Mathematical Surveys and Monographs, Volume: 156
2009; approx. 258 pp; hardcover
ISBN-13: 978-0-8218-4897-5
Expected publication date is October 23, 2009.
This book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied include Korteweg-de Vries, Benjamin-Ono, and Schrodinger equations. Many special Jacobian elliptic functions play a role in these examples.
The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena.
Graduate students and research mathematicians interested in nonlinear wave phenomena.
History, basic models, and travelling waves
Introduction and a brief review of the history
Basic models
Solitary and periodic travelling wave solutions
Well-posedness and stability definition
Initial value problem
Definition of stability
Stability theory
Orbital stability--the classical method
Grillakis-Shatah-Strauss's stability approach
The Concentration-Compactness Principle in stability theory
Existence and stability of solitary waves for the GBO equations
More about the Concentration-Compactness Principle
Instability of solitary wave solutions
Stability of periodic travelling waves
Stability of cnoidal waves
Appendices
Sobolev spaces and elliptic functions
Operator theory
Bibliography
Index
Contemporary Mathematics, Volume: 494
2009; 195 pp; softcover
ISBN-13: 978-0-8218-4745-9
Expected publication date is October 3, 2009.
This book contains the proceedings of the research conference, "Imaging Microstructures: Mathematical and Computational Challenges", held at the Institut Henri Poincare, on June 18-20, 2008.
The problems that appear in imaging microstructures pose significant challenges to our community. The methods involved come from a wide range of areas of pure and applied mathematics. The main purpose of this volume is to review the state-of the-art developments from analytic, numerical, and physics perspectives.
Graduate students and research mathematicians interested in partial differential equations, inverse problems, and applied mathematics.
Contemporary Mathematics, Volume: 495
2009; 382 pp; softcover
ISBN-13: 978-0-8218-4782-4
Expected publication date is October 4, 2009.
This volume is a collection of papers from the International Conference on Tropical and Idempotent Mathematics, held in Moscow, Russia in August 2007. This is a relatively new branch of mathematical sciences that has been rapidly developing and gaining popularity over the last decade.
Tropical mathematics can be viewed as a result of the Maslov dequantization applied to "traditional" mathematics over fields. Importantly, applications in econophysics and statistical mechanics lead to an explanation of the nature of financial crises. Another original application provides an analysis of instabilities in electrical power networks.
Idempotent analysis, tropical algebra, and tropical geometry are the building blocks of the subject. Contributions to idempotent analysis are focused on the Hamilton-Jacobi semigroup, the max-plus finite element method, and on the representations of eigenfunctions of idempotent linear operators. Tropical algebras, consisting of plurisubharmonic functions and their germs, are examined. The volume also contains important surveys and research papers on tropical linear algebra and tropical convex geometry.
Graduate students and research mathematicians interested in modern mathematics, including tropical methods and their applications.
Graduate Studies in Mathematics, Volume: 107
2009; approx. 675 pp; hardcover
ISBN-13: 978-0-8218-4815-9
Expected publication date is November 15, 2009.
Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle.
This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hypersurfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations.
The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry.
Graduate students and research mathematicians interested in differential geometry.