Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Perturbation Theory for PDEs

  • Dario Bambusi
Reference work entry

Article Outline


Definition of the Subject


The Hamiltonian Formalism for PDEs

Normal Form for Finite Dimensional Hamiltonian Systems

Normal Form for Hamiltonian PDEs: General Comments

Normal Form for Resonant Hamiltonian PDEs and its Consequences

Normal Form for Nonresonant Hamiltonian PDEs

Non Hamiltonian PDEs

Extensions and Related Results

Future Directions



Normal Form Nonlinear Wave Equation Small Denominator Localize Coefficient Smooth Vector Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.


  1. 1.
    Arnold V (1984) Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Mir, MoscowGoogle Scholar
  2. 2.
    Bambusi D (1999) Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equation. Math Z 130:345–387MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bambusi D (1999) On the Darboux theorem for weak symplectic manifolds. Proc Amer Math Soc 127(11):3383–3391MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bambusi D (2003) Birkhoff normal form for some nonlinear PDEs. Comm Math Phys 234:253–283MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bambusi D (2005) Galerkin averaging method and Poincaré normal form for some quasilinear PDEs. Ann Sc Norm Super Pisa Cl Sci 4(5):669–702MathSciNetMATHGoogle Scholar
  6. 6.
    Bambusi D (2008) A Birkhoff normal form theorem for some semilinear PDEs. In: Craig W (ed) Hamiltonian dynamical systems and applications. SpringerGoogle Scholar
  7. 7.
    Bambusi D, Carati A, Penati T (2007) On the relevance of boundary conditions for the FPU paradox. Preprint Instit Lombardo Accad Sci Lett Rend A (to appear)Google Scholar
  8. 8.
    Bambusi D, Carati A, Ponno A (2002) The nonlinear Schrödinger equation as a resonant normal form. DCDS‑B 2:109–128MathSciNetMATHGoogle Scholar
  9. 9.
    Bambusi D, Delort JM, Grébert B, Szeftel J (2007) Almost global existence for Hamiltonian semi‐linear Klein–Gordon equations with small Cauchy data on Zoll manifolds. Comm Pure Appl Math 60(11):1665–1690Google Scholar
  10. 10.
    Bambusi D, Giorgilli A (1993) Exponential stability of states close to resonance in infinite‐dimensional Hamiltonian systems. J Statist Phys 71(3–4):569–606MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bambusi D, Grebert B (2003) Forme normale pour NLS en dimension quelconque. Compt Rendu Acad Sci Paris 337:409–414MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Bambusi D, Grébert B (2006) Birkhoff normal form for partial differential equations with tame modulus. Duke Math J 135(3):507–567Google Scholar
  13. 13.
    Bambusi D, Muraro D, Penati T (2008) Numerical studies on boundary effects on the FPU paradox. Phys Lett A 372(12):2039–2042MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Bambusi D, Nekhoroshev NN (1998) A property of exponential stability in the nonlinear wave equation close to main linear mode. Phys D 122:73–104MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Bambusi D, Nekhoroshev NN (2002) Long time stability in perturbations of completely resonant PDE's, Symmetry and perturbation theory. Acta Appl Math 70(1–3):1–22MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Bambusi D, Paleari S (2001) Families of periodic orbits for resonant PDE's. J Nonlinear Sci 11:69–87MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Bambusi D, Ponno A (2006) On metastability in FPU. Comm Math Phys 264(2):539–561MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Bambusi D, Sacchetti A (2007) Exponential times in the one‐dimensional Gross–Petaevskii equation with multiple well potential. Commun Math Phys 234(2):136Google Scholar
  19. 19.
    Berti M, Bolle P (2003) Periodic solutions of nonlinear wave equations with general nonlinearities. Commun Math Phys 243:315–328MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Berti M, Bolle P (2006) Cantor families of periodic solutions for completely resonant nonlinear wave equations. Duke Math J 134(2):359–419MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Bourgain J (1996) Construction of approximative and almost‐periodic solutions of perturbed linear Schrödinger and wave equations. Geom Funct Anal 6:201–230MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Bourgain J (1996) On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE. Int Math Res Not 6:277–304CrossRefGoogle Scholar
  23. 23.
    Bourgain J (1997) On growth in time of Sobolev norms of smooth solutions of nonlinear Schrödinger equations in \({R\sp D}\). J Anal Math 72:299–310MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Bourgain J (1998) Quasi‐periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equation. Ann Math 148:363–439MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Bourgain J (2000) On diffusion in high‐dimensional Hamiltonian systems and PDE. J Anal Math 80:1–35MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Bourgain J (2005) Green's function estimates for lattice Schrödinger operators and applications. In: Annals of Mathematics Studies, vol 158. Princeton University Press, PrincetonGoogle Scholar
  27. 27.
    Bourgain J (2005) On invariant tori of full dimension for 1D periodic NLS. J Funct Anal 229(1):62–94MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Bourgain J, Kaloshin V (2005) On diffusion in high‐dimensional Hamiltonian systems. J Funct Anal 229(1):1–61MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Chernoff PR, Marsden JE (1974) Properties of infinite dimensional Hamiltonian systems In: Lecture Notes in Mathematics, vol 425. Springer, BerlinMATHGoogle Scholar
  30. 30.
