A Birkhoff-Lewis-type theorem for some Hamiltonian PDEs

SIAM Journal on Mathematical Analysis

Volumn 37, Issue 1, 2005, Pages 83-102

  Bambusi, Dario ^a   Berti, Massimiliano ^b  

^a Dipartimento di Matematica   (Italy)

^b SISSA   (Italy)

Author keywords

Birkhoff normal form; Infinite dimensional Hamiltonian systems; Periodic solutions; Perturbation theory; Variational methods

Indexed keywords

BIRKHOFF NORMAL FORM; PERIODIC SOLUTIONS; VARIATIONAL METHODS;

NONLINEAR CONTROL SYSTEMS; PERTURBATION TECHNIQUES;

THEOREM PROVING;

EID: 31144456366     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141003436107     Document Type: Article

References (21)

1. A. AMBROSETTI, V. COTI-ZELATI, AND I. EKELAND, Symmetry breaking in Hamiltonian systems, J. Differential Equations, 67 (1987), pp. 165-184.
2. D. BAMBUSI, Lyapunov center theorems for some nonlinear PDE's: A simple proof, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29 (2000), pp. 823-837.
3. D. BAMBUSI, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 234 (2003), pp. 253-285.
4. D. BAMBUSI AND B. GREBERT, Forme normale pour NLS en dimension quelconque, C.R. Acad. Sci. Paris Ser. I Math., 337 (2003), pp. 409-414.
5. D. BAMBUSI AND S. PALEARI, Families of periodic solutions for resonant PDE's, J. Nonlinear Science, 11 (2001), pp. 69-87.
6. M. BERTI, L. BIASCO, AND E. VALDINOCI, Periodic orbits close to elliptic tori and applications to the three-body problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (2004), pp. 87-138.
7. M. BERTI AND P. BOLLE, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys., 243 (2003), pp. 315-328.
8. M. BERTI AND P. BOLLE, Multiplicity of periodic solutions of nonlinear wave equations, Nonlinear Anal., 56 (2004), pp. 1001-1046.
9. G. D. BIRKHOFF AND D. C. LEWIS, On the periodic motions near a given periodic motion of a dynamical system, Ann. Mat. Pura Appl., (4), 12 (1933), pp. 117-133.
10. J. BOURGAIN, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), pp. 629-639.
11. H. BREZIS, Analyse fonctionelle, Masson, Paris, 1993.
12. W. CRAIG AND C. E. WAYNE, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), pp. 1409-1498.
13. J. GBNG AND J. You, KAM tori of Hamiltonian perturbations of 1D linear beam equations, J. Math. Anal. Appl., 277 (2003), pp. 104-121.
14. S. B. KUKSIN, Nearly integrable infinite-dimensional Hamiltonian systems, Springer-Verlag, Berlin, 1993.
15. S. B. KUKSIN AND J. PÖSCHBL, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), pp. 149-179.
16. D. C. LEWIS, Sulle oscillazioni periodiche d'un sistema dinamico, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (1934), pp. 234-237.
17. J. MOSER, Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff, in Geometry and Topology (Proc. III Lat. Am. Sch. Math., Inst. Mat. Pura Aplicada CNP, Rio de Janeiro, 1976), Lecture Notes in Math., 597, Springer-Verlag, Berlin, 1977, pp. 464-494.
18. J. PÖSCHEL, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), pp. 119-148.
19. J. PÖSCHEL, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), pp. 269-296.
20. J. PÖSCHEL, On the construction of almost periodic solutions for a nonlinear Schrödinger equation, Ergodic Theory Dynam. Systems, 22 (2002), pp. 1537-1549.
21. C. E. WAYNE, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127 (1990), pp. 479-528.