Finite Perturbations Research Articles

Some advanced chirped pulses for generalized mixed nonlinear Schrödinger dynamical equation

In an optical fibre, we investigate the propagation properties of nonlinear periodic waves (PW) for a generalized mixed nonlinear Schrödinger (GMNLS) equation. The Jacobi elliptic (JE) solutions will be used to find the nonlinear chirp. The chirp varies with two intensity-dependent chirping terms are included in the linear section of the pulse chirp. The presence of the newly discovered periodic waves will be discussed in terms of fibre parameter conditions. The long-wave limit generates a wide range of solitary pulse forms, including topological, non topological, dark, kink, hyperbolic and periodic solitary waves (SW). Our findings show that for periodic and solitary waves, a nonlinear chirp obtains. Finally, under finite perturbations, the stability of these nonlinearly chirped solutions will be quantitatively investigated.

Q-Deformed solitary pulses in the higher-order nonlinear Schrödinger equation with cubic-quintic nonlinear terms

We investigate the existence and propagation properties of deformed solitary pulses in a non-Kerr medium described by the higher-order nonlinear Schrödinger equation with cubic-quintic nonlinear terms and third-order dispersion. Two different types of exact analytical q-deformed soliton solutions have been derived by means of the ansatz method. The results show that both width and amplitude of the soliton structures are influenced by the deformed factor. It is found that the introduced deformed factor lets the soliton solution deviates from the standard profile. The requirements on the parameters of the non-Kerr material for the existence of these localized structures are presented. By employing numerical simulations, we demonstrate the stability of these deformed soliton solutions under the finite perturbations. Finally, the collision between similar soliton pulses is also investigated.

Solitary and periodic waves in quadratic-cubic non-centrosymmetric waveguides

We present a wide class of solitary and periodic waves in a non-centrosymmetric waveguide exhibiting second- and third-order nonlinearities. We show the existence of bright, gray, and W-shaped solitary waves as well as periodic waves for the quadratic-cubic nonlinear Schrödinger equation. We also obtained the exact analytical algebraic-type solitary waves of this model, including bright and W-shaped waves. The results illustrate the propagation of potentially rich set of nonlinear structures through the optical system. Such privileged nonlinear waves characteristically exist due to a balance among diffraction, quadratic and cubic nonlinearities. The stability of the solutions is discussed numerically under finite initial perturbations. These results can be used in studying the propagation of shape-preserved and periodic waves in optical waveguides with quadratic-cubic nonlinearities.

Ambipolar diffusion: Self-similar solutions and MHD code testing

Context. Ambipolar diffusion is a process occurring in partially ionised astrophysical systems that imparts a complicated mathematical and physical nature to Ohm’s law. The numerical codes that solve the magnetohydrodynamic (MHD) equations have to be able to deal with the singularities that are naturally created in the system by the ambipolar diffusion term. Aims. The global aim is to calculate a set of theoretical self-similar solutions to the nonlinear diffusion equation with cylindrical symmetry that can be used as tests for MHD codes which include the ambipolar diffusion term. Methods. First, following the general methods developed in the applied mathematics literature, we obtained the theoretical solutions as eigenfunctions of a nonlinear ordinary differential equation. Phase-plane techniques were used to integrate through the singularities at the locations of the nulls, which correspond to infinitely sharp current sheets. In the second half of the paper, we consider the use of these solutions as tests for MHD codes. To that end, we used the Bifrost code, thereby testing the capabilities of these solutions as tests as well as (inversely) the accuracy of Bifrost’s recently developed ambipolar diffusion module. Results. The obtained solutions are shown to constitute a demanding, but nonetheless viable, test for MHD codes that incorporate ambipolar diffusion. Detailed tabulated runs of the solutions have been made available at a public repository. The Bifrost code is able to reproduce the theoretical solutions with sufficient accuracy up to very advanced diffusive times. Using the code, we also explored the asymptotic properties of our theoretical solutions in time when initially perturbed with either small or finite perturbations. Conclusions. The functions obtained in this paper are relevant as physical solutions and also as tests for general MHD codes. They provide a more stringent and general test than the simple Zeldovich-Kompaneets-Barenblatt-Pattle solution.

Finite rank perturbations of normal operators: Spectral subspaces and Borel series

We characterize the spectral subspaces associated to closed sets of rank-one perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space by means of functional equations involving Borel series. As a particular instance, if T=DΛ+u⊗v is a rank-one perturbation of a diagonalizable normal operator DΛ with respect to a basis E=(en)n≥1 and the vectors u and v have Fourier coefficients (αn)n≥1 and (βn)n≥1 with respect to E, respectively, it is shown that T has non-trivial closed invariant subspaces provided that either (αn)n≥1∈ℓ1 or (βn)n≥1∈ℓ1. Likewise, analogous results hold for finite rank perturbations of DΛ. Moreover, such operators T have non-trivial closed hyperinvariant subspaces whenever they are not a scalar multiple of the identity extending previous theorems of Foiaş, Jung, Ko and Pearcy [8] and of Fang and J. Xia [6] on an open question of at least forty years.

