Auswahl der wissenschaftlichen Literatur zum Thema „Envelope vector soliton“

The dynamics of envelope solitons in a system of coupled anharmonic chains are addressed. Mathematically, the system is equivalent to the vector soliton propagation model in a single-mode fiber with low birefringence in the presence of coherent and incoherent interactions. It is numerically and analytically shown that multi-component soliton entries can behave as free scalar solitons with arbitrary velocities and amplitudes. The appropriate exact multi-soliton solutions are provided. They can be presented as a linear interference of degenerate vector solitons known before. Furthermore, the interference idea is transferred to other vector integrable systems, including the Manakov model.

We study nonlinear self-interactions including modulational instability in the case of a plane electromagnetic pulse propagating through a magnetized cold plasma at an arbitrary oblique angle to the external magnetic field. For intended applications to pulsar magnetospheres, the magnetic field is so large that both the electron- and ion-cyclotron frequencies are enormous compared with the plasma frequency or the frequency ω of the wave itself. The plasma is assumed to contain two singly charged species, either electrons and positrons or electrons and ions. (No approximation is made with respect to the mass ratio.) We restrict ourselves to the case eE0/mω ≪ 1 (i.e. the wave amplitude E0 excites the electrons to weakly, but not fully, relativistic velocities). We consider a pulse whose linear polarization is in the plane of the wave vector and the magnetic field. (The orthogonal polarization is purely electromagnetic, and induces no motion along magnetic field lines.) The pulse is assumed to be modulated along the direction of the group velocity vector. We show, using a self-consistent multiple-scales solution, that the envelope obeys the nonlinear Schrödinger equation, and from the coefficients of this equation we derive the conditions for modulational instability. Computation of the nonlinear coefficients requires detailed consideration of ponderomotive, relativistic and harmonic effects, all of which, in the ‘weakly relativistic’ case considered here, enter at the same order in the approximation scheme. Unlike the case of propagation parallel to a strong magnetic field, in oblique propagation we find a wide parameter range for modulational instability and soliton formation on time scales appropriate for pulsar micropulses.

AbstractAn asymmetric pair of coupled nonlinear Schrödinger (CNLS) equations has been derived through a multiscale perturbation method applied to a plasma fluid model, in which two wavepackets of distinct (carrier) wavenumbers ($$k_1$$ k 1 and $$k_2$$ k 2 ) and amplitudes ($$\Psi _1$$ Ψ 1 and $$\Psi _2$$ Ψ 2 ) are allowed to co-propagate and interact. The original fluid model was set up for a non-magnetized plasma consisting of cold inertial ions evolving against a $$\kappa$$ κ -distributed electron background in one dimension. The reduction procedure resulting in the CNLS equations has provided analytical expressions for the dispersion, self-modulation and cross-coupling coefficients in terms of the two carrier wavenumbers. These coefficients present no symmetry whatsoever, in the general case (of different wavenumbers). The possibility for coupled envelope (vector soliton) solutions to occur has been investigated. Although the CNLS equations are asymmetric and non-integrable, in principle, the system admits various types of vector soliton solutions, physically representing nonlinear, localized electrostatic plasma modes, whose areas of existence is calculated on the wavenumbers’ parameter plane. The possibility for either bright (B) or dark (D) type excitations for either of the (2) waves provides four (4) combinations for the envelope pair (BB, BD, DB, DD), if a set of explicit criteria is satisfied. Moreover, the soliton parameters (maximum amplitude, width) are also calculated for each type of vector soliton solution, in its respective area of existence. The dependence of the vector soliton characteristics on the (two) carrier wavenumbers and on the spectral index $$\kappa$$ κ characterizing the electron distribution has been explored. In certain cases, the (envelope) amplitude of one component may exceed its counterpart (second amplitude) by a factor 2.5 or higher, indicating that extremely asymmetric waves may be formed due to modulational interactions among copropagating wavepackets. As $$\kappa$$ κ decreases from large values, modulational instability occurs in larger areas of the parameter plane(s) and with higher growth rates. The distribution of different types of vector solitons on the parameter plane(s) also varies significantly with decreasing $$\kappa$$ κ , and in fact dramatically for $$\kappa$$ κ between 3 and 2. Deviation from the Maxwell-Boltzmann picture therefore seems to favor modulational instability as a precursor to the formation of bright (predominantly) type envelope excitations and freak waves.

