Finite Initial Perturbations Research Articles

Chirped periodic and localized waves of the (1+2)-dimensional chiral nonlinear Schrödinger equation

We study the existence and stability of nonlinearly chirped periodic waves and soliton structures in an optical medium wherein the pulse propagation is governed by the (1+2)-dimensional chiral nonlinear Schrödinger equation. An exact periodic wave solution is presented for the model equation in the presence of all physical processes by using the complex envelope traveling-wave ansatz. A class of optical gray-type solitons is obtained in the special long wave limit. The properties of these structures such as the velocity and wave number are determined by the system parameters. It is found that the frequency chirp accompanying these optical waves is inversely proportional to the intensity of the wave and its amplitude can be controlled by choosing the dispersion parameter appropriately. In addition, the stability of these waveforms is discussed numerically under some initial perturbations. The results show that those nonlinear waves can propagate in a stable fashion in the nonlinear medium under finite initial perturbations, such as amplitude and white noise.

Propagation of chirped gray solitons in weakly nonlocal media with parabolic law nonlinearity and spatio-temporal dispersion

In our research, we have examined the existence and stability of chirped periodic and solitary waves in a weakly nonlocal nonlinear medium, which exhibits various types of effects, including inter-modal dispersion, nonlinear dispersion, detuning, spatio-temporal, and parabolic law nonlinearity. By studying the nonlinear Schrödinger equation that describes the field dynamics in this system, a class of nonlinearly chirped periodic waves is derived in the presence of all physical processes. In addition, we have obtained solitary waves of the gray type in the long-wave limit of these nonlinear waveforms. We have found that the frequency chirp associated with these optical waves depends on their intensity and its magnitude can be controlled by manipulating the nonlinear dispersion parameter. Furthermore, we have numerically studied the stability of the gray soliton solution under finite initial perturbations. Our results indicate that the nonlinear waves we have identified represent new types of extremely robust chirped localized structures in weakly nonlocal nonlinear parabolic law media.

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.

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.

Chirped self-similar solitary waves for the generalized nonlinear Schrödinger equation with distributed two-power-law nonlinearities.

We investigate the propagation characteristics of the chirped self-similar solitary waves in non-Kerr nonlinear media within the framework of the generalized nonlinear Schrödinger equation with distributed dispersion, two-power-law nonlinearities, and gain or loss. This model contains many special types of nonlinear equations that appear in various branches of contemporary physics. We extend the self-similar analysis presented for searching chirped self-similar structures of the cubic model to a more general problem involving two nonlinear terms of arbitrary power. A variety of exact linearly chirped localized solutions with interesting properties are derived in the presence of all physical effects. The solutions comprise bright, kink and antikink, and algebraic solitary wave solutions, illustrating the potentially rich set of self-similar pulses of the model. It is shown that these optical pulses possess a linear chirp that leads to efficient compression or amplification, and thus are particularly useful in the design of optical fiber amplifiers, optical pulse compressors, and solitary wave based communication links. Finally, the stability of the self-similar solutions is discussed numerically under finite initial perturbations.

Propagation of chirped gray optical dips in nonlinear metamaterials

Abstract We investigate the existence and stability properties of nonlinearly chirped solitary waves on a continuous-wave background in nonlinear metamaterials with higher-order effects such as pseudo-quintic nonlinearity and self-steepening effect. Novel classes of chirped gray solitary pulses (dark pulses with nonzero minimum intensity) are derived by employing the traveling-wave method. The conditions for the formation of these localized pulses as well as the nonlinear chirp associated to them are presented. The stability of the obtained structures is also demonstrated numerically under finite initial perturbations. The results show that the envelope pulses obtained here represent new types of extremely robust nonlinear chirped structures in highly nonlinear metamaterials.

Propagation properties of dipole-managed solitons through an inhomogeneous cubic–quintic–septic medium

The presence of inhomogeneities in an optical nonlinear material may significantly change the physical features of propagating envelopes. In this paper, we discuss the propagation of very short pulses in an inhomogeneous highly nonlinear single-mode fiber within the context of a higher-order nonlinear Schrödinger equation exhibiting a diversity of important physical effects and spatially inhomogeneous coefficients. Additional effects to the cubic model include distributed third- and fourth-order dispersion, self-steepening, self-frequency shift due to stimulated Raman scattering, quintic–septic non-Kerr nonlinearities, the time derivative of non-Kerr nonlinear terms, and gain or loss term. By adopting a complex amplitude ansatz solution that is composed of the product of bright and dark solitary waves, the exact dipole soliton solution is derived. The conditions on the inhomogeneous fiber parameters for the existence of dipole structures are also reported. It is shown that the soliton dynamics in the inhomogeneous fiber media can be effectively controlled by choosing the parameters associated with the group velocity and the gain or loss term appropriately. Different from the higher-order nonlinear Schrödinger equation with constant coefficients, dipole solitons in variable-coefficient model have shown novel and interesting features. Finally, the stability of the dipole solitons is discussed numerically under finite initial perturbations.