Self-stabilisation of Cellular Automata on Tilings

Given a finite set of local constraints, we seek a cellular automaton (i.e., a local and uniform algorithm) that self-stabilises on the configurations that satisfy these constraints. More precisely, starting from a finite perturbation of a valid configuration, the cellular automaton must eventually fall back into the space of valid configurations where it remains still. We allow the cellular automaton to use extra symbols, but in that case, the extra symbols can also appear in the initial finite perturbation. For several classes of local constraints (e.g., k-colourings with k ≠ 3, and North-East deterministic constraints), we provide efficient self-stabilising cellular automata with or without additional symbols that wash out finite perturbations in linear or quadratic time, but also show that there are examples of local constraints for which the self-stabilisation problem is inherently hard. We note that the optimal self-stabilisation speed is the same for all local constraints that are isomorphic to one another. We also consider probabilistic cellular automata rules and show that in some cases, the use of randomness simplifies the problem. In the deterministic case, we show that if finite perturbations are corrected in linear time, then the cellular automaton self-stabilises even starting from a random perturbation of a valid configuration, that is, when errors in the initial configuration occur independently with a sufficiently low density.

Chirped optical soliton propagation in birefringent fibers modeled by coupled Fokas-Lenells system

The propagation of ultrashort light pulses in a birefringent optical fiber exhibiting spatiotemporal dispersion, cross- and self-phase modulation, self-steepening, and group-velocity dispersion effects is addressed. The evolution of light pulses in such system is described by the coupled Fokas-Lenells equations which offer an accurate description of pulse dynamics in the femtosecond range when certain terms of the next asymptotic order beyond those necessary for the nonlinear Schrödinger equation are sustained. We report the first analytical demonstration of the propagation of chirped solitons in a birefringent fiber medium governed by the coupled Fokas-Lenells equations. The formation of those nonlinearly chirped solitons in the optical material may be attributed to the presence of self-steepening process. The results show that the frequency chirp associated with each of the two field components is directly proportional to the total intensity of the pulse. The chirped solitons for the system including the dark-dark and bright-bright soliton pairs in the presence of all fiber parameters are retrieved. The chirps accompanying the soliton pairs are also determined. The existence constraints of these chirped solitons are presented. In addition, the stability of the chirped solutions with respect to the finite perturbations is studied numerically.

Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions

Abstract In this article, we consider a class of finite rank perturbations of Toeplitz operators that have simple eigenvalues on the unit circle. Under a suitable assumption on the behavior of the essential spectrum, we show that such operators are power bounded. The problem originates in the approximation of hyperbolic partial differential equations with boundary conditions by means of finite difference schemes. Our result gives a positive answer to a conjecture by Trefethen, Kreiss and Wu that only a weak form of the so-called uniform Kreiss–Lopatinskii condition is sufficient to imply power boundedness.

Pulse train generation in a distributed gain fibre amplifier

This work shows the generation of an array of chirped compressed periodic waves with distributed gain in a nonlinear fibre. In particular, with suitable tailoring of the gain to vary along the longitudinal distance while the dispersion and nonlinearity parameters are kept constant. Exact analytical equations using the self-similar analysis technique were obtained to describe this process. In addition, the stability of these generated periodic waves is studied under finite perturbations.

Chirped periodic and solitary waves in nonlinear negative index materials

Propagation of ultrashort pulses at least a few tens of optical cycles in duration through a negative index material is investigated theoretically based on the generalized nonlinear Schrödinger equation with pseudo-quintic nonlinearity and self-steepening effect. Novel periodic waves of different forms are shown to exist in the system in the presence of higher-order effects. It is found that such periodic structures exhibit an interesting chirping property which depends on the light field intensity. The nonlinearity in pulse chirp is found to be caused by the presence of self-steepening effect in the negative index medium. The solutions also comprise dark, bright, and kink solitary waves. Stability of the solitary wave solutions are proved analytically using the theory of nonlinear dispersive waves. The stability of the solutions is numerically studied under finite initial perturbations.