    Cohen D, Hairer E, Lubich C (2008) Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions. Arch Ration Mech Anal 187(2)341–368MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Craig W (1996) Birkhoff normal forms for water waves In: Mathematical problems in the theory of water waves (Luminy, 1995), Contemp Math, vol 200. Amer Math Soc, Providence, RI, pp 57–74CrossRefGoogle Scholar
  32. 32.
    Craig W (2006) Surface water waves and tsunamis. J Dyn Differ Equ 18(3):525–549MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Craig W, Guyenne P, Kalisch H (2005) Hamiltonian long-wave expansions for free surfaces and interfaces. Comm Pure Appl Math 58(12):1587–1641MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Craig W, Wayne CE (1993) Newton's method and periodic solutions of nonlinear wave equations. Comm Pure Appl Math 46:1409–1498MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Dell'Antonio GF (1989) Fine tuning of resonances and periodic solutions of Hamiltonian systems near equilibrium. Comm Math Phys 120(4):529–546MathSciNetCrossRefGoogle Scholar
  36. 36.
    Delort JM, Szeftel J (2004) Long-time existence for small data nonlinear Klein–Gordon equations on tori and spheres. Int Math Res Not 37:1897–1966MathSciNetCrossRefGoogle Scholar
  37. 37.
    Delort J-M, Szeftel J (2006) Long-time existence for semi‐linear Klein–Gordon equations with small Cauchy data on Zoll manifolds. Amer J Math 128(5):1187–1218MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Dyachenko AI, Zakharov VE (1994) Is free‐surface hydrodynamics an integrable system? Phys Lett A 190:144–148MathSciNetCrossRefGoogle Scholar
  39. 39.
    Eliasson HL, Kuksin SB (2006) KAM for non‐linear Schroedinger equation. Ann of Math. Preprint (to appear)Google Scholar
  40. 40.
    Fassò F (1990) Lie series method for vector fields and Hamiltonian perturbation theory. Z Angew Math Phys 41(6):843–864Google Scholar
  41. 41.
    Foias C, Saut JC (1987) Linearization and normal form of the Navier–Stokes equations with potential forces. Ann Inst H Poincaré Anal Non Linéaire 4:1–47Google Scholar
  42. 42.
    Gentile G, Mastropietro V, Procesi M (2005) Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions. Comm Math Phys 256(2):437–490MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Grébert B (2007) Birkhoff normal form and Hamiltonian PDES. Partial differential equations and applications, 1–46 Sémin Congr, 15 Soc Math France, ParisGoogle Scholar
  44. 44.
    Hairer E, Lubich C (2006) Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equationsGoogle Scholar
  45. 45.
    Kappeler T, Pöschel J (2003) KdV & KAM. Springer, BerlinGoogle Scholar
  46. 46.
    Klainerman S (1983) On almost global solutions to quasilinear wave equations in three space dimensions. Comm Pure Appl Math 36:325–344MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Krol MS (1989) On Galerkin–averaging method for weakly nonlinear wave equations. Math Meth Appl Sci 11:649–664MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Kuksin SB (1987) Hamiltonian perturbations of infinite‐dimensional linear systems with an imaginary spectrum. Funct Anal Appl 21:192–205MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Kuksin SB (1993) Nearly integrable infinite‐dimensional Hamiltonian systems. Springer, BerlinMATHGoogle Scholar
  50. 50.
    Kuksin SB, Pöschel J (1996) Invariant Cantor manifolds of quasi‐periodic oscillations for a nonlinear Schrödinger equation. Ann Math 143:149–179Google Scholar
  51. 51.
    Lidskij BV, Shulman EI (1988) Periodic solutions of the equation \({u_{tt}-u_{xx}+u^3=0}\). Funct Anal Appl 22:332–333MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Marsden J (1972) Darboux's theorem fails for weak symplectic forms. Proc Amer Math Soc 32:590–592MathSciNetMATHGoogle Scholar
  53. 53.
    Nekhoroshev NN (1977) Exponential estimate of the stability of near integrable Hamiltonian systems. Russ Math Surv 32(6):1–65MATHCrossRefGoogle Scholar
  54. 54.
    Nikolenko NV (1986) The method of Poincaré normal form in problems of integrability of equations of evolution type. Russ Math Surv 41:63–114MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Paleari S, Bambusi D, Cacciatori S (2001) Normal form and exponential stability for some nonlinear string equations. ZAMP 52:1033–1052MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Pals H (1996) The Galerkin–averaging method for the Klein–Gordon equation in two space dimensions. Nonlinear Anal TMA 27:841–856MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Pöschel J (2002) On the construction of almost‐periodic solutions for a nonlinear Schrödinger equation. Ergod Th Dyn Syst 22:1–22Google Scholar
  58. 58.
    Soffer A, Weinstein MI (1999) Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent Math 136(1):9–74MathSciNetCrossRefGoogle Scholar
  59. 59.
    Stroucken ACJ, Verhulst F (1987) The Galerkin–averaging method for nonlinear, undamped continuous systems. Math Meth Appl Sci 335:520–549MathSciNetCrossRefGoogle Scholar
  60. 60.
    Wayne CE (1990) Periodic and quasi‐periodic solutions of nonlinear wave equations via KAM theory. Comm Math Phys 127:479–528MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Weinstein A (1969) Symplectic structures on Banach manifolds. Bull Amer Math Soc 75:1040–1041MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Zakharov VE (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. Appl Mech Tech Phys 2:190–194Google Scholar
  63. 63.
    Zehnder E (1978) C L Siegel's linearization theorem in infinite dimensions. Manuscripta Math 23:363–371MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Dario Bambusi
    • 1
  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItalia