Results on Normal Forms for FPU Chains

In this paper we prove, among other results, that near the equilibirum position, any periodic FPU chain with an odd number N of particles admits a Birkhoff normal form up to order 4, whereas any periodic FPU chain with N even admits a resonant normal form up to order 4. This resonant normal form of order 4 turns out to be completely integrable. Further, for N odd, we obtain an explicit formula of the Hessian of its Hamiltonian at the fixed point. Supported in part by the Swiss National Science Foundation. Supported in part by the Swiss National Science Foundation, the programme SPECT and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652).     

Energy localization and transport in two-dimensional Fermi–Pasta–Ulam lattices

We investigate energy localization and transport in the form of discrete breathers and their movability in two-dimensional Fermi–Pasta–Ulam(FPU) lattices. We study the dynamics of the two-dimensional Fermi–Pasta–Ulam(FPU) lattice, incorporating the complicated effects of geometry, long-range interactions as well as nonlinear dispersion. We obtain several exact discrete breather(DB) solutions, such as the odd-parity and even-parity DBs, compact-like DBs and moving DBs for various geometries of the two-dimensional FPU chain. We show that DBs also exist in the same lattice in presence of next-nearest neighbour interaction. Large-amplitude exact subsonic travelling kink-soliton solutions are obtained in such a periodic chain in presence of long-range nonlinear dispersive interaction in the long-wavelength and weakly nonlinear limit. Such a two-dimensional FPU lattice admits finite amplitude nonlinear sinusoidal wave (NSW) solutions with short commensurate as well as incommensurate wavelengths for different geometries of the chain. The usefulness of these solutions for energy localization and transport in various physical systems are discussed.     

Symmetries of some classes of dynamical systems

In this paper a three-dimensional system with five parameters is considered. For some particular values of these parameters, one finds known dynamical systems. The purpose of this work is to study some symmetries of the considered system, such as Lie-point symmetries, conformal symmetries, master symmetries and variational symmetries. In order to present these symmetries we give constants of motion. Using Lie group theory, Hamiltonian and bi-Hamiltonian structures are given. Also, symplectic realizations of Hamiltonian structures are presented. We have generalized some known results and we have established other new results. Our unitary presentation allows the study of these classes of dynamical systems from other points of view, e.g. stability problems, existence of periodic orbits, homoclinic and heteroclinic orbits.     

Spin Chain Models with Spectral Curves from M Theory

We construct the integrable model corresponding to the ?= 2 supersymmetric SU(N) gauge theory with matter in the antisymmetric representation, using the spectral curve found by Landsteiner and Lopez through M Theory. The model turns out to be the Hamiltonian reduction of a N+2 periodic spin chain model, which is Hamiltonian with respect to the universal symplectic form we had constructed earlier for general soliton equations in the Lax or Zakharov–Shabat representation. Received: 22 December 1999 / Accepted: 3 March 2000     

Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications

The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and only if the corresponding Lie algebra of symmetries is abelian. The theorem on symmetries has applications to the characterization problem, to the integrable hierarchies problem, to the necessary conditions for the strong dynamical compatibility problem, and to the problem on master symmetries. The invariant necessary conditions for the non-degenerate C-integrability in Kolmogorov's sense of a given dynamical system V are derived. It is proved that the C-integrable Hamiltonian system is non-degenerate in the iso-energetic sense if and only if the corresponding Lie algebra of the iso-energetic conformal symmetries is abelian. An extended concept of integrability of Hamiltonian systems on the symplectic manifolds M ⁿ , n= 2k, is introduced. The concept of integrability describes the Hamiltonian systems that have quasi-periodic dynamics on tori or on toroidal cylinders of an arbitrary dimension . This concept includes, as a particular case, all Hamiltonian systems that are integrable in Liouville's classical sense, for which . The A-B-C-cohomologies are introduced for dynamical systems on smooth manifolds. Received: 16 January 1996 / Accepted: 3 July 1996     

Reduction of the semisimple 1:1 resonance

The method of “averaging” is often used in Hamiltonian systems of two degrees of freedom to find periodic orbits. Such periodic orbits can be reconstructed from the critical points of an associated “reduced” Hamiltonian on a “reduced space”. This paper details the construction of the reduced space and the reduced Hamiltonian for the semisimple 1:1 resonance case. The reduced space will be a 2-sphere in R³, and the reduced differential equations will be Euler's equations restricted to this sphere. The orbit projection from the energy surface in phase space to this sphere will be the Hopf map. The results of the paper are related to problems in physics on “degeneracies” due to symmetries of classical two-dimensional harmonic oscillators and their quantum analogues for the hydrogen atom.     