МЕХАНИЗМЫ ПЕРЕДАЧИ СПИН-СПИНОВОГО ВЗАИМОДЕЙСТВИЯ УГЛЕРОД-УГЛЕРОД В РЯДУ ПРОИЗВОДНЫХ БИЦИКЛОБУТАНА

Within the framework of the self-consistent theory of finite perturbations SCPT INDO, the calculation of the spin-spin interaction constants 13C-13C with preliminary optimization of geometric parameters in 17 bicyclo derivatives is carried out[1.1.0.] of Bhutan. There is a good correspondence of the calculated constants with the experimental values known in the literature and measured in this work. Based on the results of the calculations performed, the study of hybridization effects in the considered series of compounds was carried out. All bicyclobutane derivatives are characterized by an abnormally low s-character of endocyclic hybrid orbitals forming a bond between bridging carbon atoms. Substitution effects have a weak effect on the s-order of the bridge connection. The presence of an unsaturated alcohol structure in the 2-position of the bicyclo-butane fragment leads to a sharp decrease in the s-character of the endocyclic hybrid orbitals of the bridged carbon atoms.

КОНСТАНТЫ СПИН-СПИНОВОГО ВЗАИМОДЕЙСТВИЯ МЕЖДУ ЯДРАМИ УГЛЕРОДА В СТЕРИЧЕСКИ НАПРЯЖЕННЫХ СИСТЕМАХ НА ПРИМЕРЕ БИЦИКЛОПЕНТАНОВ И БИЦИКЛОГЕКСАНОВ

Within the framework of the self-consistent theory of finite perturbations SCPT INDO, the calculation of the spin-spin interaction constants 13C-13C with preliminary optimization of geometric parameters in 16 bicyclo derivatives is carried out[2.1.0] pentane and 11 bicyclo derivatives[2.2.0]hexane. According to the results of the calculations, the bridge connection in the studied series of compounds is characterized by an unusually low s-character of hybrid orbitals, amounting to 4.8-8.8 % in bicyclo derivatives[2.1.0] pentane and 11.3-12.9 % in bicyclo derivatives[2.2.0]hexane, which corresponds to the orbital hybridization from sp10. 30 to sp19. 92 in bicyclo[2.1.0]pentanes and from sp6. 76 to sp7. 89 in bicyclo[2.2.0]hexanes. The spin-spin interaction constants between bridged and non-bridged, or between two non-bridged carbon atoms in the studied series of compounds correlate with the corresponding spin-spin interaction constants 13C-13C in isostructured cyclopropanes and cyclobutanes, which indicates close hybridization

A simple scenario of the laminar breakdown in liquid metal flows

In the article, authors present a numerical method for modelling a laminar-turbulent transition in magnetohydrodynamic flows. The small magnetic Reynolds number approach is considered. Velocity, pressure and electrical potential are decomposed to the sum of state values and finite amplitude perturbations. A solver based on the Nektar++ framework is described. The authors suggest using small-length local perturbations as a transition trigger. They can be imposed by blowing or by electrical enforcing. The stability of the Hartmann flow and the flow in the bend are considered as examples. Tables 4, Figs 19, Refs 28.

Subcritical asymmetric Rayleigh breakup of a charged drop induced by finite amplitude perturbations in a quadrupole trap.

The breakup pathway of Rayleigh fission of a charged drop is unequivocally demonstrated by continuous, high-speed imaging of a drop levitated in an AC quadrupole trap. The experimental observations consistently exhibited asymmetric, subcritical Rayleigh breakup with an upward (i.e., opposite to the direction of gravity) ejection of a jet from the levitated drop. These experiments supported by numerical calculations show that the gravity induced downward shift of the equilibrium position of the drop in the trap causes significant, large amplitude shape oscillations superimposed over the center-of-mass oscillations. The shape oscillations result in sufficient deformations to act as triggers for the onset of instability below the Rayleigh limit (a subcritical instability). The concurrently occurring, center-of-mass oscillations, which are out of phase with the applied voltage, are shown to lead to an asymmetric breakup such that the Rayleigh fission occurs upwards via the ejection of a jet at the pole of the deformed drop. As an important application, it follows by inference that the nanodrop generation in electrospray devices will occur, more as a rule rather than as an exception, via asymmetric, subcritical Rayleigh fission events of microdrops due to inherent directionality provided by the external electric fields.

Stability of a three-dimensional plate made of a nonlinear viscoelastic material under superimposed finite deformations

In this paper, the stability of a three-dimensional plate made of a nonlinear viscoelastic material under finite perturbations is considered. A fairly broad computational experiment has been performed. The permissible boundaries of the region with respect to the final, initial perturbations are established for the given parameters of loading and structures. Finite sequences of bifurcation points are constructed, which confirm, in contrast to stability under small perturbations, the existence of a sequence of stable equilibrium states. New phenomena and characteristic effects are established.

