A Dynamical Perspective on the ɸ4 Model: Past, Present and Future

Description

This book presents a careful selection of the most important developments of the \phi^4 model, offering a judicious summary of this model with a view to future prospects and the challenges ahead. Over the past four decades, the \phi^4 model has been the basis for a broad array of developments in the physics and mathematics of nonlinear waves. From kinks to breathers, from continuum media to discrete lattices, from collisions of solitary waves to spectral properties, and from deterministic to stochastic models of \phi^4 (and \phi^6, \phi^8, \phi^12 variants more recently), this dynamical model has served as an excellent test bed for formulating and testing the ideas of nonlinear science and solitary waves. Front Matter ....Pages i-xxi Historical Overview of the \(\phi ^4\) Model (David K. Campbell)....Pages 1-22 Some Recent Developments on Kink Collisions and Related Topics (Tomasz Romańczukiewicz, Yakov Shnir)....Pages 23-49 Collective Coordinate Methods and Their Applicability to \(\phi ^4\) Models (Herbert Weigel)....Pages 51-74 Mathematical Analysis of Fractal Kink-Antikink Collisions in the \(\phi ^4\) Model (R. H. Goodman)....Pages 75-91 Stochastic Dynamics of \(\phi ^4\) Kinks: Numerics and Analysis (Grant Lythe)....Pages 93-110 Discrete Variants of the \(\phi ^4\) Model: Exceptional Discretizations, Conservation Laws and Related Topics (Sergey V. Dmitriev, Panayotis G. Kevrekidis)....Pages 111-136 Discrete Breathers in \(\phi ^4\) and Related Models (Jesús Cuevas–Maraver, Panayotis G. Kevrekidis)....Pages 137-162 \(\phi ^{4}\) Continuum Breathers: Forty Years After (John P. Boyd)....Pages 163-186 The Continuing Story of the Wobbling Kink (Igor Barashenkov)....Pages 187-212 \(\phi ^4\) Solitary Waves in a Parabolic Potential: Existence, Stability, and Collisional Dynamics (Ryan M. Ross, Panayotis G. Kevrekidis, David K. Campbell, Robert Decker, Aslihan Demirkaya)....Pages 213-234 The \(\phi ^4\) Model in Higher Dimensions (Jean-Guy Caputo)....Pages 235-252 Higher-Order Field Theories: \(\phi ^6\), \(\phi ^8\) and Beyond (Avadh Saxena, Ivan C. Christov, Avinash Khare)....Pages 253-279 Effects of Discrete Breathers on Thermal Transport in the \(\phi ^4\) Lattice (Daxing Xiong, Sergey V. Dmitriev)....Pages 281-308 Back Matter ....Pages 309-311