Periodicity and Quasi-Periodicity for Super-Integrable Hamiltonian Systems

Classical trajectories are calculated for two Hamiltonian systems with ring shaped potentials. Both systems are super-integrable, but not maximally super-integrable, having four globally defined single valued integrals of motion each. All finite trajectories are quasi-periodical; they become truly periodical if a commensurability condition is imposed on an angular momentum component

Nonlinear spectral analysis of Peregrine solitons observed in optics and in hydrodynamic experiments

The data recorded in optical fiber [1] and in hydrodynamic [2] experiments reported the pioneering observation of nonlinear waves with spatiotemporal localization similar to the Peregrine soliton are examined by using nonlinear spectral analysis. Our approach is based on the integrable nature of the one-dimensional focusing nonlinear Schrodinger equation (1D-NLSE) that governs at leading order the propagation of the optical and hydrodynamic waves in the two experiments. Nonlinear spectral analysis provides certain spectral portraits of the analyzed structures that are composed of bands lying in the complex plane. The spectral portraits can be interpreted within the framework of the so-called finite gap theory (or periodic inverse scattering transform). In particular, the number N of bands composing the nonlinear spectrum determines the genus g = N - 1 of the solution that can be viewed as a measure of complexity of the space-time evolution of the considered solution. Within this setting the ideal, rational Peregrine soliton represents a special, degenerate genus 2 solution. While the fitting procedures employed in [1] and [2] show that the experimentally observed structures are quite well approximated by the Peregrine solitons, nonlinear spectral analysis of the breathers observed both in the optical fiber and in the water tank experiments reveals that they exhibit spectral portraits associated with more general, genus 4 finite-gap NLSE solutions. Moreover, the nonlinear spectral analysis shows that the nonlinear spectrum of the breathers observed in the experiments slowly changes with the propagation distance, thus confirming the influence of unavoidable perturbative higher order effects or dissipation in the experiments

Soliton and rogue wave statistics in supercontinuum generation in photonic crystal fibre with two zero dispersion wavelengths

International audienceStochastic numerical simulations are used to study the statistical properties of supercontinuum spectra generated in photonic crystal fibre with two zero dispersion wavelengths. For picosecond pulse excitation, we examine how the statistical properties of solitons generated on the long wavelength edge of the supercontinuum (``optical rogue waves") are modified by energy transfer to dispersive waves across the second zero dispersion wavelength. The soliton statistics (characterized in terms of peak power, wavelength and pulse duration) are shown to be strongly modified by the mechanism of dispersive wave generation, with the detailed form of the probability distribution depending strongly on input pulse energy

80 GHz waveform generated by the optical Fourier synthesis of four spectral sidebands

International audienceUsing the linear phase shaping of a simple four-line optical frequency comb, we experimentally demonstrate the generation of various optical waveforms such as parabolic, triangular or flat-top pulse trains at a repetition rate of 80 GHz. The initial 80 GHz comb is obtained through the nonlinear spectral broadening of a 40 GHz carrier-suppressed sinusoidal beating in a highly nonlinear fiber. Proof-of-principle experiments are reported for two distinct configurations of the waveform generated: continuous trains and bunches of shaped pulses

On the use of the group SO(4,2) in atomic and molecular physics

In this paper the dynamical noninvariance group SO(4,2) for a hydrogen-like atom is derived through two different approaches. The first one is by an established traditional ascent process starting from the symmetry group SO(3). This approach is presented in a mathematically oriented original way with a special emphasis on maximally superintegrable systems, N-dimensional extension and little groups. The second approach is by a new symmetry descent process starting from the noninvariance dynamical group Sp(8,R) for a four-dimensional harmonic oscillator. It is based on the little known concept of a Lie algebra under constraints and corresponds in some sense to a symmetry breaking mechanism. This paper ends with a brief discussion of the interest of SO(4,2) for a new group-theoretical approach to the periodic table of chemical elements. In this connection, a general ongoing programme based on the use of a complete set of commuting operators is briefly described. It is believed that the present paper could be useful not only to the atomic and molecular community but also to people working in theoretical and mathematical physics.Comment: 31 page

Roadmap on optical rogue waves and extreme events

The pioneering paper 'Optical rogue waves' by Solli et al (2007 Nature 450 1054) started the new subfield in optics. This work launched a great deal of activity on this novel subject. As a result, the initial concept has expanded and has been enriched by new ideas. Various approaches have been suggested since then. A fresh look at the older results and new discoveries has been undertaken, stimulated by the concept of 'optical rogue waves'. Presently, there may not by a unique view on how this new scientific term should be used and developed. There is nothing surprising when the opinion of the experts diverge in any new field of research. After all, rogue waves may appear for a multiplicity of reasons and not necessarily only in optical fibers and not only in the process of supercontinuum generation. We know by now that rogue waves may be generated by lasers, appear in wide aperture cavities, in plasmas and in a variety of other optical systems. Theorists, in turn, have suggested many other situations when rogue waves may be observed. The strict definition of a rogue wave is still an open question. For example, it has been suggested that it is defined as 'an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses'. This definition (as suggested by a peer reviewer) is clear at the intuitive level and can be easily extended to the case of spatial beams although additional clarifications are still needed. An extended definition has been presented earlier by N Akhmediev and E Pelinovsky (2010 Eur. Phys. J. Spec. Top. 185 1-4). Discussions along these lines are always useful and all new approaches stimulate research and encourage discoveries of new phenomena. Despite the potentially existing disagreements, the scientific terms 'optical rogue waves' and 'extreme events' do exist. Therefore coordination of our efforts in either unifying the concept or in introducing alternative definitions must be continued. From this point of view, a number of the scientists who work in this area of research have come together to present their research in a single review article that will greatly benefit all interested parties of this research direction. Whether the authors of this 'roadmap' have similar views or different from the original concept, the potential reader of the review will enrich their knowledge by encountering most of the existing views on the subject. Previously, a special issue on optical rogue waves (2013 J. Opt. 15 060201) was successful in achieving this goal but over two years have passed and more material has been published in this quickly emerging subject. Thus, it is time for a roadmap that may stimulate and encourage further research.Peer ReviewedPostprint (author's final draft

Ondes scélérates et autres manifestations extrêmes dans les fibres optiques

National audienceNous présentons divers exemples d'évènements rares optiques qui peuvent apparaître lors de la propagation d'impulsions dans des fibres optiques. Ainsi, différents types de structures optiques se démarquant très nettement du comportement statistique moyen ont été mis en évidence dans les processus de génération de supercontinua optiques ainsi que dans les phases d'amplification paramétrique ou Raman

Optical besselon waves

We theoretically describe a new type of an optical waveform, the ‘besselon’, which is synthesised by the line-by-line application of π/2-spectral phase shifts to sinusoidally phase-modulated continuous-wave light

All-fiber transform-limited spectral compression by self-phase modulation of amplitude shaped pulses

International audienceWe demonstrate efficient spectral compression of picosecond pulses in an all-fiber configuration at telecommunication wavelengths. Thanks to parabolic pulse shaping, a spectral compression by a factor 12 is achieved with an enhanced Strehl ratio