Higher order local Dirichlet integrals and de Branges-Rovnyak spaces

We investigate expansive Hilbert space operators T that are finite rank perturbations of isometric operators. If the spectrum of T is contained in the closed unit disc D‾, then such operators are of the form T=U⊕R, where U is isometric and R is unitarily equivalent to the operator of multiplication by the variable z on a de Branges-Rovnyak space H(B). In fact, the space H(B) is defined in terms of a rational operator-valued Schur function B. In the case when dim⁡ker⁡T⁎=1, then H(B) can be taken to be a space of scalar-valued analytic functions in D, and the function B has a mate a defined by |B|2+|a|2=1 a.e. on ∂D. We show the mate a of a rational B is of the form a(z)=a(0)p(z)q(z), where p and q are appropriately derived from the characteristic polynomials of two associated operators. If T is a 2m-isometric expansive operator, then all zeros of p lie in the unit circle, and we completely describe the spaces H(B) by use of what we call the local Dirichlet integral of order m at the point w∈∂D.

On steady alternate bars forced by a localized asymmetric drag distribution in erodible channels

Studying the effect of different in-stream fluvial turbines siting on river morphodynamics allowed us to witness the onset of a time-averaged, large-scale, alternate distortion of bed elevations, which could not be exclusively related to the turbine rotor blockage. The longitudinal profiles of this two-dimensional bathymetric perturbation resemble those of steady fluvial bars. In this contribution we generalize the problem addressing a spatially impulsive, asymmetric distribution of drag force in the channel cross-section. This is experimentally investigated through the deployment of differently sized grids perpendicular to the flow, and analytically explored as a finite perturbation of an open channel flow over an erodible sediment layer, as described by a coupled flow–sediment shallow water equation. The steady solutions of this fluvial morphodynamic problem, physically represented by alternate bars scaling with the channel width, highlight the importance of the resonant conditions in defining the spatial extent of the bed deformation. The equations further suggest that in very shallow flows any asymmetric obstruction may lead to an upstream propagation of the steady bars, consistent with previous studies on the effects of channel curvature. In broad terms, this study provides the preliminary framework to control the onset of river meandering through imposed finite perturbations of the cross-section. In a more applied sense, it provides a tool to predict non-local scour–deposition patterns associated with the deployment of energy converters or other flow obstructions.

Propagation of chirped periodic and localized waves with higher-order effects through optical fibers

We study the propagation properties of nonlinear periodic waves in an optical fiber medium exhibiting Kerr dispersion and quintic nonlinearity. The quintic derivative nonlinear Schrödinger equation is applied to model the evolution of femtosecond light waves in such system. Unlike periodic structures in Kerr media, the novel waves possess a nonlinear chirp which varies with their intensity. In addition to the linear part, the pulse chirp also includes two intensity dependent chirping terms. The conditions on fiber parameters for the existence of the newly found periodic waves are presented. A wide variety of solitary pulse shapes including the bright, dark, kink, anti-kink, and gray solitary waves are obtained in the long-wave limit. Our results show that Kerr dispersion plays a crucial role in introducing a nonlinear chirp for the periodic and solitary waves. Finally, the stability of these nonlinearly chirped solutions is numerically studied under the finite perturbations.

Transient growth, edge states, and repeller in rotating solid and fluid.

For the classical problem of the rotation of a solid, we show a somehow surprising behavior involving large transient growth of perturbation energy that occurs when the moment of inertia associated to the unstable axis approaches the moment of inertia of one of the two stable axes. In that case, small but finite perturbations around this stable axis may induce a total transfer of energy to the unstable axis, leading to relaxation oscillations where the stable and unstable manifolds of the unstable axis play the role of a separatrix, an edge state. For a fluid in solid-body rotation, a similar linear and nonlinear dynamics apply to the transfer of energy between three inertial waves respecting the triadic resonance condition. We show that the existence of large transient energy growth and of relaxation oscillations may be physically interpreted as in the case of a solid by the existence of two quadratic invariants, the energy and the helicity in the case of a rotating fluid. They occur when two waves of the triad have helicities that tend towards each other, when their amplitudes are set such that they have the same energy. We show that this happens when the third wave has a vanishing frequency which corresponds to a nearly horizontal wave vector. An inertial wave, perturbed by a small-amplitude wave with a nearly horizontal wave vector, will then be periodically destroyed, its energy being transferred entirely to the unstable wave, although this perturbation is linearly stable, resulting in relaxation oscillations of wave amplitudes. In the general case we show that the dynamics described for particular triads of inertial waves is valid for a class of triadic interactions of waves in other physical problems, where the physical energy is conserved and is linked to the classical conservation of the so-called pseudomomentum, which singles out the role of waves with vanishing frequency.

Finite Rank Perturbations of Linear Relations and Matrix Pencils

We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least n. In the operator case, it was recently proved that the difference of these numbers is independent of n and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by n+1 and that this bound is sharp. The reason for this behavior is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.