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Patterns and Interfaces in Dissipative Dynamics

Springer Verlag, 2006
Series in Synergetics

Spontaneous pattern formation in nonlinear dissipative systems far from equilibrium is a paradigmatic case of emergent behaviour associated with complex systems. It is encountered in a great variety of settings, both in nature and technology, and has numerous applications ranging from nonlinear optics through solid and fluid mechanics, physical chemistry and chemical engineering to biology. Nature creates its variety of forms through spontaneous pattern formation and self-assembly, and this strategy is likely to be imitated by future biomorphic technologies. This book is a first-hand account by one of the leading players in this field, which gives in-depth descriptions of analytical methods elucidating the complex evolution of nonlinear dissipative systems, and brings the reader to the forefront of current research. The introductory chapter on the theory of dynamical systems is written with a view to applications of its powerful methods to spatial and spatio-temporal patterns. It is followed by two chapters treating moving interfaces, based largely on reaction-diffusion and phase-separating systems. The following two chapters on amplitude equations for patterns and waves describe universal phenomena generated by representative equations which can be derived for a variety of non-equilibrium systems originating in fluid mechanics, physical chemistry or nonlinear optics. This book addresses graduate students and non-specialists from the many related areas of applied mathematics, physical chemistry, chemical engineering and biology, as well as the seasoned scientist in search of a modern source of reference.

Vortices in Nonlinear Fields
From liquid crystals to superfluids,
From non-equilibrium patterns to cosmic strings

Oxford University Press, 1999
International Series of Monographs on Physics, v.100
ISBN 0 19 850167 6

Symmetry breaking is responsible for the astounding variety of natural phenomena derived from a few simple and symmetric basic laws. An ordered state formed as a result of a symmetry-breaking transition in an extended system contains, as a rule, topological defects, and dynamics of further evolution is determined to a large extent by the dynamics of defects.

The book is centered on the dynamics of vortices - a most common variety of defects. The book considers from a unified point of view the structure and dynamics of vortices in a variety of nonlinear field models with spontaneously broken symmetry. Point vortices or vortex lines correspond, in different physical settings, to quantized vortices in superfluids or superconductors, dislocations in non-equilibrium patterns, rotating spiral waves, disclinations in liquid crystals, singularities in optical fields or strings in relativistic field theories. The study of vortex dynamics is largely motivated by our interest in the phenomenon of weak turbulence which can be understood in terms of motion and interaction of stable localized structures: defects in the prevailing ordered state.

The book is unique in its multidisciplinary scope, and at the same time is self-contained and methodically unified. The emphasis is on analytical methods that allow to gain understanding of the common theoretical structure of defect dynamics in various applications. In each particular setting, the effort is made to describe material from different fields in common language and at the same time to justify a physical model at a certain level of approximation.

From the Miracle to the Number
Popular science book (in Russian)

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