Inhomogeneous Nonlinear Media Research Articles

Dynamics of dipole solitons on a continuous-wave background in an inhomogeneous nonlinear medium with higher-order dispersion

Dynamics of dipole solitons on a continuous-wave background in an inhomogeneous nonlinear medium with higher-order dispersion

Numerical investigation on nonautonomous optical rogue waves and Modulation Instability analysis for a nonautonomous system

In this paper, we report existence of optical rogue waves in the focussing non—autonomous nonlinear Schrödinger equation (NLSE) through numerical studies of modulation instability (MI). The dynamics of non-autonomous rogue waves discussed and its associated modulation instability through linear stability analysis taken place followed by pulse splitting behaviour due to non—autonomous coefficient. We prove that the excitation of rogue waves with certain conditions in the base band modulation instability regime. The above analysis of complex dynamics in terms of MI processes has allowed to experiments to excite the nonlinear superposition of rogue wave solutions using a modulated plane wave optical field injected into optical fiber which offer the evidence for excitation of nonautonomous rogue waves in an inhomogeneous nonlinear medium. It is identified from the results frequency modulation on a wavefield induces modulation instability as a result of rogue waves. We analyze the dependence of parameters coefficient of group velocity dispersion(GVD) and nonlinearity (α(z)) and non—autonomous coefficient (β(z)) and the instability of rogue waves. Our work suggests that the presence of non-autonomous coefficients can have a significant impact on the emergence of extreme events, particularly in relation to their self—steepening nature.

Propagation of solitons in inhomogeneous birefringent nonlinear dispersive media

We study the existence and wave-speed management of nonlinear localized waves in an inhomogeneous birefringent nonlinear dispersive medium in the presence of external potential. The coupled nonlinear Schrödinger equation with varying coefficients which can be applied to model the interaction among the modes in nonlinear optics and in Bose–Einstein condensation is considered. By means of a complex transformation, we derive new types of periodic nonlinear waves for the present coupled system which are expressed in terms of Jacobi elliptic functions. Novel soliton pulse solutions including bright-dipole and bright–dark solutions are identified in the long-wave limit of these periodic solutions. As an application of these soliton pulses, we discuss their wave-speed management for the different choices of dispersion parameter. The results show that the varied dispersion parameter can be used to control effectively the wave speed of propagating localized waves.

Multi-pole solitons in an inhomogeneous multi-component nonlinear optical medium

Investigations on nonlinear optics are active, with the applications in the modulators, fiber lasers, optical sensors, etc. In this paper, we focus our attention on a system for the ultra-short pulses in an inhomogeneous multi-component nonlinear optical medium. Starting from the existing Lax pair and one-fold Darboux transformation (DT), we construct the N-fold DT of that system, which involves N distinct spectral parameters, where N is a positive integer. The N-fold generalized DT with one spectral parameter is obtained through resorting to the Taylor-series-expansion coefficients of a special solution to that Lax pair. Double-pole soliton solutions of that system are derived via that N-fold generalized DT with N=2. With the aid of the N-fold DT, an N-fold Darboux matrix is constructed, based on which the multi-pole soliton solutions in the determinant form with respect to the electromagnetic field E are determined. Graphically, we find that those double-pole soliton solutions are a kind of the bound-state soliton solutions which represent the elastic interactions between the two solitons. Effects of the coefficients in that system on the double-pole soliton are shown via choosing the trigonometric, linear and quadratic functions. Furthermore, we present the triple-pole soliton and quadruple-pole soliton with respect to E. Our results might be useful in understanding the ultra-short pulses in the nonlinear optical media.

Extension of the stability criterion for dissipative vector solitons of a laser coupled two-dimensional Ginzburg–Landau Equation generated from vector asymmetric inputs

Pulses propagating in inhomogeneous nonlinear media with linear/nonlinear gain and loss described by the vector (2+1)−dimensional cubic-quintic Ginzburg−−Landau equations (CQCGLEs) are considered. The evolution and the stability of the vector dissipative optical solitons generated from an elliptic input are studied. Based on the variational approach, we analyze the influence of various physical parameters on the dynamics of the propagating signal and its relevant parameters. Following a suitable choice of the test function, the stability analysis has been fulfilled by the analysis of the total energy. Numerical results have confirmed the analytical prediction of the collision on the stability of the vector dissipative solitons.

Flat-floor bubbles, dark solitons, and vortices stabilized by inhomogeneous nonlinear media

We consider one- and two-dimensional (1D and 2D) optical or matter-wave media with a maximum of the local self-repulsion strength at the center, and a minimum at periphery. If the central area is broad enough, it supports ground states in the form of flat-floor \textquotedblleft bubbles", and topological excitations, in the form of dark solitons in 1D and vortices with winding number $m$ in 2D. Unlike bright solitons, delocalized bubbles and dark modes were not previously considered in this setting. The ground and excited states are accurately approximated by the Thomas-Fermi expressions. The 1D and 2D bubbles, as well as vortices with $m=1$, are completely stable, while the dark solitons and vortices with $m=2$ have nontrivial stability boundaries in their existence areas. Unstable dark solitons are expelled to the periphery, while unstable double vortices split in rotating pairs of unitary ones. Displaced stable vortices precess around the central point.

Determining Temperature Fields in a Spatially Inhomogeneous Nonlinear Medium

We propose a method of determining temperature fields in a spatially inhomogeneous medium with temperature-dependent thermophysical properties of the material. For this purpose, point and nonlocal transformations of the nonstationary heat conduction equation are used. Examples of applying the theory for various boundary conditions in the spherical symmetric case are given.

Fractional Schrödinger equation in gravitational optics

This paper addresses issues surrounding the concept of fractional quantum mechanics, related to lights propagation in inhomogeneous nonlinear media, specifically restricted to a so-called gravitational optics. Besides Schrödinger–Newton equation, we have also concerned with linear and nonlinear Airy beam accelerations in flat and curved spaces and fractal photonics, related to nonlinear Schrödinger equation, where impact of the fractional Laplacian is discussed. Another important feature of the gravitational optics’ implementation is its geometry with the paraxial approximation, when quantum mechanics, in particular, fractional quantum mechanics, is an effective description of optical effects. In this case, fractional-time differentiation reflexes this geometry effect as well.

Quartic and dipole solitons in a highly dispersive optical waveguide with self-steepening nonlinearity and varying parameters

We show theoretically that highly dispersive optical media characterized by a Kerr nonlinear response may support the existence of quartic and dipole solitons in the presence of the self-steepening effect. The existence and stability properties of these localized pulses are examined in the presence of all the material parameters. Regimes for the modulation instability of a continuous-wave signal propagating inside the nonlinear medium are investigated and an analytic expression for the gain spectrum is obtained and shown to be dependent on the self-steepening parameter in addition to second- and fourth-order group velocity dispersion parameters. Self-similar soliton solutions are constructed for a generalized nonlinear Schr\odinger equation with distributed second-, third-, and fourth-order dispersions, self-steepening nonlinearity, and gain or loss describing ultrashort pulse propagation in the inhomogeneous nonlinear media via the similarity transformation method. The evolutional dynamics of the self-similar structures are investigated in a periodic distributed waveguide system and an exponential dispersion decreasing waveguide.

Manipulation of vector solitons in a system of inhomogeneous coherently coupled nonlinear Schrödinger models with variable nonlinearities

We investigate non-autonomous solitons in a general coherently coupled nonlinear Schrödinger (CCNLS) system with temporally modulated nonlinearities and with an external harmonic oscillator potential. This general CCNLS system encompasses three distinct types of CCNLS equations that describe the dynamics of beam propagation in an inhomogeneous Kerr-like nonlinear optical medium for different choices of nonlinear polarizations owing to the anisotropy of the medium. We identify a generalized similarity transformation to relate the considered model into the standard integrable homogeneous coupled nonlinear evolution equations with constant nonlinearities, accompanied by a constraint relation expressed in the form of the Riccati equation. With the help of a non-standard Hirota’s bilinearization method and exact soliton solutions, we explore the impact of varying nonlinearities and refractive index in the propagation and collisions analytically by reverse engineering. Interestingly, we show the emergence of several modulated solitonic phenomena such as periodic oscillation, amplification, compression, tunneling/cross-over, excitons, as well as their combined effect in the single-soliton propagation and two-soliton collisions with appropriate forms of nonlinearity. Notably, we identify a tool to transform the nature of soliton collisions with certain type of inhomogeneous nonlinearities. The results could be of significant interest to the studies on management of nonlinear waves in the contexts like nonlinear optics and can also be extended to Bose–Einstein condensates and super-fluids.

Stability analysis for moving dissipative solitons in two-dimensional dynamical model

Pulse propagating in inhomogeneous nonlinear media with linear/nonlinear gain and loss described by the asymmetrical (2 + 1)-dimensional cubic-quintic Ginzburg-Landau equation is considered. The evolution and the stability of the dissipative optical solitons generated from an asymmetric input with respect to two transverse coordinates x and y are studied. Our approach is based on the variational method. This approach allows us to analyze the influence of various physical parameters on the dynamics of the propagating signal and its relevant parameters. According to the parameters of the system and a suitable choice of the test function, a domain of dissipative parameters for stable solitonic solutions is determined. Bifurcation diagrams related to the existence of the stationary solutions presented show a good agreement between analytical and numerical results.

Nonlinear propagation of coupled surface TE and leaky TM electromagnetic waves

ABSTRACT Propagation of the coupled electromagnetic wave, which is a superposition of TE surface and TM leaky waves, in the Goubau line (a perfectly conducting cylinder covered by a concentric dielectric layer) filled with nonlinear inhomogeneous medium is studied (if the permittivity is linear, the coupled wave does not exist). Nonlinear coupled TE–TM wave is characterised by two (independent) frequencies and two (coupled) propagation constants (propagation constants). The physical problem is reduced to a nonlinear two-parameter transmission eigenvalue problem for Maxwell's equations. Existence of coupled TE–TM waves is proved. Intervals of localisation of propagation constants are found.

Regimes of two-color light bullet formation in a gradient waveguide

In this paper we concentrate on the remarkable role of a gradient waveguide in two-color light bullet formation. We study generation of the second harmonic in such an inhomogeneous nonlinear medium, taking into account diffraction and relatively weak temporal dispersion. Using the averaged Lagrangian method we consider all possible combinations of the range of group velocities (normal or anomalous dispersion) and waveguide geometry (focusing or defocusing waveguide). Stability conditions for a propagating two-color light bullet are derived analytically. We demonstrate the formation of a stable two-component light bullet in a parabolic planar quadratically nonlinear waveguide either at anomalous or at normal group velocity dispersion. We discuss also the results of numerical simulation confirming our analytical findings. Besides that, simulation allows us to expand the scope of the study and to show light bullet propagation at a certain phase mismatching and the formation of a stable coupled localized structure from a signal at the fundamental frequency as well.

Enhancement of ABCs efficiency at computer simulation of optical pulse interaction with inhomogeneous nonlinear medium

We focus our attention on enhancing of computation performance at numerical simulation of a laser pulse interaction with inhomogeneous nonlinear medium (optical periodic structure or photonic crystal) which is described in the framework of 1D Schrödinger equation without using slowly varying envelope in spatial coordinate. It allows taking into account the optical pulse reflection from medium inhomogeneous. In practice, as a rule at computer simulation the domain size before photonic crystal is many times greater than the photonic crystal size. Decreasing this domain one can essentially increase a computation performance. This aim is achieved by re-statement of the problem: instead of initial distribution of complex amplitude along spatial coordinate before the photonic crystal we consider its specifying as a boundary condition (BC), which results in the pulse propagation in both directions of the propagation axis. Taking this into account and that a part of a laser pulse reflects from faces of photonic crystal layers, we use the artificial boundary conditions (ABCs). To avoid an influence of waves propagated in negative direction of spatial coordinate on the laser pulse interaction in photonic crystal (this influence appears due to imperfection of ABC). We introduce in consideration some number of artificial waves related with the equation under consideration. With this aim, firstly, we provide a computation of the laser pulse propagation in a linear medium and storage the complex amplitude values at chosen section of a coordinate, along which a laser light propagates. Then we use this complex amplitude values as a left boundary condition for 1D nonlinear Schrödinger equation in domain containing a nonlinear photonic crystal. To decrease amplitude of the wave reflected from artificial boundaries. We demonstrate also that instead of a complex amplitude values storage one can use the corresponding analytical expression for the linear Schrödinger equation solution, corresponding to a laser pulse propagation in a linear domain before the photonic crystal. We discuss also various ways for full transform of the optical pulse energy for reflected wave into artificial waves. This results essentially increasing of computation efficiency.

On the existence of the nonlinear leaky TE-polarized waves in a metal–dielectric cylindrical waveguide

We consider propagation of leaky waves in the Goubau line (a perfectly conducting cylinder covered by a concentric dielectric layer) filled with nonlinear inhomogeneous medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green’s function of an auxiliary boundary value problem on an interval. The existence of propagating nonlinear leaky waves for the chosen nonlinearity (Kerr law) is proved using the method of contraction. Conditions under which a finite number of waves can propagate are obtained and the intervals of localization of the corresponding propagation constants are determined. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. In numerical experiments, two types of nonlinearities are considered and compared: Kerr nonlinearity and nonlinearity with saturation. New propagation regimes are discovered.

