Spectral And Dynamical Stability Of Nonlinear Waves By Todd Kapitula And Keith Promislow
by Todd Kapitula /
2013 / English / PDF
2.9 MB Download
This book unifies the dynamical systems and functional analysis approaches to the linear and nonlinear stability of waves. It synthesizes fundamental ideas of the past 20+ years of research, carefully balancing theory and application. The book isolates and methodically develops key ideas by working through illustrative examples that are subsequently synthesized into general principles. Many of the seminal examples of stability theory, including orbital stability of the KdV solitary wave, and asymptotic stability of viscous shocks for scalar conservation laws, are treated in a textbook fashion for the first time. It presents spectral theory from a dynamical systems and functional analytic point of view, including essential and absolute spectra, and develops general nonlinear stability results for dissipative and Hamiltonian systems. The book is intended for first or second year graduate students in mathematics, or those with equivalent mathematical maturity. It is highly illustrated and there are many exercises scattered throughout the text that highlight and emphasize the key concepts. Upon completion of the book, the reader will be in an excellent position to understand and contribute to current research in nonlinear stability. Introduction Background Material and Notation Essential and Absolute Spectra Asymptotic Stability of Waves in Dissipative Systems Orbital Stability of Waves in Hamiltonian Systems Point Spectrum: Reduction to Finite-Rank Eigenvalue Problems Point Spectrum: Linear Hamiltonian Systems The Evans Function for Boundary-Value Problems The Evans Function for Sturm–Liouville Operators on the Real Line The Evans Function for nth-Order Operators on the Real Line
