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# Nearly integrable infinite-dimensional Hamiltonian systems

Written in English

## Subjects:

• Hamiltonian systems.,
• Schrödinger equation.

Edition Notes

Includes bibliographical references (p. 96-100) and index.

## Book details

Classifications The Physical Object Statement Sergej B. Kuksin. Series Lecture notes in mathematics ;, 1556, Lecture notes in mathematics (Springer-Verlag) ;, 1556. LC Classifications QA614.83 .K85 1993 Pagination xxvii, 101 p. ; Number of Pages 101 Open Library OL1426115M ISBN 10 0387571612 LC Control Number 93037172

: Nearly Integrable Infinite-Dimensional Hamiltonian Systems (Lecture Notes in Mathematics) (): Kuksin, Sergej B.: BooksCited by:   [Kuk93] Kuksin, S. B., Nearly Integrable Infinite Dimensional Hamiltonian Systems, Lecture Notes in Mathematics, (Springer, Berlin, ).

[Mel65] Mel’nikov, V. K., ‘ On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function ’, Soviet Math. Dokl. 6 (), – Author: Joackim Bernier, Erwan Faou, Benoît Grébert. Nearly Integrable Infinite-Dimensional Hamiltonian Systems (Lecture Notes in Mathematics series) by Sergej B.

Kuksin. The book is devoted to partial differential equations ofHamiltonian form, close to integrable equations. Nearly Integrable Infinite-Dimensional Hamiltonian Systems. Authors; Sergej B. Kuksin; Book. The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations.

For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly. The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations. For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly persist in the perturbed one.

The theorem. Nearly integrable infinite-dimensional Hamiltonian systems. The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations.

For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly persist in the perturbed one.

Author: Sergej B Kuksin. Periodic solutions of nearly integrable Hamiltonian systems bifurcating from infinite-dimensional tori Dedicated to Shair Ahmad, on the occasion of his 85th birthday Author links open overlay Nearly integrable infinite-dimensional Hamiltonian systems book Alessandro Fonda a Giuliano Klun b Andrea Sfecci aAuthor: Alessandro Fonda, Giuliano Klun, Andrea Sfecci.

To the best of my knowledge, the complete understanding of what is an integrable system for the case of three (3D) or more independent variables is still missing. In particular, for the case of three independent variables (a.k.a.

3D or (2+1)D) the overwhelming majority of examples are generalizations of the systems with two independent variables.

We prove the existence of periodic solutions of some infinite-dimensional nearly integrable Hamiltonian systems, bifurcating from infinite-dimensional tori, by the use of a Nearly integrable infinite-dimensional Hamiltonian systems book of the Poincaré–Birkhoff Theorem.

©htsreserved. Introduction. Sergej B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Mathematics, vol.Springer-Verlag, Berlin, MR [22]. Once again KAM theory is committed in the context of nearly integrable Hamiltonian systems.

While elliptic and hyperbolic tori determine the distribution of maximal invariant tori, they themselves form n-parameterwithout the need for untypical conditions or external parameters, torus bifurcations of high co-dimension may be found in a single given Hamiltonian : Springer-Verlag Berlin Heidelberg.

Integrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties, singularites, and topological by: Nearly integrable infinite-dimensional Hamiltonian systems.

[Sergej B Kuksin] -- The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations. For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed. Books. Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in MathematicsSpringer ; Analysis of Hamiltonian PDEs, Clarendon Press, Oxford ; with A.

Shirkyan: Mathematics of two-dimensional turbulence, Cambridge University Press ; References. Download PDF: Sorry, we are unable to provide the full text but you may find it at the following location(s): (external link)Author: Sergei Kuksin.

Nearly Integrable Infinite-Dimensional Hamiltonian Systems. Author: Sergej B. Kuksin In this book we construct the mathematical apparatus of classical mechanics from the very beginning; thus, the reader is not assumed to have any previous knowledge beyond standard courses in analysis (differential and integral calculus, differential.

Perturbation Theory for Nearly Integrable Systems. S.S. Ab dullaev: Construction of Mappings for Hamiltonian Systems and Their Applications, Lect. Notes Phys.21–37 (). A foundational result for integrable systems is the Frobenius theorem, which effectively states that a system is integrable only if it has a foliation; it is completely integrable if it has a foliation by maximal integral manifolds.

1 General dynamical systems. 2 Hamiltonian systems and Liouville integrability. 3 Action-angle variables. Books. Publishing Support.

Login. Kuksin S B Nearly integrable infinite dimensional Hamiltonian systems Lecture Notes in Mathematics vol (Berlin: Xu J, Qiu Q and You J A KAM theorem of degenerate infinite dimensional Hamiltonian systems (II) Sci.

China Ser. A 39 Google ScholarCited by: Get this from a library. Nearly integrable infinite dimensional Hamiltonian systems.

[Sergej B Kuksin]. Real dynamical systems are generally not integrable. In many cases the deviation of a system from an integrable one is small and can be considered as perturbation of the integrable system.

To study perturbed systems special theoretical methods, known as perturbation theory, have been developed.