Nonlinear excitation of the ablative Rayleigh-Taylor instability for all wave numbers.

Small-scale perturbations in the ablative Rayleigh-Taylor instability (ARTI) are often neglected because they are linearly stable when their wavelength is shorter than a linear cutoff. Using two-dimensional (2D) and three-dimensional (3D) numerical simulations, it is shown that linearly stable modes of any wavelength can be destabilized. This instability regime requires finite amplitude initial perturbations and linearly stable ARTI modes to be more easily destabilized in 3D than in 2D. It is shown that for conditions found in laser fusion targets, short wavelength ARTI modes are more efficient at driving mixing of ablated material throughout the target since the nonlinear bubble density increases with the wave number and small-scale bubbles carry a larger mass flux of mixed material.

Bäcklund transformation, analytic soliton solutions and numerical simulation for a (2+1)-dimensional complex Ginzburg–Landau equation in a nonlinear fiber

In this paper, we investigate a nonlinear fiber described by a (2[Formula: see text]+[Formula: see text]1)-dimensional complex Ginzburg–Landau equation with the chromatic dispersion, optical filtering, nonlinear and linear gain. Bäcklund transformation in the bilinear form is constructed. With the modified bilinear method, analytic soliton solutions are obtained. For the soliton, the amplitude can decrease or increase when the absolute value of the nonlinear or linear gain is enlarged, and the width can be compressed or amplified when the absolute value of the chromatic dispersion or optical filtering is enhanced. We study the stability of the numerical solutions numerically by applying the increasing amplitude, embedding the white noise and adding the Gaussian pulse to the initial values based on the analytic solutions, which shows that the numerical solutions are stable, not influenced by the finite initial perturbations.

Self-similar localized pulses for the nonlinear Schrödinger equation with distributed cubic-quintic nonlinearity

We obtain the exact analytical self-similar bright-, dark-, and kink-solitary-wave solutions of the nonlinear Schr\odinger equation with localized inhomogeneous cubic-quintic nonlinearity by employing a similarity transformation. This equation could be a model equation of stable pulse propagation beyond ultrashort range in optical fiber communication systems in inhomogeneous media. We have investigated that the self-similar bright- and dark-solitary-wave solitons show interesting compression and amplification features that can be controlled by suitably changing the phase modulation parameter and consequently varying the nonlinear parameter. We unearth a surprising connection between optical self-similar localized waves in graded-index inhomogeneous media and solitons in homogeneous media with the same type of cubic-quintic nonlinearity. Finally, the stability of the solutions is discussed numerically under finite initial perturbations.

Vector–soliton bound states for the coupled mixed derivative nonlinear Schrödinger equations in optical fibers

In this paper, the vector–soliton bound states (VSBSs) are investigated for the coupled mixed derivative nonlinear Schrödinger equations, which can describe the pulse propagation in the femtosecond regime of birefringent optical fibers. Symmetric and asymmetric VSBSs with the periodic collisions are obtained and analyzed. The intensity profile and collision period of the bound state are related to the ratio of the real parts of two wavenumbers. The separation factors are introduced in the solutions to linearly adjust the soliton separations. Amplitude-changing collisions between the VSBS and vector bright solitons are also obtained. Via the split-step Fourier method, stability analysis shows that the VSBSs can resist the finite initial perturbations. In addition, the derivative cubic nonlinearity and cubic nonlinearity terms are found to both have no influence on the types of VSBSs, but leads to one-sided compression, center shifts and amplitude decrease of the VSBSs. Energy-exchanging phenomena during the amplitude-changing collisions are not affected, either.

Exact spatial soliton solution for nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index

In this paper, we analyze (2+1)-dimensional nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index in the transverse direction, and obtain an exact solution in explicit form using an ansatz method. Finally, the stability of the solution is discussed numerically, and the result reveals that the solution is stable to the finite initial perturbations.

Soliton Solution, Bäcklund Transformation, and Conservation Laws for the Sasa-Satsuma Equation in the Optical Fiber Communications

Under investigation in this paper, with symbolic computation, is the Sasa-Satsuma (SS) equation which can describe the propagation of ultra short pulses in optical fiber communications. By virtue of the Ablowitz-Kaup-Newell-Segur procedure, the Lax pair for the SS equation is directly established. Based on such a Lax pair, a Bäcklund transformation is constructed, through which the explicit onesoliton solution is derived.Meanwhile, an infinite number of conservation laws is provided to indicate the integrability of the SS equation in the Liouville sense. To further understand the stability of the one-soliton solution, we employ the split-step Fourier method to simulate the propagation of the soliton pulses under the finite initial perturbations. In addition, the interaction of two adjacent pulses with different separation distances is investigated through numerical simulation. Analytic and numerical results discussed in this paper are expected to be applied to the description of the optical pulse propagation.

