Mathematics of Complexity and Dynamical Systems

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

Perturbation Theory for PDEs

  • Dario Bambusi
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-1806-1_84

Article Outline

Glossary

Definition of the Subject

Introduction

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

Bibliography

Keywords

Manifold 
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Bibliography

  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