Abstract
We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems Δu=ɛf x (u, y), whereu ∶y ε ℝM →u(y) ε ℝN,f=f(x, y) is a real-analytic periodic function and ɛ is a small parameter. Kolmogorov's theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusT M+1. In the autonomous case,f=f(x), the above result holds for any ɛ.
Similar content being viewed by others
References
Encyclopaedia of Math. Sciences, Vol. 3,Dynamical Systems, V. I. Arnold (ed.), Springer-Verlag, Berlin 1988.
Bruno, A. D.,Convergence of transformations of differential equations to normal form, Dokl. Akad. Nauk SSR165, 987–989 (1965);Analytic form of differential equations, Trans. Moscow Math. Soc.25, 131–288 (1971) and26, 199–239 (1972).
Bollobas, B.,Graph Theory, Springer-Verlag, Berlin 1979.
Chierchia, L.,Birth and death of invariant tori for Hamiltonian systems, inThe Geometry of Hamiltonian Systems, T. Ratiu (ed.), Springer-Verlag, Berlin 1991.
Chierchia, L. and Falcolini, C.,A direct proof of a theorem by Kolmogorov in Hamiltonian systems, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. IV,XXI, Fasc.4, 541–593 (1994).
Chierchia, L. and Zehnder, E.,Asymptotic expansions of quasiperiodic solutions. Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. IV,XVI, Fasc.2, 245–258 (1989).
Eliasson, L. H.,Absolutely convergent series expansions for quasi periodic motions, Reports Department of Math., Univ of Stockholm, Sweden, No. 2, 1–31 (1988);Generalization of an estimate of small divisors by Siegel, inAnalysis, et cetera, P. H. Rabinowitz and E. Zehnder (eds.), Academic Press, New York 1990.
Gallavotti, G.,Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable hamiltonian systems. A review. Rev. on Math. Phys.6, 343–411 (1994).
Gallavotti, G. and Gentile, G.,Majorant series convergence for twistless KAM tori. Ergod. Th. & Dynam. Sys.15, 857–869 (1995).
Goulden, I. P. and Jackson, D. M.,Combinatorial Enumeration, Wiley Interscience Series in Discrete Math. 1983.
John, F.,Partial Differential Equations, Springer-Verlag, Berlin 1982.
Moser, J.,Minimal solutions of variational problems. Ann. Inst. Henri Poincaré (Anal. Nonlinéaire)3, 229–262 (1986).
Moser, J.,Minimal foliations on a torus. Four lectures at CIME Conf. on Topics in Calculus of Variations. M. Giaquinta (ed.), Springer Lect. Notes in Maths.1365, (1987).
Moser, J.,A stability theorem for minimal foliations on a torus. Ergod. Th. & Dynam. Sys.8*, 251–281 (1988).
Moser, J.,Quasi-periodic solutions of nonlinear elliptic partial differential equations. Bol. Soc. Brasil. Mat. (N.S.)20, 29–45 (1989).
Siegel, C. L.,Iterations of analytic functions, Annals of Math.43, No. 4, 607–612 (1942).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chierchia, L., Falcolini, C. A note on quasi-periodic solutions of some elliptic systems. Z. angew. Math. Phys. 47, 210–220 (1996). https://doi.org/10.1007/BF00916825
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00916825