This book examines the new powerful method discovered by the author for the study of nonlinear dynamical systems relying on reduction of nonlinear differential equations to the linear abstract Schroedinger-like equation in Hilbert space. Besides the possibility of unification of many apparently completely different techniques, the introduced "quantal" Hilbert space formalism enables new original methods to be discovered for solving nonlinear problems arising in investigation of ordinary and partial differential equations as well as difference equations. Applications covered in the book...