Stability of Nonlinear Waves in Hamiltonian Dynamial Systems

Anna Geyer (Author), Dmitry E. Pelinovsky (Author)

9781470475529

Paperback / softback, published 30 September 2025

380 pages
25.4 x 17.8 x 2.1 cm, 0.748 kg

This monograph offers a comprehensive and accessible treatment of both classical and modern approaches to the stability analysis of nonlinear waves in Hamiltonian systems. Starting with a review of stability of equilibrium points and periodic orbits in finite-dimensional systems, it advances to the infinite-dimensional setting, addressing orbital stability and linearization techniques for spatially decaying and spatially periodic solutions of nonlinear dispersive wave equations, such as the nonlinear Schrodinger, Korteweg-de Vries, and Camassa-Holm equations. The book rigorously develops foundational tools, such as the Vakhitov-Kolokolov slope criterion, the Grillakis-Shatah-Strauss approach, and the integrability methods, but it also introduces innovative adaptations of the stability analysis in problems where conventional methods fall short, including instability of peaked traveling waves and stability of solitary waves over nonzero backgrounds. Aimed at graduate students and researchers, this monograph consolidates decades of research and presents recent advancements in the field, making it an indispensable resource for those studying the stability of nonlinear waves in Hamiltonian systems.

• Stability in finite-dimensional systems
• Stability of solitary waves
• Stability of periodic waves
• Orbital stability in integrable Hamiltonian systems
• Spectral stability in integrable Hamiltonian systems
• Stability of peaked waves
• Stability of domain walls and black solitons
• Jacobi elliptic functions and integrals
• Spectral theory for linear operators
• Bibliography
• Index