Description
Book SynopsisThis book is the first monograph on a 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 Schrödinger-like equation in Hilbert space. Besides the possibility of unification of many apparently completely different techniques, the “quantal” Hilbert space formalism introduced 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 include symmetries and first integrals, linearization transformations, Bäcklund transformations, stroboscopic maps, functional equations involving the case of Feigenbaum-Cvitanovic renormalization equations and chaos.
Table of ContentsHilbert Space Formation for Ordinary Differential Equations - Evolution Equation in Hilbert Space; Operator Evolution Equations; Symmetries and First Integrals; Alternative Approaches; Hilbert Space Formulation for Partial Differential Equations - Evolution Equation in Hilbert Space; Operator Evolution Equations; Symmetries and First Integrals; Hilbert Space Formulation for Difference Equations - Evolution Equation in Hilbert Space; Functional Equations; Applications - First Integrals; Linearization Transformations; Backlund Transformations; Feigenbaum-Cvitanovic Renormalization Equations; Chaos and Lyapunov Exponents. Appendices: Hilbert Spaces; Basic Notions of Quantum Mechanics; Bose Operators and Coherent States; Position and Momentum Operators; Functional Derivatives.
