@article{ASNSP_2005_5_4_4_669_0, author = {Bambusi, Dario}, title = {Galerkin averaging method and Poincar\'e normal form for some quasilinear PDEs}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 4}, number = {4}, year = {2005}, pages = {669-702}, zbl = {1170.35317}, mrnumber = {2207739}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2005_5_4_4_669_0} }
Bambusi, Dario. Galerkin averaging method and Poincaré normal form for some quasilinear PDEs. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 4, pp. 669-702. http://www.numdam.org/item/ASNSP_2005_5_4_4_669_0/
[Bam03a] An averaging theorem for quasilinear Hamiltonian PDEs, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (2003), 685-712. | MR 2015427 | Zbl 1031.37056
,[Bam03b] Birkhoff normal form for some nonlinear PDEs, Commun. Math. Phys. 234 (2003), 253-285. | MR 1962462 | Zbl 1032.37051
,[Bam03c] Birkhoff normal form for some quasilinear Hamiltonian PDEs, preprint, 2003. | MR 2227839
,[BCP02] The nonlinear Schrödinger equation as a resonant normal form, Discrete Contin. Dyn. Syst. Ser. B 2 (2002), 109-128. | MR 1877043 | Zbl 1068.37056
, and ,[BG03] Forme normale pour NLS en dimension quelconque, C. R. Math. Acad. Sci. Paris 337 (2003), 409-414. | MR 2015085 | Zbl 1030.35143
and ,[BG04] Birkhoff normal form for PDEs with tame modulus, Duke Math. J. (2004), to appear. | MR 2272975 | Zbl 1110.37057
and ,[BN98] A property of exponential stability in nonlinear wave equations near the fundamental linear mode, Phys. D 122 (1998), 73-104. | MR 1650123 | Zbl 0937.35010
and ,[Bou96] Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal. 6 (1996), 201-230. | MR 1384610 | Zbl 0872.35007
,[Cra96] Birkhoff normal forms for water waves, In: “Mathematical problems in the theory of water waves” (Luminy, 1995), Vol. 200, Contemp. Math., Amer. Math. Soc., Providence, RI, 1996, 57-74. | MR 1410500 | Zbl 0953.76009
,[CS93] Numerical simulation of gravity waves, J. Comput. Phys. 108 (1993), 73-83. | MR 1239970 | Zbl 0778.76072
and[CSS97] The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 615-667. | Numdam | MR 1470784 | Zbl 0892.76008
, and ,[CW95] An integrable normal form for water waves in infinite depth, Phys. D 84 (1995), 513-531. | MR 1336546 | Zbl 0883.35092
and ,[DZ94] Is free-surface hydrodynamics an integrable system? Phys. Lett. A 190 (1994), 144-148. | MR 1283779 | Zbl 0961.76511
and ,[FS87] Linearization and normal form of the Navier-Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 1-47. | Numdam | MR 877990 | Zbl 0635.35075
and ,[FS91] Asymptotic integration of Navier-Stokes equations with potential forces. I, Indiana Univ. Math. J. 40 (1991), 305-320. | MR 1101233 | Zbl 0739.35066
and ,[GP88] Estimates for normal forms of differential equations near an equilibrium point, Z. Angew. Math. Phys. 39 (1988), 713-732. | MR 963640 | Zbl 0685.58028
and ,[Kat75] Quasi-linear equations of evolution, with applications to partial differential equations, In: “Spectral Theory and Differential Equations” (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, 25-70. | MR 407477 | Zbl 0315.35077
,[Kro89] On a Galerkin-averaging method for weakly nonlinear wave equations, Math. Methods Appl. Sci. 11 (1989), 649-664. | MR 1011811 | Zbl 0707.35095
,[Mat01] Time-averaging under fast periodic forcing of parabolic partial differential equations: exponential estimates, J. Differential Equations 174 (2001), 133-180. | MR 1844527 | Zbl 1023.35055
,[MS03] Exponential averaging for Hamiltonian evolution equations, Trans. Amer. Math. Soc. 355 (2003), 747-773. | MR 1932724 | Zbl 1008.37043
and ,[Nik86] The method of Poincaré normal forms in problems of integrability of equations of evolution type, Uspekhi Mat. Nauk 41 (1986), 109-152, 263. | MR 878327 | Zbl 0632.35026
,[Pal96] The Galerkin-averaging method for the Klein-Gordon equation in two space dimensions, Nonlinear Anal. 27 (1996), 841-856. | MR 1402170 | Zbl 0861.35101
,[PB05] Korteweg-de Vries equation and energy sharing in Fermi-Pasta-Ulam, Chaos 15 (2005), 015107, 5. | MR 2133458 | Zbl 1080.37073
and ,[Sha85] Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), 685-696. | MR 803256 | Zbl 0597.35101
,[SV87] The Galerkin-averaging method for nonlinear, undamped continuous systems, Math. Methods Appl. Sci. 9 (1987), 520-549. | MR 1200364 | Zbl 0638.35057
and ,[SW00] Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model In: “International Conference on Differential Equations”, Vol. 1, 2 (Berlin, 1999), World Sci. Publishing, River Edge, NJ, 2000, 390-404. | MR 1870156 | Zbl 0970.35126
and ,[Zak68] Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2 (1968), 190-194.
[Zeh78] C. L. Siegel's linearization theorem in infinite dimensions, Manuscripta Math. 23 (1977/78), 363-371. | MR 501144 | Zbl 0374.47037
,