The book is devoted to several topical questions in modern mathematical and theoretical physics, astrophysics, geophysics, and cosmology that remain unsolved within the framework of the standard approaches. To them, one can attribute unexplained properties of the magnetic fields of stars and planets, puzzles of the Earth's atmosphere, the phenomenon of ball lightning, the problem of a qualitative description for nuclear forces and their well-known property of saturation, enigmatic properties of spiral galaxies, the problem of the cosmological singularity, mysteries of the dark matter and dark energy, amongst others. To find theoretical ways for understanding such phenomena, new nonlinear generalizations of the classical field theories and advanced methods to solve nonlinear equations arising in them are studied and presented in this book.

The monograph is devoted to a number of poorly explored problems of non-linear electrodynamics, nuclear physics, and general relativity that are very important for astrophysics, geophysics, and cosmology. In it new theoretical methods are proposed to solve the problems under consideration and to explain a number of puzzling experimental and observational data In the book the authors study the non-linear equations of the Yang-Mills theory and their applications to several unsolved problems of electrodynamics. They find and investigate a non-linear generalisation of the Yukawa meson theory to describe the phenomenon of nuclear saturation and various properties of nuclei. They propose a generalised Dirac's equation to describe the quark structure and anomalous magnetic moments of nucleons. They consider a number of unsolved problems of general relativity and study anomalous properties of gravitational waves. Finally, they find and investigate a generalisation of the Einstein gravitational equations based on Weyl's geometry to explain puzzling properties of spiral galaxies and to apply it to several unsolved problems of cosmology.