Nonlinear waves and solitons on contours and closed surfaces (Q639235)

This book is an update of the first edition (reviewed in Zbl 1167.35038), and deals with models of nonlinear media filling closed (compact) curves and surfaces. The main subject is a systematic construction and study of solitons states in such systems. Both the first and second editions of the book are written in a self-consistent form, so that one can find the necessary mathematical techniques, which are presented under the same cover as the material dealing with physical models and physically relevant solutions. In this respect, the book may be used as a basis for a graduate course on the theory of nonlinear waves and solitons. The book is divided into three large parts. The first introduces the mathematical background, the second presents the main results for solitons on closed curves and surfaces (i.e., one- and two-dimensional solitons, respectively), and the third part offers applications to diverse physically relevant settings, on various spatial scales, in microscopic and macroscopic physics alike. In particular, the first examples of solitons solutions are introduced in terms of nonlinear kinematics of curves moving in the two-dimensional plane. Subsequently, this concept is extended for the motion of two-dimensional surfaces in the three-dimensional space. Elements of differential geometry, necessary for the development of these topics, are also included in a sufficiently detailed form. The second part is focused on nonlinear hydrodynamics in terms of Lagrangian and Eulerian descriptions of the inviscid fluid motion in compact domains (for instance, liquid drops). The third part comprises a number of interesting physical applications, such as surfaces of nuclei, quantum-Hall liquids, magnetohydrodynamics, and others. In particular, the consideration of physical applications is expanded in comparison with the first edition of the book. The book is intended for graduate students and researchers in mathematics, physics and engineering. This second edition has been thoroughly revised, expanded and updated accordingly.