Non-autonomous solitons in inhomogeneous nonlinear media with distributed dispersion

We show that the methodology based on the powerful method of the so-called hidden symmetry reductions (HSR) provides a systematic way to analytically derive non-autonomous temporal soliton solutions of a one-dimensional inhomogeneous nonlinear Schrodinger equation with a linear gain or loss, a linear density profile, and distributed dispersion. The fundamental innovation of our approach is to notice that the ansatz used in the powerful method of the HSR contains a free parameter that can be used to control the wave propagations related to both the rogue and the symmetric cnoidal solitons raising the possibility of relative experiments and potential applications in nonlinear optics and Bose–Einstein condensates (BECs). We show that for BECs with the time-dependent s-wave scattering length in linear potential when the loss or the gain of atoms is taken into consideration our algorithm is a useful tool to control non-autonomous bright rogue and symmetric cnoidal solitons of BECs. Our results show that the nonlinearity and the linear trapping potential play important roles in the evolutional characteristics such as amplitude, width, trajectory, linear chirp phase, phase offset, and phase shift. Our analysis might have the applications in the theory of nonlinear transmission networks for achieving pulse compression.

Excitation manipulation of three-dimensional completely localized rogue waves in a partially nonlocal and inhomogeneous nonlinear medium

We study a (2 + 1)-dimensional variable-coefficient-coupled partially nonlocal nonlinear Schrodinger equation with nonlinearities localized in x and y-directions and non-localized in z-direction, and set up a one-to-one connection between it and the standard nonlinear Schrodinger equation. Using this one-to-one connection, and with the help of bilinear method, we analytically derive first-order and second-order rogue wave solutions including rogue wave triplet solution, which are completely localized in three-dimensional space. By modulating the maximal value of the effective propagation distance, and comparing this maximal value with the top (peak) position of rogue wave excitation in the exponential diffraction decreasing system, we discuss the excitation manipulation of three-dimensional rogue waves, including original excitation, top excitation, tail excitation and fast complete excitation of single rogue wave and rogue wave triplet.

Influence of pseudo-stimulated-Raman-scattering on the modulational instability in an inhomogeneous nonlinear medium

We study the influence of the pseudo-Raman effect on the modulation instability (MI) in an inhomogeneous nonlinear medium. The system is governed by the extended nonlinear Schrodinger (NLS) equation, which is derived from a Zakharov-type system for the interaction between high-frequency and low-frequency waves. The resulting inhomogeneous NLS equation includes a pseudo-stimulated-Raman-scattering term. The model may apply to the propagation of waves in plasmas, surface waves in the ocean, and optical beams in nonlinear waveguides. This nonautonomous model is converted into an autonomous equation by a similarity transformation, allowing us to find MI regions of the system. The results suggest new possibilities for controlling multi-soliton patterns generated by the MI. The MI patterns are also produced by numerical simulations that confirm the analytical results.

Preventable interaction of attosecond soliton in an inhomogeneous lossy fiber: Application to dispersion and nonlinearity management

In this paper, inhomogeneous sixth order nonlinear Schrödinger equation with variable dispersion, nonlinearity and loss/gain is considered which describes the propagation of attosecond soliton in an inhomogeneous nonlinear fiber medium. Multi-soliton solutions are obtained by the Darboux transformation technique based on the newly constructed Lax pair. In order to study the impact of variable coefficients on soliton dynamics in an inhomogeneous fiber, various forms of inhomogeneous profiles for variable coefficients are considered. By symbolic computation, characteristics of these coefficients are discussed including the soliton amplification, soliton compression, boomerang soliton and breathing soliton. We further discuss the preventable interaction between the attosecond solitons which are attained through manipulating the control parameters in various inhomogeneous functions. Our results may be useful for the experimental control and manipulation of the attosecond soliton in inhomogeneous optical fiber system.

Modulation of localized solutions in an inhomogeneous saturable nonlinear Schrödinger equation

In this paper we study the modulation of localized solutions by an inhomogeneous saturable nonlinear medium. Throughout an appropriate ansatz we convert the inhomogeneous saturable nonlinear Schrodinger equation in a homogeneous one. Then, via a variational approach we construct localized solutions of the autonomous equation and we present some modulation patterns of this localized structures. We have checked the stability of such solutions through numerically simulations.