The aim of this book is to introduce the mathematics of continuous breathers, one of the most beautiful mathematical and physical objects appearing in the basic description of phenomena associated with waves and dispersion. This book serves as the first instalment in this area with a rigorous flavour and attention to deep properties of breather solutions.It also intends to serve as the starting point for a theory that has developed in many directions over the past twenty years and is now under intense scrutiny and active research. Accordingly, what is presented here are the first steps the initial results needed to understand the main goals and challenges in the study of breathers. Balancing rigor with accessibility, this text serves mathematical, physical, and computational interests. By blending geometric intuition, analysis, and explicit constructions, it highlights the complexity and elegance of breathers, offering a coherent overview and inspiration for future research.
