Nonlinear Oscillations of Hamiltonian PDEs (eBook)

Massimiliano Berti (Autor)

eBook Download: PDF

2007
XIV, 180 Seiten
Birkhäuser Boston (Verlag)
978-0-8176-4681-3 (ISBN)

Infos

Many partial differential equations (PDEs) that arise in physics can be viewed as infinite-dimensional Hamiltonian systems. This monograph presents recent existence results of nonlinear oscillations of Hamiltonian PDEs, particularly of periodic solutions for completely resonant nonlinear wave equations. The text serves as an introduction to research in this fascinating and rapidly growing field. Graduate students and researchers interested in variational techniques and nonlinear analysis applied to Hamiltonian PDEs will find inspiration in the book.

---

Many partial differential equations (PDEs) that arise in physics can be viewed as infinite-dimensional Hamiltonian systems. This monograph presents recent existence results of nonlinear oscillations of Hamiltonian PDEs, particularly of periodic solutions for completely resonant nonlinear wave equations.After introducing the reader to classical finite-dimensional dynamical system theory, including the Weinstein Moser and Fadell Rabinowitz bifurcation results, the author develops the analogous theory for nonlinear wave equations. The theory and applications of the Nash Moser theorem to a class of nonlinear wave equations is also discussed together with other basic notions of Hamiltonian PDEs and number theory. The main examples of Hamiltonian PDEs presented include: the nonlinear wave equation, the nonlinear Schrodinger equation, beam equations, and the Euler equations of hydrodynamics.This text serves as an introduction to research in this fascinating and rapidly growing field. Graduate students and researchers interested in variational techniques and nonlinear analysis applied to Hamiltonian PDEs will find inspiration in the book.

Inhalt

Preface 6
Contents 14
1 Finite Dimension 16
2 Infinite Dimension 43
3 A Tutorial in Nash–Moser Theory 72
4 Application to the NonlinearWave Equation 85
5 Forced Vibrations 122
Appendix A Hamiltonian PDEs 149
Appendix B Critical Point Theory 155
Appendix C Free Vibrations of NonlinearWave Equations: A Global Result 164
Appendix D Approximation of Irrationals by Rationals 169
Appendix E The Banach Algebra Property of Xs,s 174
Solutions 176
References 178
Index 186

Produkt-Details

• Erscheint lt. Verlag: 5.10.2007
• Reihe/Serie: Progress in Nonlinear Differential Equations and Their Applications
• Reihe/Serie: Progress in Nonlinear Differential Equations and Their Applications
• Zusatzinfo: XIV, 180 p. 10 illus.
• Verlagsort: Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Bifurcations • critical point theory • Dynamical system • Nash-Moser theorem • Number Theory • partial differential equation • Partial differential equations • wave equation
• ISBN-10: 0-8176-4681-7 / 0817646817
• ISBN-13: 978-0-8176-4681-3 / 9780817646813