Fractional Zaslavsky and Henon Discrete Maps

This paper is devoted to the memory of Professor George M. Zaslavsky passed away on November 25, 2008. In the field of discrete maps, George M. Zaslavsky introduced a dissipative standard map which is called now the Zaslavsky map. G. Zaslavsky initialized many fundamental concepts and ideas in the fractional dynamics and kinetics. In this paper, starting from kicked damped equations with derivatives of non-integer orders we derive a fractional generalization of discrete maps. These fractional maps are generalizations of the Zaslavsky map and the Henon map. The main property of the fractional differential equations and the correspondent fractional maps is a long-term memory and dissipation. The memory is realized by the fact that their present state evolution depends on all past states with special forms of weights.Comment: 26 pages, LaTe

Universality in Systems with Power-Law Memory and Fractional Dynamics

There are a few different ways to extend regular nonlinear dynamical systems by introducing power-law memory or considering fractional differential/difference equations instead of integer ones. This extension allows the introduction of families of nonlinear dynamical systems converging to regular systems in the case of an integer power-law memory or an integer order of derivatives/differences. The examples considered in this review include the logistic family of maps (converging in the case of the first order difference to the regular logistic map), the universal family of maps, and the standard family of maps (the latter two converging, in the case of the second difference, to the regular universal and standard maps). Correspondingly, the phenomenon of transition to chaos through a period doubling cascade of bifurcations in regular nonlinear systems, known as "universality", can be extended to fractional maps, which are maps with power-/asymptotically power-law memory. The new features of universality, including cascades of bifurcations on single trajectories, which appear in fractional (with memory) nonlinear dynamical systems are the main subject of this review.Comment: 23 pages 7 Figures, to appear Oct 28 201

Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.Comment: 35 pages, 4 figures, LaTe

A time-fractional Borel-Pompeiu formula and a related hypercomplex operator calculus

In this paper we develop a time-fractional operator calculus in fractional Clifford analysis. Initially we study the $L_p$-integrability of the fundamental solutions of the multi-dimensional time-fractional diffusion operator and the associated time-fractional parabolic Dirac operator. Then we introduce the time-fractional analogues of the Teodorescu and Cauchy-Bitsadze operators in a cylindrical domain, and we investigate their main mapping properties. As a main result, we prove a time-fractional version of the Borel-Pompeiu formula based on a time-fractional Stokes' formula. This tool in hand allows us to present a Hodge-type decomposition for the forward time-fractional parabolic Dirac operator with left Caputo fractional derivative in the time coordinate. The obtained results exhibit an interesting duality relation between forward and backward parabolic Dirac operators and Caputo and Riemann-Liouville time-fractional derivatives. We round off this paper by giving a direct application of the obtained results for solving time-fractional boundary value problems.The work of M. Ferreira, M.M. Rodrigues and N. Vieira was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT–Fundação para a Ciência e a Tecnologia, within project UID/MAT/04106/2019. The work of the authors was supported by the project New Function Theoretical Methods in Computational Electrodynamics / Neue funktionentheoretische Methoden für instationäre PDE, funded by Programme for Cooperation in Science between Portugal and Germany (“Programa de Ações Integradas Luso-Alemãs 2017” - DAAD-CRUP - Acção No. A-15/17 / DAAD-PPP Deutschland-Portugal, Ref: 57340281). N. Vieira was also supported by FCT via the FCT Researcher Program 2014 (Ref: IF/00271/2014).publishe

Geometry and field theory in multi-fractional spacetime

We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can define two models. In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional. At the UV critical value, the model is rendered power-counting renormalizable. However, this is not the most fundamental regime. Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order. In doing so, we obtain a hierarchy of scales characterizing different geometric regimes. At very small scales, discrete symmetries emerge and the notion of a continuous spacetime begins to blur, until one reaches a fundamental scale and an ultra-microscopic fractal structure. This fine hierarchy of geometries has implications for non-commutative theories and discrete quantum gravity. In the latter case, the present model can be viewed as a top-down realization of a quantum-discrete to classical-continuum transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and improved (especially section 4.5), typos corrected, references added; v4: further typos correcte

Analysis of laser radiation using the Nonlinear Fourier transform

Modern high-power lasers exhibit a rich diversity of nonlinear dynamics, often featuring nontrivial co-existence of linear dispersive waves and coherent structures. While the classical Fourier method adequately describes extended dispersive waves, the analysis of time-localised and/or non-stationary signals call for more nuanced approaches. Yet, mathematical methods that can be used for simultaneous characterisation of localized and extended fields are not yet well developed. Here, we demonstrate how the Nonlinear Fourier transform (NFT) based on the Zakharov-Shabat spectral problem can be applied as a signal processing tool for representation and analysis of coherent structures embedded into dispersive radiation. We use full-field, real-time experimental measurements of mode-locked pulses to compute the nonlinear pulse spectra. For the classification of lasing regimes, we present the concept of eigenvalue probability distributions. We present two field normalisation approaches, and show the NFT can yield an effective model of the laser radiation under appropriate signal normalisation conditions

Nested Algebraic Bethe Ansatz for Open Spin Chains with Even Twisted Yangian Symmetry

We present a nested algebraic Bethe ansatz for a one dimensional open spin chain whose boundary quantum spaces are irreducible so2n- or sp2n-representations and the monodromy matrix satisfies the defining relations of the Olshanskii twisted Yangian Y±(gl2n). We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a so2n- or sp2n-symmetric open spin chain to that of a gln-symmetric periodic spin chain. We explicitly derive the structure of the Bethe vectors and the nested Bethe equations

Gain through losses in nonlinear optics

Instabilities of uniform states are ubiquitous processes occurring in a variety of spatially extended nonlinear systems. These instabilities are at the heart of symmetry breaking, condensate dynamics, self-organization, pattern formation and noise amplification across diverse disciplines, including physics, chemistry, engineering and biology. In nonlinear optics, modulation instabilities are generally linked to the so-called parametric amplification process, which occurs when certain phase-matching or quasi-phase-matching conditions are satisfied. In the present review article, we summarize the principle results on modulation instabilities and parametric amplification in nonlinear optics, with special emphasis on optical fibres. We then review state-of-the-art research about a peculiar class of modulation instabilities and signal amplification processes induced by dissipation in nonlinear optical systems. Losses applied to certain parts of the spectrum counterintuitively lead to the exponential growth of the damped mode themselves, causing gain through losses. We discuss the concept of imaging of losses into gain, showing how to map a given spectral loss profile into a gain spectrum. We demonstrate with concrete examples that dissipation-induced modulation instability, apart from being of fundamental theoretical interest, may pave the way towards the design of a new class of tuneable fibre-based optical amplifiers, optical parametric oscillators, frequency comb sources and pulsed lasers