A dynamical perspective on the ɸ4 model : past present and future This book presents a careful selection of the most important developments of the 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 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 4 (and 6 8 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 Historical Perspectives of \\phi^4 Model Some Recent Developments on Kink Collisions Statistical Mechanics and Stochastic Aspects of \\phi^4 Models Discrete Variants of \\phi^4 Model Discrete Breathers in \\phi^4 and Related Models \\phi^4 Breathers: History and Recent Developments Decay of Bions in \\phi^4 Models \\phi^4 Model with a Potential \\phi^4 Model in Higher Dimensions Beyond the \\phi^4 Model: \\phi^6 \\phi^8 etc. . 2.3.2 From Internal Modes to Oscillons and Back2.4 Production of the Kink Antikink Pairs from Radiation; 2.5 Kink Antikink Scattering in the φ6 Model; 2.6 The Role of Quasinormal Modes in Topological Defect Collisions; 2.7 Kink Boundary Scattering in the φ4 Model; 2.8 Radiative Decay of the Internal Oscillating Mode; 2.9 Conclusions; References; 3 Collective Coordinate Methods and Their Applicability to φ4 Models; 3.1 Motivation; 3.2 Kink Antikink Scattering; 3.2.1 Dynamical Kink Antikink Interaction; 3.2.2 Bouncing Solutions and Bounce Windows; 3.3 Collective Coordinates 3.5.4 Mapping Collective Coordinates and Solution to Full Field Equation3.6 Comparison: ɸ6 Model; 3.7 Conclusion and Critical Analysis; References; 4 Mathematical Analysis of Fractal Kink Antikink Collisions in the ɸ4 Model; 4.1 Introduction; 4.2 Analysis of Collective Coordinate Models; 4.2.1 The ODE Model; 4.2.2 Derivation of the Discrete Map Approximation; 4.2.3 Interpretation and Analysis of the Map; 4.3 Computational Exploration of Map (4.10); 4.4 Conclusions and Outlook; References; 5 Stochastic Dynamics of ɸ4 Kinks: Numerics and Analysis; 5.1 Introduction; 5.1.1 Some Definitions Intro; Preface; Contents; Contributors; Acronyms; 1 Historical Overview of the φ4 Model; 1.1 Pre history: The Klein Gordon Equation(s) and φ4 Theory; 1.2 Applications of φ4 Theory in Physics; 1.3 The Saga of φ4 Versus Sine Gordon; 1.4 Concluding Remarks; References; 2 Some Recent Developments on Kink Collisions and Related Topics; 2.1 Introduction; 2.2 Solitons and Perturbations; 2.2.1 Spectral Structure of Small Perturbations; 2.3 Interplay Between the States of Perturbative and Non perturbative Sectors; 2.3.1 The Effect of Negative Radiation Pressure 5.1.2 Structure of the Chapter5.2 Stationary Densities; 5.3 Numerical Solution; 5.4 Brownian Motion of Kinks; 5.5 Nucleation of Kink Antikink Pairs; 5.6 Discussion; References; 6 Discrete Variants of the ɸ4 Model: Exceptional Discretizations Conservation Laws and Related Topics; 6.1 Introduction; 6.2 Klein Gordon Field with ɸ4 Potential and Its Standard Discretization; 6.3 Approaches to Derive Discrete ɸ4 Models Free of the Static Peierls Nabarro Potential; 6.3.1 Speight Ward Hamiltonian Model; 6.3.2 Momentum Conserving Discretizations; 6.3.3 Discretized First Integral Approach