AbstractThe interaction between two co-propagating electromagnetic pulses in a magnetized plasma is considered, from first principles, relying on a fluid-Maxwell model. Two circularly polarized wavepackets by same group velocities are considered, characterized by opposite circular polarization, to be identified as left-hand- or right hand circularly polarized (i.e. LCP or RCP, respectively). A multiscale perturbative technique is adopted, leading to a pair of coupled nonlinear Schrödinger-type (NLS) equations for the modulated amplitudes of the respective vector potentials associated with the two pulses. Systematic analysis reveals the existence, in certain frequency bands, of three different types of vector soliton modes: an LCP-bright/RCP-bright coupled soliton pair state, an LCP-bright/RCP-dark soliton pair, and an LCP-dark/RCP-bright soliton pair. The value of the magnetic field plays a critical role since it determines the type of vector solitons that may occur in certain frequency bands and, on the other hand, it affects the width of those frequency bands that are characterized by a specific type of vector soliton (type). The magnetic field (strength) thus arises as an order parameter, affecting the existence conditions of each type of solution (in the form of an envelope soliton pair). An exhaustive parametric investigation is presented in terms of frequency bands and in a wide range of magnetic field (strength) values, leading to results that may be applicable in beam-plasma interaction scenarios as well as in space plasmas and in the ionosphere.

Ce travail est consacré à l'étude des ondes modulées se propageant le long de métamatériaux mécaniques flexibles nonlinéaires (FlexMM). Ces structures sont des matériaux architecturés constitués d'éléments souples très déformables connectés à des éléments plus rigides. Leur capacité à subir de grandes déformations locales favorise l'apparition de phénomènes d'ondes non linéaires. En utilisant une approche par éléments discrets, nous formulons des équations discrètes non linéaires qui décrivent les déplacements longitudinaux et rotationnels de chaque cellule unitaire et leur couplage mutuel. Une analyse multi-échelles est employée afin d'obtenir une équation de Schrödinger non linéaire (NLS) effective décrivant les ondes modulées pour le degré de liberté rotationnel du FlexMM. En nous appuyant sur l'équation NLS, nous identifions divers types de phénomènes d'ondes non linéaires dans le FlexMM. En particulier, nous avons observé que des ondes planes faiblement non linéaires peuvent être modulationellement stables ou instables en fonction des paramètres du système et de l'excitation utilisée. De plus, nous avons trouvé que les FlexMMs supportent des solitons-enveloppe vectoriels où le degré de liberté rotationnel des unités peut prendre la forme de solitons dits "bright" ou "dark" et, en raison du couplage, le degré de liberté de déplacement longitudinal présente un comportement de type "kink". Enfin, nous abordons le phénomène de "catastrophe de gradient", qui prédit l'émergence de structures similaires aux solitons de Peregrine dans la limite semi-classique de l'équation NLS, dans la structure FlexMM. Grâce à nos prédictions analytiques et à l'utilisation de simulations numériques, nous pouvons déterminer les conditions requises et les valeurs des paramètres physiques pour observer ces phénomènes dans les FlexMMs
This work is dedicated to the investigation of modulated waves propagating along nonlinear flexible mechanical metamaterials (FlexMM). These structures are architected materials consisting of highly deformable soft elements connected to stiffer ones. Their capacity to undergo large local deformations promotes the occurrence of nonlinear wave phenomena. Using a lump element approach, we formulate nonlinear discrete equations that describe the longitudinal land rotational displacements of each unit cell and their mutual coupling. A multiple scales analysis is employed in order to derive an effective nonlinear Schrödinger (NLS) equation describing envelope waves for the rotational degree of freedom of FlexMM. Leveraging on the NLS equation we identify various type of nonlinear waves phenomena in FlexMM. In particular we observed that weakly nonlinear plane waves can be modulationally stable or unstable depending of the system and excitation parameters. Moreover we have found that the FlexMMs support envelope vector solitons where the units rotational degree of freedom might take the form of bright or dark soliton and due to coupling, the longitudinal displacement degree of freedom has a kink-like behavior. Finally, we address the phenomenon of "gradient catastrophe", which predicts the emergence of Peregrine soliton-like structures in the semiclassical limit of the NLS equation, in FlexMM. Through our analytical predictions and by using numerical simulations, we can determine the required conditions and the values of the physical parameters in order to observe these phenomena in FlexMMs

Experimentalists have produced all-optical switches capable of 100-fs responses [1]. Also, there are experimental observations [2] and theoretical calculations [3] of spatial soliton interactions. To adequately model such effects, nonlinearities in optical materials [4] (both instantaneous and dispersive) must be included. In principle, the behavior of electromagnetic fields in nonlinear dielectrics can be determined by solving Maxwell's equations subject to the assumption that the electric polarization has a nonlinear relation to the electric field. However, until our previous work [5 - 8], the resulting nonlinear Maxwell's equations have not been solved directly. Rather, approximations have been made that result in a class of generalized nonlinear Schrodinger equations (GNLSE) [9] that solve only for the envelope of the optical pulses.

Temporal optical solitons may exist in media where the dispersion-induced chirp is balanced by nonlinear self-phase modulation due to the Kerr effect. Usually, the parabolic approximation is employed for the frequency dependence of the wave vector (dispersion relation) resulting in the nonlinear Schrödinger for the field envelope. The propagation of temporal solitons in optical fibers can be reasonably described by this provided that the mean frequency is not too close to the zero-dispersion point. But, if the parabolic approximation fails higher-order terms of the expansion must be taken into account [1, 2] or the dispersion relation of the material must be approximated in a different way. Here we choose the latter option.