They are based on the assumption that the Author: Sadrilla S. Abdullaev. to the classical theory of integrable Hamiltonian systems (with an eye to Hamiltonian perturbation theory, which was part of the courses but is not covered in these notes), that is, the Liouville–Arnol’d theorem, the construction of the action–angle coordinates, and the geometric structure of the ﬁbration by the invariant Size: KB.

the system () has an invariant torus with as its frequency. Remark In [] the authors only obtained the existence of invariant tori for Hamiltonian systems (), while the frequency of the persisting invariant tori may have some in [], instead of proving Theorem directly, we are going to deduce it from another KAM theorem, which is concerned with perturbations of a Cited by: 1.

We study the persistence of lower-dimensional tori in Hamiltonian systems of the form, where (x,y,z)∈Tn×Rn×R2m, ε is a small parameter, and M(ω) can be singular. Hou and J. You An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation.

Invariant tori of nearly integrable Hamiltonian systems with degeneracy, Mathematische Zeitschrift, Vol,A KAM Theorem of Degenerate Infinite Dimensional Hamiltonian Systems(I.

For the last years, interest among mathematicians and physicists in infinite-dimensional Hamiltonian systems and Hamiltonian partial differential equations has been growing strongly, and many papers and a number of books have been written on integrable Hamiltonian PDEs.

During the last decade though, the interest has shifted steadily towards non-integrable Hamiltonian PDEs. where is some function of, known as the Hamilton function, or Hamiltonian, of the system (1).A Hamiltonian system is also said to be a canonical system and in the autonomous case (when is not an explicit function of) it may be referred to as a conservative system, since in this case the function (which often has the meaning of energy) is a first integral (i.e.

the energy is conserved during. Instabilities in nearly integrable systems, usually called Arnold diffusion, take place along resonances and by means of a framework of partially hyperbolic invariant objects and their homoclinic and heteroclinic connections.

the Klein-Gordon and the wave equations can be seen as infinite dimensional Hamiltonian systems. Using dynamical. Symplectic theory of completely integrable Hamiltonian systems In memory of Professor J.J. Duistermaat () Alvaro Pelayo and San Vu˜ Ngo´.c Abstract This paper explains the recent developments on the symplectic theory of Hamiltonian completely integrable systems on symplectic 4-manifolds, compact or not.

One fundamental ingredient of. KAM for the nonlinear Schrödinger equation. Pages from Volume [K88] S. Kuksin, "Perturbation of conditionally periodic solutions of infinite-dimensional Hamiltonian systems," Izv.

Akad. Nauk SSSR Ser. Mat., vol. 52, iss S. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, New York: Springer-Verlag, Cited by: § 1. The topology of the state space of an integrable system 28 § 2. Proof of the theorem on non-integrability 30 §3.

Unsolved problems 32 Chapter IV. Non-integrability of nearly integrable Hamiltonian systems 33 § 1. Poincare's method 33 §2. The creation of isolated periodic solutions-an obstruction to integrability 36 § 3.

In this paper, we prove that the net of transition chain is δ-dense for nearly integrable positive definite Hamiltonian systems with 3 degrees of freedom in the cusp-residual generic sense in C r-topology, r ≥ 6. The main ingredients of the proof existed in [CZ, C17a, C17b].Cited by: 2.

Averaging principle for the KdV equation with a small initial value. Kuksin S B Nearly Integrable Infinite-Dimensional Hamiltonian Systems (Lecture Notes in Mathematics vol ) (Berlin: Nekhoroshev N An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems Usp.

Mat. Nauk 32 5–Cited by: 2. We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangean tori by glueing together local KAM conjugacies with help of a partition of unity.

In this way we find a global Whitney smooth conjugacy between a nearly-integrable system and an integrable one. algebraically completely integrable Hamiltonian system.

Following [Ma1] we emphasize the deformation-theoretic construction, in which the Poisson structure on an open subset of the system is obtained via symplectic reduction from the cotangent bundle T∗U Σ,D of the moduli space UΣ,D of stable bundles with a level-D structure.

Acta Math. VolumeNumber 2 (), Perturbation theory for infinite-dimensional integrable systems on the line. A case study. Percy Deift and Xin ZhouCited by: The Mathematical Sciences Research Institute (MSRI), founded inis an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions.

The Institute is located at 17 Gauss Way, on the University of California, Berkeley campus, close to Grizzly Peak, on the. INVARIANT TORI IN NON-DEGENERATE NEARLY INTEGRABLE HAMILTONIAN SYSTEMS Received May Invariant tori for analytic nearly integrable Hamiltonian systems are constructed under rather weak suﬃcient conditions being even necessary in the case of maximal invariant tori.

All small devisors are controlled by a general. For the last years, interest among mathematicians and physicists in infinite-dimensional Hamiltonian systems and Hamiltonian partial differential equations has been growing strongly, and many papers and a number of books have been written on integrable Hamiltonian PDEs.

Nearly integrable Hamiltonian systems, exponentially small splitting of separatrices, Melnikov method, complex matching. The author has been partially supported by the Spanish MCyT/FEDER grant MTM and the Catalan SGR grant SGR MARCEL GUARDIA.

We consider the problem of Arnold's diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known.Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems Physica D (3&4) () T.

Kapitula, J. N. Kutz, and B. Sandstede The Evans function for nonlocal equations Indiana U. Math. J. 53(4) () T. Kapitula and P. Kevrekidis Linear stability of perturbed Hamiltonian systems: theory and a.We prove the existence of periodic solutions of some infinite-dimensional nearly integrable Hamiltonian systems, bifurcating from infinite-dimensional tori, by the use of .

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