Exponential averaging for Hamiltonian evolution equations

We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrödinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.

     

Polynomial entropies for Bott integrable Hamiltonian systems

In this paper, we study the entropy of a Hamiltonian flow in restriction to an energy level where it admits a first integral which is nondegenerate in the sense of Bott. It is easy to see that for such a flow, the topological entropy vanishes. We focus on the polynomial and the weak polynomial entropies h_pol and h _pol ^* . We show that, under natural conditions on the critical levels of the Bott first integral and on the Hamiltonian function H, h _pol ^* ∈ {0, 1} and h_pol ∈ {0, 1, 2}. To prove this result, our main tool is a semi-global desingularization of the Hamiltonian system in the neighborhood of a polycycle.     

Arnold diffusion for nearly integrable Hamiltonian systems

In this paper, we prove that the nearly integrable system of the form ■ admits orbits that pass through any finitely many prescribed small balls on the same energy level H^-1 (E)provided that E > min h, if h is convex, and εP is typical. This settles the Arnold diffusion conjecture for convex systems in the smooth category. We also prove the counterpart of Arnold diffusion for the Riemannian metric perturbation of the flat torus.     

Nonlinear continuous integrable Hamiltonian couplings

Based on a kind of special non-semisimple Lie algebras, a scheme is presented for constructing nonlinear continuous integrable couplings. Variational identities over the corresponding loop algebras are used to furnish Hamiltonian structures for the resulting continuous integrable couplings. The application of the scheme is illustrated by an example of nonlinear continuous integrable Hamiltonian couplings of the AKNS hierarchy of soliton equations.     

Hamiltonian structures for integrable models of field theory

Theoretical and Mathematical Physics -     

Polynomial entropies and integrable Hamiltonian systems

We introduce two numerical conjugacy invariants of dynamical systems — the polynomial entropy and the weak polynomial entropy — which are well-suited for the study of “completely integrable” Hamiltonian systems. These invariants describe the polynomial growth rate of the number of balls (for the usual “dynamical” distances) of covers of the ambient space. We give explicit examples of computation of these polynomial entropies for generic Hamiltonian systems on surfaces.     

Global properties of integrable Hamiltonian systems

This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems. We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems which involve the Chern and Lagrange class. Our approach, which uses simple ideas from differential geometry and algebraic topology, reveals the fundamental role of the integer affine structure on the base space of these bundles. We provide a geometric proof of the classification of Lagrangian bundles with fixed integer affine structure by their Lagrange class.      

Analytic-non-integrability of an integrable analytic Hamiltonian system

We introduce the polynomial Hamiltonian and we prove that the associated Hamiltonian system is Liouville-C^∞-integrable, but fails to be real-analytically integrable in any neighbourhood of an equilibrium point. The proof only uses power series expansions, and is elementary.     

Biasymptotic solutions of perturbed integrable Hamiltonian systems

We prove that small perturbations of a real analytic integrable Hamiltonian system ind degrees of freedom generically have biasymptotic orbits which are obtained as intersections of the stable and unstable manifolds of invariant hyperbolic tori of dimensiond–1. Hence, these solutions will be forward and backward asymptotic to such a torus and not to a periodic solution. The generic condition, which is open and dense, is given by an explicit condition on the averaged perturbation.     

Generic perturbations of linear integrable Hamiltonian systems

In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem that does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is nonresonant is more subtle. Our second result shows that for a generic perturbation the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of ε ^?1) interval of time and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).     

Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.     

Soliton dynamics for near integrable differential-difference equations

In this talk, we consider both the spectral and the direct perturbation methods for studying perturbations of an integrable differential-difference equation, the Toda lattice. Both methods employ the formalism of inverse scattering to represent the corrections in terms of an appropriate basis of squared eigenfunctions.Mathematical Sciences Department, The University of North Carolina at Wilmington, Wilmington, NC 28403 USA. E-mail: herman@seq.cms.uncwil.edu. Translated from Teoreticheskaya i Matermaticheskaya Fizika, Vol. 99, No. 2, pp. 315–321, May, 1994.