Abstract
In this paper, we consider the persistence of invariant tori in infinite-dimensional Hamiltonian systems
where \(\theta \in \mathbb {T}^\Lambda \), \(I\in \mathbb {R}^\Lambda \), the frequency \({\omega }=(\cdots ,{\omega }_\lambda ,\cdots )_{\lambda \in \Lambda }\) is regarded as parameters varying freely over some subset \(\ell ^\infty (\Lambda ,\mathbb {R})\) of the parameter space \(\mathbb {R}^\Lambda \), \({\omega }=(\cdots ,{\omega }_\lambda ,\cdots )_{\lambda \in \Lambda }\) is a bilateral infinite sequence of rationally independent frequency, in other words, any finite segments of \({\omega }=(\cdots ,{\omega }_\lambda ,\cdots )_{\lambda \in \Lambda }\) are rationally independent.
Similar content being viewed by others
References
Arnold, V.I.: Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Uspekhi Matematicheskikh Nauk 18, 13–40 (1963)
Biasco, L., Massetti, J., Procesi, M.: An abstract Birkhoff normal form theorem and exponential type stability of the 1D NLS. Commun. Math. Phys. 375, 2089–2153 (2020)
Bourgain, J.: On invariant tori of full dimension for 1D periodic NLS. J. Funct. Anal. 229, 62–94 (2005)
Cheng, C.Q., Sun, Y.S.: Existence of KAM tori in degenerate Hamiltonian systems. J. Differ. Equ. 114, 288–335 (1994)
Chierchia, L., Perfetti, P.: Maximal almost-periodic solutions for Lagrangian equations on infinite dimensional tori. In: Seminar on Dynamical Systems. Birkhäuser, Basel (1994)
Chow, S.-N., Li, Y., Yi, Y.: Persistence of invariant tori on submanifolds in Hamiltonian systems. J. Nonlinear Sci. 12, 585–617 (2002)
Eliasson, L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Annali della Scuola Normale Superiore di Pisa 15, 115–147 (1988)
Fröhlich, J., Spencer, T., Wayne, C.E.: Localization in disordered, nonlinear dynamical systems. J. Stat. Phys. 42, 247–274 (1986)
Huang, P., Li, X.: Persistence of invariant tori in integrable Hamiltonian systems under almost periodic perturbations. J. Nonlinear Sci. 28, 1865–1900 (2018)
Kolmogorov, A.N.: On the conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk SSSR 98, 525–530 (1954)
Kuksin, S.B.: Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funct. Anal. Appl. 21(3), 192–205 (1987)
Moser, J.: Convergent series expansions for quasi-periodic motions. Math. Ann. 169, 136–176 (1967)
Montalto, R., Procesi, M.: Linear Schrödinger equation with an almost periodic potential. SIAM J. Math. Anal. 53, 386–434 (2021)
Pöschel, J.: On elliptic lower dimensional tori in Hamiltonian systems. Math. Z. 202, 559–608 (1989)
Pöschel, J.: Small divisors with spatial structure in infinite dimensional Hamiltonian systems. Commun. Math. Phys. 127, 351–393 (1990)
Pöschel, J.: A lecture on the classical KAM theorem. Proc. Symp. Pure Math. 69, 701–732 (2001)
Pöschel, J.: On the construction of almost periodic solutions for a nonlinear Schrödinger equation. Ergod. Theory Dyn. Syst. 22(5), 1537–1549 (2002)
Rüssmann, H.: Nondegeneracy in the perturbation theory of integrable dynamical systems. In: Stochastics Algebra and Analysis in Classical and Quantum Dynamics (Marseille 1988), pp. 211–223. Springer, Dordrecht (1990)
Rüssmann, H.: Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Regul. Chaotic Dyn. 6, 119–204 (2001)
Sevryuk, M.B.: KAM-stable Hamiltonians. J. Dyn. Control Syst. 1, 351–366 (1995)
Sevryuk, M.B.: Partial preservation of frequencies in KAM theory. Nonlinearity 19, 1099–1140 (2006)
Vittot, M., Bellissard, J.: Invariant tori for an infinite lattice of coupled classical rotators. CPT-Marseille, Preprint (1985)
Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127(3), 479–528 (1990)
Wu, Y., Yuan, X.: On the Kolmogorov theorem for some infinite-dimensional Hamiltonian systems of short range. Nonlinear Anal. 202, Paper No. 112120, 34 (2021)
Xu, J., You, J., Qiu, Q.: Invariant tori for nearly integrable Hamiltonian systems with degeneracy. Math. Z. 226, 375–387 (1997)
Xu, J., You, J.: Persistence of the non-twist torus in nearly integrable Hamiltonian systems. Proc. Am. Math. Soc. 138, 2385–2395 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Partially supported by the National Natural Science Foundation of China (11901131), Science and Technology Foundation of Guizhou Province ([2020]1Y006).
Appendix A: Some Useful Lemmas
Appendix A: Some Useful Lemmas
In this appendix, we give some useful lemmas, which are used in proving Theorem 3.1.
Lemma A.1
(Lemma B.1 in [13])
-
(i)
Let \(\mu _1, \mu _2>0\). Then
$$\begin{aligned} \sup \limits _{\begin{array}{c} \ell \in \mathbb {Z}_*^\infty \\ |\ell |_\eta <\infty \end{array}}\prod \limits _{i} (1+\langle i\rangle ^{\mu _1} |\ell _i|^{\mu _2})e^{-\rho |\ell |_\eta }\le \exp \Bigg ({\tau \over {\rho ^{1\over \eta }}}\ln {\tau \over \rho }\Bigg ) \end{aligned}$$for some constant \(\tau =\tau (\eta , \mu _1, \mu _2)>0\).
-
(ii)
Let \(\rho >0\). Then \(\sum \nolimits _{\ell \in \mathbb {Z}_*^\infty }e^{-\rho |\ell |_\eta }\lesssim \exp \Bigg ({\tau \over {\rho ^{1\over \eta }}}\ln {\tau \over \rho }\Bigg )\) for some constant \(\tau =\tau (\eta )>0\).
Lemma A.2
(Cauchy estimates, Lemma 2.7 in [13]) Let \(\sigma ,\rho >0\) and \(u\in \mathcal {H}(\mathbb {T}_{\sigma +\rho }^\infty ,X).\) Then for any \(k\in \mathbb {N}\), the kth differential \(d_\varphi ^k u\) satisfies the estimate
Lemma A.3
(Lemma 10 in [15]) Suppose that for some \(v\ge w\),
Then
for \(0<\rho _0\le \rho <r\le \rho _0-s_0\) and \(0<s\le s_0/2\), where \(\Phi \) denotes the time-1-map of the hamiltonian vectorfield \(X_F\).
Lemma A.4
(Lemma 11 in [15]) Suppose f is real analytic from \({\mathcal {W}}_h\) into \(\mathbb {C}^\Lambda \). If
on \({\mathcal {W}}_h\), then f has a real analytic inverse \(\varphi \) on \({\mathcal {W}}_{h/4}\). Moreover,
on this domain.
Rights and permissions
About this article
Cite this article
Huang, P. Persistence of Invariant Tori in Infinite-Dimensional Hamiltonian Systems. Qual. Theory Dyn. Syst. 21, 50 (2022). https://doi.org/10.1007/s12346-022-00581-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-022-00581-2