Roberto De Leo

A symmetry breaking mechanism for selecting the speed of relativistic solitons

We propose a mechanism for fixing the velocity of relativistic solitons based on the breaking of the Lorentz symmetry of the sine-Gordon (SG) model. The proposal is first elaborated for a molecular chain model as the simple pendulum limit of a double pendulums chain. It is then generalized to a full class of two-dimensional field theories of the sine-Gordon type. From a phenomenological point of view, the mechanism allows one to select the speed of a SG soliton just by tuning elastic couplings constants and kinematical parameters. From a fundamental, field-theoretical point of view we show that the characterizing features of relativistic SG solitons (existence of conserved topological charges and stability) may be still preserved even if the Lorentz symmetry is broken and a soliton of a given speed is selected.

Numerical Analysis of the Novikov Problem of a Normal Metal in a Strong Magnetic Field

We present the results of our numerical exploration of the fractal structure found by S.P. Novikov in an elementary multivalued Poisson dynamical system on the 3-torus coming from the problem of the dependence of magnetoresistance on the direction of the magnetic field in a normal metal.

PROPAGATION OF TWIST SOLITONS IN FULLY INHOMOGENEOUS DNA CHAINS

In the framework of a recently introduced model of DNA torsional dynamics, we argued — on the basis of perturbative considerations — that an inhomogeneous DNA chain could support long-lived soliton-type excitations due to the peculiar geometric structure of DNA and the effect of this on nonlinear torsional dynamics. Here we consider an inhomogeneous version of this model of DNA torsional dynamics, and investigate numerically the propagation of solitons in a DNA chain with a real base sequence (corresponding to the Human Adenovirus 2); this implies inhomogeneities of up to 50% in the base masses and inter-pair interactions. We find that twist solitons propagate for considerable distances (2–10 times their diameters) before stopping due to phonon emission. Our results show that twist solitons may exist in realistic DNA chain models, and on a more general level that solitonic propagation can take place in highly inhomogeneous media. The most relevant feature for general nonlinear dynamics is that we identify the physical mechanisms allowing this behavior and thus the class of models candidate to support long-lived soliton-type excitations in the presence of significant inhomogeneities.

Twist solitons in complex macromolecules: from DNA to polyethylene

Abstract DNA torsion dynamics is essential in the transcription process; simple models for it have been proposed by several authors, in particular Yakushevich (Y model). These are strongly related to models of DNA separation dynamics such as the one first proposed by Peyrard and Bishop (and developed by Dauxois, Barbi, Cocco and Monasson among others), but support topological solitons. We recently developed a “composite” version of the Y model, in which the sugar–phosphate group and the base are described by separate degrees of freedom. This at the same time fits experimental data better than the simple Y model, and shows dynamical phenomena, which are of interest beyond DNA dynamics. Of particular relevance are the mechanism for selecting the speed of solitons by tuning the physical parameters of the non-linear medium and the hierarchal separation of the relevant degrees of freedom in “master” and “slave”. These mechanisms apply not only do DNA, but also to more general macromolecules, as we show concretely by considering polyethylene.

Sine-Gordon solitons, auxiliary fields and singular limit of a double pendulums chain

We consider the continuum version of an elastic chain supporting topological and non-topological degrees of freedom; this generalizes a model for the dynamics of DNA recently proposed and investigated by ourselves. In a certain limit, the non-topological degrees of freedom are frozen, and the model reduces to the sine-Gordon equations and thus supports well-known topological soliton solutions. We consider a (singular) perturbative expansion around this limit and study in particular how the non-topological field assumes the role of an auxiliary field. This provides a more general framework for the slaving of this degree of freedom on the topological one, already observed elsewhere in the context of the mentioned DNA model; in this framework one expects such a phenomenon to arise in a quite large class of field-theoretical models.

Solitons in a double pendulums chain model, and DNA roto-torsional dynamics

Abstract It was first suggested by Englander et al to model the nonlinear dynamics of DNA relevant to the transcription process in terms of a chain of coupled pendulums. In a related paper [4] we argued for the advantages of an extension of this approach based on considering a chain of double pendulums with certain characteristics. Here we study a simplified model of this kind, focusing on its general features and nonlinear travelling wave excitations; in particular, we show that some of the degrees of freedom are actually slaved to others, allowing for an effective reduction of the relevant equations

Topology of Plane Sections of Periodic Polyhedra with an Application to the Truncated Octahedron

The main results of A. Zorich and I. Dynnikov regarding plane sections of periodic surfaces are extended to the piecewise linear case. As an application, the stereographic map of a truncated octahedron, extended to all of ℝ3 by periodicity, is analyzed numerically.The main results of A. Zorich and I. Dynnikov regarding plane sections of periodic surfaces are extended to the piecewise linear case. As an application, the stereographic map of a truncated octahedron, extended to all of ℝ3 by periodicity, is analyzed numerically.

Solvability of the cohomological equation for regular vector fields on the plane

We consider planar vector fields without zeroes ξ and study the image of the associated Lie derivative operators Lξ acting on the space of smooth functions. We show that the cokernel of Lξ is infinite-dimensional as soon as ξ is not topologically conjugate to a constant vector field and that, if the topology of the integral trajectories of ξ is “simple enough” (e.g. if ξ is polynomial) then ξ is transversal to a Hamiltonian foliation. We use this fact to find a large explicit subalgebra of the image of Lξ and to build an embedding of

Composite model for DNA torsion dynamics.

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