Abstract
In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus
where \({G(x,u)=u^4+ O(u^5)}\). Namely, we show that, for generic m, many of the small amplitude invariant finite dimensional tori of the linear equation \({(*)_{G=0}}\), written as the system
persist as invariant tori of the nonlinear equation \({(*)}\), re-written similarly. The persisted tori are filled in with time-quasiperiodic solutions of \({(*)}\). If \({d\ge2}\), then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensional Hamiltonian PDEs behave in a chaotic way.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V.I. Arnold. Mathematical Methods in Classical Mechanics, 3rd edn. Springer, Berlin (2006).
Bambusi D.: Birkhoff normal form for some nonlinear PDEs. Communications in Mathematical Physics 234, 253–283 (2003)
Bambusi D., Grébert B.: Birkhoff normal form for PDE’s with tame modulus. Duke Mathematical Journal 135(3), 507–567 (2006)
Berti M., Bolle P.: Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity 25, 2579–2613 (2012)
Berti M., Bolle P.: Quasi-periodic solutions with Sobolev regularity of NLS on \({\mathbb{T}^d}\) and a multiplicative potential. Journal of the European Mathematical Society 15, 229–286 (2013)
Bobenko A. I., Kuksin S. B.: The nonlinear Klein–Gordon equation on an interval as a perturbed Sine–Gordon equation. Commentarii Mathematici Helvetici 70, 63–112 (1995)
Bourgain J.: Construction of approximative and almost-periodic solutions of perturbed linear Schrödinger and wave equations. GAFA 6, 201–235 (1995)
Bourgain J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Shödinger equation. Annals of Mathematics 148, 363–439 (1998)
J. Bourgain. Green’s function estimates for lattice Schrödinger operators and applications. Annals of Mathematical Studies, Princeton (2004).
Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.: Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Inventiones Mathematicae 181, 31–113 (2010)
W. Craig. Problèmes de Petits Diviseurs dans les Équations aux Dérivées Partielles. Panoramas et Synthèses, Société Mathématique de France (2000).
Eliasson L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Annali della Scoula Normale Superiore di Pisa 15, 115–147 (1988)
L.H Eliasson. Perturbations of Linear Quasi-Periodic Systems. In: Dynamical Systems and Small Divisors (Cetraro, Italy, 1998), 1–60, Lect. Notes Math. 1784, Springer (2002).
L.H Eliasson. Almost reducibility of linear quasi-periodic systems. In: Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), 679–705, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI (2001).
L.H. Eliasson, B. Grébert and S.B. Kuksin. KAM for the non-linear Beam equation 1: small-amplitude solutions. arXiv:1412.2803v3.
L.H. Eliasson, B. Grébert and S.B. Kuksin. KAM for the nonlinear beam equation 2: a normal form theorem. arXiv:1502.02262.
Eliasson L.H., Kuksin S.B.: Infinite Töplitz–Lipschitz matrices and operators. Zeitschrift für angewandte Mathematik und Physik 59, 24–50 (2008)
Eliasson L.H., Kuksin S.B.: On reducibility of Schrödinger equations with quasiperiodic in time potentials. Communications in Mathematical Physics 286(1), 125–135 (2009)
Eliasson L.H., Kuksin S.B.: KAM for the nonlinear Schrödinger equation. Annals of Mathematics 172, 371–435 (2010)
B. Grébert and É. Paturel. KAM for the Klein Gordon equation on \({\mathbb{S}^d}\), Bollettino dellUnione Matematica Italiana 9 (2016), 237–288.
Geng J., Xu X., You J.: An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation. Advances in Mathematics 226, 5361–5402 (2011)
Geng J., You J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Communications in Mathematical Physics 262, 343–372 (2006)
Geng J., You J.: KAM tori for higher dimensional beam equations with constant potentials. Nonlinearity 19, 2405–2423 (2006)
G. H. Hardy and E. M. Wright. An Introduction to the Theory of Number. Oxford University Press, Oxford (2008).
Hörmander L.: Note on Hölder Estimates. The boundary problem of physical geodesy. Archive for Rational Mechanics and Analysis 62, 1–52 (1976)
S. Krantz and H. Parks. A Premier of Real Analytic Functions. Birkhäuser, Basel (2002).
Kuksin S. B.: Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Functional Analysis and Its Applications 21, 192–205 (1987)
S. B. Kuksin. Nearly Integrable Infinite-dimensional Hamiltonian Systems. Lecture Notes in Mathematics, 1556. Springer, Berlin (1993).
S. B. Kuksin. Analysis of Hamiltonian PDEs. Oxford University Press, Oxford (2000).
Kuksin S. B., Pöschel J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Annals of Mathematics 143, 149–179 (1996)
J. Moser and C. L. Siegel. Lectures on Celestial Mechanics. Springer, Berlin (1971).
Pöschel J.: Quasi-periodic solutions for a nonlinear wave equation. Commentarii Mathematici Helvetici 71, 269–296 (1996)
Procesi C., Procesi M.: A normal form of the nonlinear Schrdinger equation with analytic non–linearities. Communications in Mathematical Physics 312, 501–557 (2012)
Procesi C., Procesi M.: A KAM Algorithm for the Resonant Nonlinear Schrödinger Equation. Advances in Mathematics 272, 399–470 (2015)
I. M. Vinogradov. An Introduction to the Theory of Numbers. Pergamon Press, London (1955).
Wang W.-M.: Energy supercritical nonlinear Schrödinger equations: Quasiperiodic solutions. Duke Mathematical Journal 165, 1129–1192 (2016)
You J.: Perturbations of lower dimensional tori for Hamiltonian systems. Journal of Differential Equations 152, 1–29 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Eliasson, L.H., Grébert, B. & Kuksin, S.B. KAM for the nonlinear beam equation. Geom. Funct. Anal. 26, 1588–1715 (2016). https://doi.org/10.1007/s00039-016-0390-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-016-0390-7