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Spectral and Dynamical Stability of Nonlinear Waves (Hardcover, 2013 ed.)
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Spectral and Dynamical Stability of Nonlinear Waves (Hardcover, 2013 ed.)
Series: Applied Mathematical Sciences, 185
Expected to ship within 12 - 17 working days
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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 structure of the linear eigenvalue problem for
Hamiltonian systems is carefully developed, including the Krein
signature and related stability indices. The Evans function for the
detection of point spectra is carefully developed through a series
of frameworks of increasing complexity. Applications of the Evans
function to the Orientation index, edge bifurcations, and large
domain limits are developed through illustrative examples. 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.
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