Discrete Solitons in Optics

Symplectic maps are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and fluid physics. Twist maps correspond to Hamiltonians for which the velocity is a monotonic function of the canonical momentum. Twist maps have a Lagrangian variational formulation. One-parameter families of twist maps typically exhibit the full range of possible dynamics-from simple or integrable motion to complex or chaotic motion. One class of orbits, the minimizing orbits, can be found throughout this transition; the properties of the minimizing orbits are discussed in detail. Among these orbits are the periodic and quasiperiodic orbits, which form a scaffold in the phase space and constrain the motion of the remaining orbits. The theory of transport deals with the motion of ensembles of trajectories. The variational principle provides an efficient technique for computing the flux escaping from regions bounded by partial barriers formed from minimizing orbits. Unsolved problems in the theory of transport include the explanation for algebraic tails in correlation functions, and its extension to maps of more than two dimensions.

Symplectic maps, variational principles, and transport

In recent years, several authors have studied "minimal" orbits of Hamiltonian systems in two degrees of freedom and of area preserving monotone twist diffeomorphisms. Here, "minimal" means action minimizing. This class of orbits has many interesting properties, as may be seen in the survey article of Bangert [4]. It is natural to ask if there is any generalization of this class of orbits to Hamiltonian systems in more degrees of freedom. In this article, we propose a generalization to periodic Hamiltonian systems in more degrees of freedom. However, we generalize not the notion of minimal orbit, but the closely related notion of minimal measure, which we introduced in [18]. We obtain two basic results here: an existence theorem for minimal measures, and a regularity theorem which asserts that the minimal measures can be expressed as (partially defined) Lipschitz sections of the tangent bundle. In the sort of generalization that we do here, a major difficulty is finding the right setting. The setting which we propose here has two important features: the results are valid for periodic positive definite Lagrangian systems, and the results are formulated in terms of invariant measures. I am indebted to J. Moser for pointing out to me several years ago that periodic positive definite Lagrangian systems in one degree of freedom provide a setting in which it is possible to formulate results which generalize both the author's results [17] (and the closely related results of Aubry and Le Dacron [1]) and the results of Hedlund [12] concerning "class A " geodesics on a Riemannian manifold diffeomorphic to the 2-torus. Indeed, Moser has proved [20] that every twist diffeomorphism is the time one map associated to a suitable periodic positive definite Lagrangian system. Denzler [10] has carried out Moser's program in one degree of freedom. This remark of Moser suggested to me that periodic positive definite Lagrangian systems should provide the right setting in more degrees of freedom. There is some earlier work in the direction of this paper. Bernstein and Katok [6] obtained results concerning periodic orbits near invariant tori, using a variational method related to the variational method of this paper.

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Action minimizing invariant measures for positive definite Lagrangian systems.

We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikov's technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.

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An analytical study of transport, mixing and chaos in an unsteady vortical flow

Wave transmission in nonlinear lattices

The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states

The ground state (at 0K) of a discrete molecular-crystal model for the Peierls chain with classical atoms and non-interacting spinless quantum electrons is calculated numerically as a function of the electron-phonon coupling for an 'irrational' electron concentration. A metal-insulator transition arising from the extinction of the Frohlich conductivity for the incommensurate system is observed when the Peierls gap is only 10% of the total unperturbed bandwidth. Beyond the critical coupling, the Frohlich mode disappears and the electrons are exponentially localised but the lattice distortion remains incommensurate (the polaron lattice). In this region, the Peierls-Nabarro barrier does not vanish and is calculated. The existence of metastable Fermi glasses is also proved, but they are shown to have an energy larger than that of the incommensurate ground state. The observed transition is identified as a transition by breaking of analyticity, which is similar to the one found previously in the Frenkel-Kontorova model. This behaviour is explained as being a consequence of the competition between the Fermi and the lattice wavevectors, which are incommensurate with each other.

http://iopscience.iop.org/article/10.1088/0022-3719/16/24/014/pdf

P.Y. Le Daeron

Papers

The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states

The metal-insulator transitions in the Peierls chain

Metal-insulator transition in incommensurate peierls chains by extinction of the frohlich conductivity