Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio

We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt{2}-1$. We show that the Poincar\'e-Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the tranversality of the splitting whose dependence on the perturbation parameter $\varepsilon$ satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.Comment: 17 pages, 2 figure

Dynamics of generalized PT-symmetric dimers with time-periodic gain–loss

A parity-time (PT)-symmetric system with periodically varying-in-time gain and loss modeled by two coupled Schrödinger equations (dimer) is studied. It is shown that the problem can be reduced to a perturbed pendulum-like equation. This is done by finding two constants of motion. Firstly, a generalized problem using Melnikov-type analysis and topological degree arguments is studied for showing the existence of periodic (libration), shift- periodic (rotation), and chaotic solutions. Then these general results are applied to the PT-symmetric dimer. It is interestingly shown that if a sufficient condition is satisfied, then rotation modes, which do not exist in the dimer with constant gain–loss, will persist. An approximate threshold for PT-broken phase corresponding to the disappearance of bounded solutions is also presented. Numerical study is presented accompanying the analytical results

QCD and strongly coupled gauge theories : challenges and perspectives

We highlight the progress, current status, and open challenges of QCD-driven physics, in theory and in experiment. We discuss how the strong interaction is intimately connected to a broad sweep of physical problems, in settings ranging from astrophysics and cosmology to strongly coupled, complex systems in particle and condensed-matter physics, as well as to searches for physics beyond the Standard Model. We also discuss how success in describing the strong interaction impacts other fields, and, in turn, how such subjects can impact studies of the strong interaction. In the course of the work we offer a perspective on the many research streams which flow into and out of QCD, as well as a vision for future developments.Peer reviewe

Transverse intersection of invariant manifolds in perturbed multi-symplectic systems

A multi-symplectic system is a PDE with a Hamiltonian structure in both temporal and spatial variables. This article considers spatially periodic perturbations of symmetric multi-symplectic systems. Due to their structure, unperturbed multi-symplectic systems often have families of solitary waves or front solutions, which together with the additional symmetries lead to large invariant manifolds. Periodic perturbations break the translational symmetry in space and might break some of the other symmetries as well. In this article, periodic perturbations of a translation invariant PDE with a one-dimensional symmetry group are considered. It is assumed that the unperturbed PDE has a three-dimensional invariant manifold associated with a solitary wave or front connection of multi-symplectic relative equilibria. Using the momentum associated with the symmetry group, sufficient conditions for the persistence of invariant manifolds and their transversal intersection are derived. In the equivariant case, invariance of the momentum under the perturbation gives the persistence of the full three-dimensional manifold. In this case, there is also a weaker condition for the persistence of a two-dimensional submanifold with a selected value of the momentum. In the non-equivariant case, the condition leads to the persistence of a one-dimensional submanifold with a seleceted value of the momentum and a selected action of the symmetry group. These results are applicable to general Hamiltonian systems with double zero eigenvalue in the linearization due to continuous symmetry. The conditions are illustrated on the example of the defocussing non-linear Schroumldinger equations with perturbations which illustrate the three cases. The perturbations are: an equivariant Hamiltonian perturbation which keeps the momentum level sets invariant; an equivariant damped, driven perturbation; and a perturbation which breaks the rotational symmetry

Localization and nonlinear transport in single walled carbon nanotube fibers

International audienc