Exact spatial soliton solution for nonlinear Schrödinger equation with a type of transverse nonperiodic modulation

In this paper, we analyze (2 + 1)D nonlinear Schrödinger (NLS) equation based on a type of nonperiodic modulation of linear refractive index in the transverse direction. We obtain an exact solution in explicit form for the (2 + 1)D nonlinear Schrödinger (NLS) equation with the nonperiodic modulation. Finally, the stability of the solution is discussed numerically, and the results reveal that the solution is stable to the finite initial perturbations.

Generation and propagation of pulse trains with ultrashort pulse separation

We investigate the nonlinear Schrödinger equation with variable coefficients by employing perturbation method. The analysis solution of the harmonic form is presented. The solution is one of forms to describe pulse trains with ultrashort pulse separation, which is about two orders of magnitude shorter than one of sech-type solitons considered before. And we could systematically adjust the perturbation parameter to obtain different pulse separation. As an example, we consider a nonlinear dispersive system with spatial parameter variations, and the results show that, the pulse train with ultrashort pulse separation presented by analysis solution may keep its shape even if the velocity is changed. The stability of the solution is discussed numerically, and the results reveal that the finite initial perturbations, such as white noise could not influence the main character of the solution. In addition, the stability of the solution is also discussed under more general conditions.

在具有空间非均匀非线性的(2+1)维非线性光学介质中光的传输

The (2+1)-dimensional nonlinear Schrodinger (NLS) equation with spatially inhomogeneous nonlinearities is investigated, which describes propagation of light in (2+1)-dimensional nonlinear optical media with inhomogeneous nonlinearities. New types of optical modes and nonlinear effects in optical media are presented numerically. The results reveal that the regular split of beam can be obtained in (2+1)-dimensional nonlinear optical media with inhomogeneous nonlinearities, by adjusting the guiding parameter. Furthermore, the stability of beam regular split is discussed numerically, and the results reveal that the beam regular split is stable to the finite initial perturbations.

Stability of chirped bright and dark soliton-like solutions of the cubic complex Ginzburg–Landau equation with variable coefficients

We consider an inhomogeneous optical fiber system described by the generalized cubic complex Ginzburg–Landau (CGL) equation with varying dispersion, nonlinearity, gain (loss), nonlinear gain (absorption) and the effect of spectral limitation. Exact chirped bright and dark soliton-like solutions of the CGL equation were found by using a suitable ansatz. Furthermore, we analyze the features of the solitons and consider the problem of stability of these soliton-like solutions under finite initial perturbations. It is shown by extensive numerical simulations that both bright and dark soliton-like solutions are stable in an inhomogeneous fiber system. Finally, the interaction between two chirped bright and dark soliton-like pulses is investigated numerically.

Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation

We consider the inhomogeneous higher-order nonlinear Schrodinger equation and explicitly present exact combined solitary wave solutions that can describe the simultaneous propagation of bright and dark solitary waves in a combined form in inhomogeneous fiber media or in optical communication links with distributed parameters. Furthermore, we analyze the features of the solutions, and numerically discuss the stabilities of these solitary waves under slight violations of the parameter conditions and finite initial perturbations. The results show that there exist combined solitary wave solutions in an inhomogeneous fiber system, and the combined solitary wave solutions are stable under slight violations of the parameter conditions and finite initial perturbations. Finally, the interaction between two neighboring combined solitary waves is numerically discussed.

Exact multisoliton solutions of the higher-order nonlinear Schrödinger equation with variable coefficients

We investigate the generalized higher-order nonlinear Schrödinger equation with variable coefficients under two sets of parametric conditions. The exact one-soliton solution is presented by the ansatz method for one set of parametric conditions. For the other, exact multisoliton solutions are presented by employing the Darboux transformation based on the Lax pair. As an example, we consider a soliton control system, and the results show that the soliton control system may relax the limitations to parametric conditions. The stability of the solution is discussed numerically; the results reveal that finite initial perturbations, such as amplitude, chirp, or white noise, could not influence the main character of the solution. In addition, the evolution of a quite arbitrary Gaussian pulse and the interaction between neighboring pulses have been studied in detail.

The relaxation of two-dimensional rolls in Rayleigh–Bénard convection

Large aspect ratio, two-dimensional, periodic convection layers containing a Boussinesq fluid of finite Prandtl number bounded by rigid or free horizontal surfaces are investigated numerically. The fluid equations are solved using both a standard pseudospectral and a Fourier integral method for the time evolution of finite initial perturbations, both random thermal perturbations and localized roll disturbances, into a final equilibrium state. The suggestion that a Fourier integral solution method is required to yield roll relaxation, the two-dimensional process increasing the convection wavelength to values larger than critical, is investigated. Roll relaxation is found for both free-slip and no-slip surfaces using either solution method as long as the initial state is chosen to be of the form of a localized roll disturbance. A wide variety of simulations are performed and roll relaxation is found to be independent of the periodic domain length, weakly dependent on the Rayleigh number and dependent upon the magnitude of the initial localized roll disturbances.