliton propagation can be used to potentially great advantage in practical long-distance systems. There is, as a result, a wealth of theoretical and experimental research in progress all over the world. At NTT (Japan) a pule-code-modulated soliton train has been transmitted at 10Gbit/s over one million kilometres with zero error! Perhaps all long-distance, large bandwidth communication problems have been solved for ever. This book gives a clear account of the theory and mathematics of solitons travelling in optical fibres. It is written by the authority on the subject.

Web-like waves, often observed on the surface of shallow water, are examples of nonlinear waves. They are generated by nonlinear interactions among several obliquely propagating solitary waves, also known as solitons. In this book, modern mathematical tools—algebraic geometry, algebraic combinatorics, and representation theory, among others—are used to analyze these two-dimensional wave patterns. The author’s primary goal is to explain some details of the classification problem of the soliton solutions of the KP equation (or KP solitons) and their applications to shallow water waves.This book is intended for researchers and graduate students.

This is the first book to treat combinatorial and geometric aspects of two-dimensional solitons. Based on recent research by the author and his collaborators, the book presents new developments focused on an interplay between the theory of solitons and the combinatorics of finite-dimensional Grassmannians, in particular, the totally nonnegative (TNN) parts of the Grassmannians.The book begins with a brief introduction to the theory of the Kadomtsev–Petviashvili (KP) equation and its soliton solutions, called the KP solitons. Owing to the nonlinearity in the KP equation, the KP solitons form very complex but interesting web-like patterns in two dimensions. These patterns are referred to as soliton graphs.  The main aim of the book is to investigate the detailed structure of the soliton graphs and to classify these graphs. It turns out that the problem has an intimate connection with the study of the TNN part of the Grassmannians. The book also provides an elementary introduction to the recent development of the combinatorial aspect of the TNN Grassmannians and their parameterizations, which will be useful for solving the classification problem.This work appeals to readers interested in real algebraic geometry, combinatorics, and soliton theory of integrable systems. It can serve as a valuable reference for an expert, a textbook for a special topics graduate course, or a source for independent study projects for advanced upper-level undergraduates specializing in physics and mathematics.