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.