The book places emphasis on both the mathematical significance and the strong physical background of wave equations. It presents the theory of wave equations in a unique way, different from the traditional descriptions provided by previous literature. The book is primarily focused on mathematical ideas and thoughts about wave equations. Starting from the modern theory of harmonic analysis, the book develops a few new tools in this field that are being used for better understanding the theory of mathematical physics underlying the well-posedness and scattering theory of wave and Klein-Gordon equations. Additionally, a significant part of this book discusses theories and methods, such as invariant and conservation laws, inward/outward energy methods, etc., that have never been covered by similar books in this field. Finally, the book briefly introduces recent developments in mathematical fields. It is specially designed for experts in mathematics and physics who deal with numerous applications of nonlinear waves in physics, engineering, biology, and other fields.

Changxing Miao: Professor of Institute of Applied Physics and Computational Mathematics, the winner of the National Science Fund for Distinguished Young Scholars, Yu Min Mathematical Science Award, Outstanding Expert of China Academy of Engineering Physics, and First Prize of Science and Technology Innovation of China Academy of Engineering Physics. His research interests include harmonic analysis, scattering theory of nonlinear dispersive equations, mathematical theory of fluid dynamics equations, etc. In recent years Professor Miao has published more than 100 papers in international first-class academic journals (such as: CPAM, PLMS, CMP, ARMA, AIM, TAMS, RMI, MZ, JFA, JMPA, SIAM, AIHP, CPDE, MRL, IUMJ, etc.) He also published six monographs in Chinese language. His work plays a significant role in the development of the mathematical fields mentioned above in China and has been highly praised by famous mathematicians, such as Constantin, Kenig and Sogge. His research team was called "one of the most dynamic and influential teams worldwide in the research field of partial differential equations" by Professor Carlos Kenig, the former president of the International Mathematical Union. Ruipeng Shen: Professor of Center for Applied Mathematics, Tianjin University. He earned his B.S at Peking University and PhD at the University of Chicago. He worked as a Post Doc fellow at Massachusetts Institute of Technology and McMaster University before he joined Tianjin University in 2015. Professor Shen works in the research fields of harmonic analysis and partial differential equations and is particularly interested in the global behaviour of solutions to the wave equations. He published in Memoirs of the American Mathematical Society, Advances in Mathematics, Mathematiche Annalen, Analysis & PDE, etc.