Modulational instability in isolated and driven Fermi–Pasta–Ulam lattices

We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi–Pasta–Ulam (FPU) lattices. The growth of the instability is followed by a process of relaxation to equipartition, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, while in the original FPU problem low frequency excitations of the lattice were considered. This relaxation process leads to the formation of chaotic breathers in both one and two space dimensions. The system then relaxes to energy equipartition, on time scales that increase as the energy density is decreased. We supplement this study by considering the nonconservative case, where the FPU lattice is homogeneously driven at high frequencies. Standing and travelling nonlinear waves and solitonic patterns are detected in this case. Finally we investigate the dynamics of the FPU chain when one end is driven at a frequency located above the zone boundary. We show that this excitation stimulates nonlinear bandgap transmission effects.     

Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand’s theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems. The demonstration relies on Perlick’s classification of Bertrand spacetimes and on the construction of a suitable, globally defined generalization of the Runge–Lenz vector.     

KAM Theory in Configuration Space and Cancellations in the Lindstedt Series

The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for any quasi-periodic solution with Diophantine frequency vector converges. If one studies the Lindstedt series by following a perturbation theory approach, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where the KAM theorem is usually formulated by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser’s modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. In this paper, we revisit Moser’s theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates, and the interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates.     

Symmetries and Boundary Conditions with a Twist

Interest in finite-size systems has risen in the last decades, due to the focus on nanotechnological applications and because they are convenient for numerical treatment that can subsequently be extrapolated to infinite lattices. Independently of the envisioned application, special attention must be given to boundary condition, which may or may not preserve the symmetry of the infinite lattice. Here, we present a detailed study of the compatibility between boundary conditions and conservation laws. The conflict between open boundary conditions and momentum conservation is well understood, but we examine other symmetries, as well: we discuss gauge invariance, inversion, spin, and particle-hole symmetry and their compatibility with open, periodic, and twisted boundary conditions. In the interest of clarity, we develop the reasoning in the framework of the one-dimensional half-filled Hubbard model, whose Hamiltonian displays a variety of symmetries. Our discussion includes analytical and numerical results. Our analytical survey shows that, as a rule, boundary conditions break one or more symmetries of the infinite-lattice Hamiltonian. The exception is twisted boundary condition with the special torsion Θ = πL/2, where L is the lattice size. Our numerical results for the ground-state energy at half-filling and the energy gap for L = 2–7 show how the breaking of symmetry affects the convergence to the L → ∞ limit. We compare the computed energies and gaps with the exact results for the infinite lattice drawn from the Bethe-Ansatz solution. The deviations are boundary-condition dependent. The special torsion yields more rapid convergence than open or periodic boundary conditions. For sizes as small as L = 7, the numerical results for twisted condition are very close to the L → ∞ limit. We also discuss the ground-state electronic density and magnetization at half filling under the three boundary conditions.     

Hamiltonian Dynamics for Einstein’s Action in <Emphasis Type="Italic">G</Emphasis>→0 Limit

The Hamiltonian analysis for the Einstein’s action in G→0 limit is performed. Considering the original configuration space without involve the usual ADM variables we show that the version G→0 for Einstein’s action is devoid of physical degrees of freedom. In addition, we will identify the relevant symmetries of the theory such as the extended action, the extended Hamiltonian, the gauge transformations and the algebra of the constraints. As complement part of this work, we develop the covariant canonical formalism where will be constructed a closed and gauge invariant symplectic form. In particular, using the geometric form we will obtain by means of other way the same symmetries that we found using the Hamiltonian analysis.     

Lie symmetry, discrete symmetry and supersymmetry of the Pauli Hamiltonian

Starting from the full group of symmetries of a system we select a discrete subset of transformations which allows to introduce the Clifford algebra of operators generating new supercharges of extended supersymmetry. The system defined by the Pauli Hamiltonian is discussed. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001 Partially supported by the KBN-Grant # 5 P03B056 20.     

q-Breathers and the Fermi-Pasta-Ulam problem

The Fermi-Pasta-Ulam (FPU) paradox consists of the non-equipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here q-breathers (QB). They are characterized by time periodicity, exponential localization in the q-space of normal modes and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.     

Directed transient long-range transport in a slowly driven Hamiltonian system of interacting particles

We study the Hamiltonian dynamics of a one-dimensional chain of linearly coupled particles in a spatially periodic potential which is subjected to a time-periodic mono-frequency external field. The average over time and space of the related force vanishes and hence, the system is effectively without bias which excludes any ratchet effect. We pay special attention to the escape of the entire chain when initially all of its units are distributed in a potential well. Moreover for an escaping chain we explore the possibility of the successive generation of a directed flow based on large accelerations. We find that for adiabatic slope-modulations due to the ac-field transient long-range transport dynamics arises whose direction is governed by the initial phase of the modulation. Most strikingly, that for the driven many particle Hamiltonian system directed collective motion is observed provides evidence for the existence of families of transporting invariant tori confining orbits in ballistic channels in the high-dimensional phase spaces.     

