October  2012, 5(5): i-iii. doi: 10.3934/dcdss.2012.5.5i

Preface

1. 

Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1

2. 

Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Blvd, Lawrence, Kansas 66045-7567

Published  January 2012

Partial differential equations viewed as dynamical systems on an infinite-dimensional space describe many important physical phenomena. Lately, an unprecedented expansion of this field of mathematics has found applications in areas as diverse as fluid dynamics, nonlinear optics and network communications, combustion and flame propagation, to mention just a few. In addition, there have been many recent advances in the mathematical analysis of differential difference equations with applications to the physics of Bose-Einstein condensates, DNA modeling, and other physical contexts. Many of these models support coherent structures such as solitary waves (traveling or standing), as well as periodic wave solutions. These coherent structures are very important objects when modeling physical processes and their stability is essential in practical applications. Stable states of the system attract dynamics from all nearby configurations, while the ability to control coherent structures is of practical importance as well. This special issue of Discrete and Continuous Dynamical Systems is devoted to the analysis of nonlinear equations of mathematical physics with a particular emphasis on existence and dynamics of localized modes. The unifying idea is to predict the long time behavior of these solutions. Three of the papers deal with continuous models, while the other three describe discrete lattice equations.

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Citation: Dmitry Pelinovsky, Milena Stanislavova, Atanas Stefanov. Preface. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : i-iii. doi: 10.3934/dcdss.2012.5.5i
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