Gray Solitary Waves Research Articles

On conservation laws, their applications in stability analysis and chirped solitary wave solutions for the generalized Schrödinger–Hirota equation and its reductions

Conserved quantities for a generalized version of the Schrödinger–Hirota (SH) equation and its reductions are deduced using an algorithmic approach which is easier to implement compared to more sophisticated mathematical methods. The conserved mass and the Hamiltonian, whenever it exists, are then used to analyse the stability of exact stationary wave solutions, by invoking the Vakhitov-Kolokolov criterion and variational methods. In addition we have focussed on two classes of chirped solitary wave solutions for reductions of the generalised SH equation, namely the algebraic and the hyperbolic classes. It is found that along with the regular dependance of the chirp on the intensity there may even be an additional dependence varying inversely on the intensity of the solitary wave. The chirped hyperbolic solutions are found to correspond to gray solitary waves for suitable parameter values which may also flip to anti-dark solitary waves depending on the ratio of the parameters involved in the solution.

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.

W-shaped, dark and grey solitary waves in the nonlinear Schrödinger equation competing dual power-law nonlinear terms and potentials modulated in time and space

ABSTRACTThis paper explores the higher-order nonlinear Schrodinger equation with dual power-law nonlinearities and external potential that can depend on space and time coordinates. We constructed exact soliton-like solutions with arbitrary power-law nonlinearities using the similarity transformation approach. We obtained a family of soliton-like solutions: the so-called W-shaped, dark and the grey solitons. These solutions are at first, found by considering just the time modulated nonlinearities, where the frame variable coincides with the characteristic coordinate ξ. Then the nonlinearities coefficients and potential are space–time modulated with a bounded function . In both cases, a numerical study based on appropriate choices of parameters is achieved. For a fixed absolute value of nonlinearities, we examine how different soliton profiles are influenced when increasing the power-law index n. These variations of parameters display some substantial changes in solitary wave profiles; paving the way for many applications.

Many-body dissipative flow of a confined scalar Bose-Einstein condensate driven by a Gaussian impurity

The many-body dissipative flow induced by a mobile aussian impurity harmonically oscillating within a cigar-shaped Bose-Einstein condensate is investigated. For very {small and large driving frequencies} the superfluid phase is preserved. Dissipation is identified, for intermediate driving frequencies, by the non-zero value of the drag force whose abrupt increase {signals the spontaneous downstream emission of an array of gray solitons. After each emission event, typically each of the solitary waves formed decays and splits into two daughter gray solitary waves that are found to be robust propagating in the bosonic background for large evolution times.} In particular, a smooth transition towards dissipation is observed, with the {\it critical} velocity for solitary wave formation depending on both the characteristics of the obstacle, namely its driving frequency and width as well as on the interaction strength. The variance of a sample of single-shot simulations indicates the fragmented nature of the system; here it is found to increase during evolution for driving frequencies where the coherent structure formation becomes significant. Finally, we demonstrate that for fairly large particle numbers in-situ single-shot images directly capture the gray soliton's decay and splitting.

Reorientational versus Kerr dark and gray solitary waves using modulation theory

We develop a modulation theory model based on a Lagrangian formulation to investigate the evolution of dark and gray optical spatial solitary waves for both the defocusing nonlinear Schrödinger (NLS) equation and the nematicon equations describing nonlinear beams, nematicons, in self-defocusing nematic liquid crystals. Since it has an exact soliton solution, the defocusing NLS equation is used as a test bed for the modulation theory applied to the nematicon equations, which have no exact solitary wave solution. We find that the evolution of dark and gray NLS solitons, as well as nematicons, is entirely driven by the emission of diffractive radiation, in contrast to the evolution of bright NLS solitons and bright nematicons. Moreover, the steady nematicon profile is nonmonotonic due to the long-range nonlocality associated with the perturbation of the optic axis. Excellent agreement is obtained with numerical solutions of both the defocusing NLS and nematicon equations. The comparisons for the nematicon solutions raise a number of subtle issues relating to the definition and measurement of the width of a dark or gray nematicon.

Relativistic solitary waves modulating long laser pulses in plasmas

This paper discusses the existence of solitary electromagnetic waves trapped in a self-generated Langmuir wave and embedded in an infinitely long circularly polarized electromagnetic wave propagating through a plasma. From a mathematical point of view they are exact solutions of the one-dimensional relativistic cold fluid plasma model with nonvanishing boundary conditions. Under the assumption of travelling wave solutions with velocity V and vector potential frequency ω, the fluid model is reduced to a Hamiltonian system. The solitary waves are homoclinic (grey solitons) or heteroclinic (dark solitons) orbits to fixed points. Using a dynamical systems description of the Hamiltonian system and a spectral method, we identify a large variety of solitary waves, including asymmetric ones, discuss their disappearance for certain parameter values and classify them according to (i) grey or dark character, (ii) the number of humps of the vector potential envelope and (iii) their symmetries. The solutions come in continuous families in the parametric V–ω plane and extend up to velocities that approach the speed of light. The stability of certain types of grey solitary waves is investigated with the aid of particle-in-cell simulations that demonstrate their propagation for a few tens of the inverse of the plasma frequency.

Gray solitary-wave solutions in nonlinear negative-index materials

We predict the existence of gray (dark) solitary waves in negative-index materials on the basis of a derived higher-order nonlinear Schrödinger equation. The conditions for the formation of three cases of gray solitary waves and exact analytical expressions are presented. Furthermore, we investigate the properties of these gray solitary waves in negative-index materials. The results show that the higher-order linear and nonlinear effects play a crucial role for the formation and properties of the second and third cases of gray solitary waves.

Bright and dark solitary wave propagation and bistability in the anomalous dispersion region of optical waveguides with third- and fifth-order nonlinearities

A unified analysis of all existing solitary wave solutions - bright, black and gray - of the cubic-quintic nonlinear Schrödinger equation (CQNLSE) is made concretely, considering a monomode step-index optical fibre with third- and fifth-order nonlinearities in the anomalous dispersion frequency region. Different aspects of the effect of the fifth-order nonlinearity, such as the existence of bright and gray solitary waves in the same parameter range and dark solitary wave bistability are considered. The influence of the distributions modified by the nonlinearity radial fields on the quintic dispersion coefficient in the CQNLSE is calculated. The results obtained are applicable to the cases of spatial solitons and optical beam self-action in highly nonlinear media.

Symbiotic solitary-wave pairs sustained by cross-phase modulation in optical fibers

A new class of solitary-wave envelope solutions is found to the problem of nonlinear (Kerr) interaction between two optical waves copropagating at different frequencies in an optical fiber. The solutions, consisting of pairs of (i) generalized bright and gray solitary waves and (ii) periodic solitary waves, are symbiotic in the sense that each wave in the pair depends crucially on the cross-phase modulation from the complementary wave. The solutions of type (i) admit bright pulse propagation in the normal-dispersion regime and dark (gray) pulse propagation in the anomalous-dispersion regime. A perturbation analysis indicates that the solutions of type (i) form boundary oscillatory states in the presence of small mismatches in group velocity and/or initial position.