The Fermi-Pasta-Ulam recurrence and related phenomena for 1<Emphasis Type="Italic">D</Emphasis> shallow-water waves in a finite basin

Different regimes of the Fermi-Pasta-Ulam (FPU) recurrence are simulated numerically for fully nonlinear “one-dimensional” potential water waves in a finite-depth flume between two vertical walls. In such systems, the FPU recurrence is closely related to the dynamics of coherent structures approximately corresponding to solitons of the integrable Boussinesq system. A simplest periodic solution of the Boussinesq model, describing a single soliton between the walls, is presented in analytic form in terms of the elliptic Jacobi functions. In the numerical experiments, it is observed that depending on the number of solitons in the flume and their parameters, the FPU recurrence can occur in a simple or complicated manner, or be practically absent. For comparison, the nonlinear dynamics of potential water waves over nonuniform beds is simulated, with initial states taken in the form of several pairs of colliding solitons. With a mild-slope bed profile, a typical phenomenon in the course of evolution is the appearance of relatively high (rogue) waves, while for random, relatively short-correlated bed profiles it is either the appearance of tall waves or the formation of sharp crests at moderate-height waves.     

Acceleration-enlarged symmetries in nonrelativistic space-time with a cosmological constant

By considering the nonrelativistic limit of de Sitter geometry one obtains the nonrelativistic space-time with a cosmological constant and Newton–Hooke (NH) symmetries. We show that the NH symmetry algebra can be enlarged by the addition of the constant acceleration generators and endowed with central extensions (one in any dimension (D) and three in D=(2+1)). We present a classical Lagrangian and Hamiltonian framework for constructing models quasi-invariant under enlarged NH symmetries that depend on three parameters described by three nonvanishing central charges. The Hamiltonian dynamics then splits into external and internal sectors with new noncommutative structures of external and internal phase spaces. We show that in the limit of vanishing cosmological constant the system reduces to the one, which possesses acceleration-enlarged Galilean symmetries.     

Homotopy Classes for Stable Periodic and Chaotic¶Patterns in Fourth-Order Hamiltonian Systems

We investigate periodic and chaotic solutions of Hamiltonian systems in ℝ⁴ which arise in the study of stationary solutions of a class of bistable evolution equations. Under very mild hypotheses, variational techniques are used to show that, in the presence of two saddle-focus equilibria, minimizing solutions respect the topology of the configuration plane punctured at these points. By considering curves in appropriate covering spaces of this doubly punctured plane, we prove that minimizers of every homotopy type exist and characterize their topological properties. Received: 6 April 1999 / Accepted: 2 May 2000     

CP, T and/or CPT violations in the K^0 - \overline K^0 system

The possible violation of the CP, T and/or CPT symmetries in the system is studied from a phenomenological point of view. With this aim, we first introduce parameters which represent the violation of these symmetries in the mixing parameters and decay amplitudes in a convenient and well-defined way and, treating these parameters as small, derive formulas which relate them to the experimentally measured quantities. We then perform numerical analyses, with the aid of the Bell–Steinberger relation, to derive constraints on these symmetry-violating parameters, firstly paying particular attention to the results reported by the KTeV Collaboration and the NA48 Collaboration, and then with the results reported by the CPLEAR Collaboration as well taken into account. A case study, in which either CPT symmetry orT symmetry is assumed, is also carried out. It is demonstrated that the CP andT symmetries are violated definitively at the level of in decays and presumably at the level of in the mixing, and that the Bell–Steinberger relation helps us to establish that CP andT violations are definitively present in mixing and to testCPT symmetry to a level of . Received: 14 February 2000 / Revised version: 24 April 2000 / Published online: 8 September 2000     

Droplet States in the XXZ Heisenberg Chain

We consider the ground states of the ferromagnetic XXZ chain with spin up boundary conditions. The ground state of this model, restricted to a sector with a fixed number of down spins, describes a droplet of down spins in an environment of up spins. We find the exact energy and the states that describe these droplets in the limit of an infinite number of down spins. We prove that there is a gap in the spectrum above the droplet states. As the XXZ Hamiltonian has a gap above the fully magnetized ground states as well, this means that the droplet states (for sufficiently large droplets) form an isolated band. The width of this band tends to zero in the limit of infinitely large droplets. We also prove the analogous results for finite chains with periodic boundary conditions and for the infinite chain. Received: 5 September 2000 / Accepted: 8 December 2000     

Continued Fractions and the d-Dimensional Gauss Transformation

In this paper we study a multidimensional continued fraction algorithm which is related to the Modified Jacobi–Perron algorithm considered by Podsypanin and Schweiger. We demonstrate that this algorithm has many important properties which are natural generalisations of properties of one-dimensional continued fractions. For this reason, we call the transformation associated to the algorithm the d-dimensional Gauss transformation. We construct a coordinate system for the natural extension which reveals its symmetries and allows one to give an explicit formula for the density of its invariant measure. We also discuss the ergodic properties of this invariant measure. Received: 4 February 2000 / Accepted: 23 May 2000