Higher-order Nonlinear Schrodinger Equation Research Articles

Rogue-wave solutions of a higher-order nonlinear Schrödinger equation for inhomogeneous Heisenberg ferromagnetic system

In this paper, a higher-order nonlinear Schrodinger equation for the inhomogeneous Heisenberg ferromagnetic spin system is studied. By virtue of the generalized Darboux transformation, higher-order rogue-wave solutions are derived. Rogue-wave propagation and interaction are analyzed. We have observed that perturbation parameter and inhomogeneities in the medium affect the propagation speed and direction of first-order rogue waves and interaction of second-order rogue waves.

Bright hump solitons for the higher-order nonlinear Schrödinger equation in optical fibers

Under investigation is the higher-order nonlinear Schrodinger equation with the third-order dispersion (TOD), self-steepening (SS) and self-frequency shift, which can be used to describe the propagation and interaction of ultrashort pulses in the subpicosecond or femtosecond regime. Through the introduction of an auxiliary function, bilinear form is derived. Bright one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. From the one-soliton solutions, we present the parametric regions for the existence of single- and double-hump solitons, and find that they are affected by the coefficients of the group velocity dispersion (GVD) and TOD. Besides, propagation of the one single- or double-hump soliton is observed. We analytically obtain the amplitudes for the single- and double-hump solitons, and calculate the interval between the two peaks for the double-hump soliton. Moreover, soliton amplitudes are related to the coefficients of the GVD, TOD and SS, while the interval between the two peaks for the double-hump soliton is dependent on the coefficients of the GVD and TOD. Interactions are seen between the (i) two single-hump solitons, (ii) two double-hump solitons, and (iii) single- and double-hump solitons. Those interactions are proved to be elastic via the asymptotic analysis.

Higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms: A model for sub-10-fs-pulse propagation

We analytically solved the higher-order nonlinear Schrodinger (HNLS) equation with non-Kerr nonlinearity under some parametric conditions and investigated explicitly bright and dark solitary wave solutions. Periodic wave solutions are also presented. The functional form of the bright and dark solitons presented are different from fundamental known sech(.) and tanh(.) respectively. We have estimated theoretically the size of the derivative non-Kerr nonlinear coefficients of the HNLS equation that agreed the reality of the waveguide made of highly nonlinear optical materials, could be used as the model parameters for sub-10fs pulse propagation.

Exact solutions of some nonlinear partial differential equations using functional variable method

The functional variable method is a powerful solution method for obtaining exact solutions of some nonlinear partial differential equations. In this paper, the functional variable method is used to establish exact solutions of the generalized forms of Klein–Gordon equation, the (2 + 1)-dimensional Camassa–Holm Kadomtsev–Petviashvili equation and the higher-order nonlinear Schrodinger equation. By using this useful method, we found some exact solutions of the above-mentioned equations. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions. It is shown that the proposed method is effective and general.

Periodic and Solitary Wave Solutions of Complex Nonlinear Evolution Equations by Using First Integral Method

Abstract In this paper, we utilize the first integral method with symbolic computation to constructing periodic and solitary wave solutions of complex nonlinear evolution equations. With the aid of Maple, new exact solutions of two nonlinear evolution equations in physics, namely, generalized Zakharov equations (GZ) and higher-order nonlinear Schrodinger equation (HONLS) describing propagation of ultrashort pulses in nonlinear optical fibers are obtained.

Application of Kudryashov method for high-order nonlinear Schrödinger equation

Higher-order nonlinear Schrodinger equation for describing the propagation of femtosecond pulses in optical fibers is studied. Kudryashov method is used for obtaining exact soliton solutions of this equation.

Magnetization Reversal through a Soliton in a Site-Dependent Weak Ferromagnet

Magnetization reversal or switching is the process by which the magnetization of a specimen is changed from one stable direction into another. Switching the magnetization of a magnetic bit through the flipping of a soliton offers the possibility of developing a new innovative approach for data storage technologies. The spin dynamics of a site-dependent ferromagnet with the antisymmetric Dzyaloshinskii-Moriya interaction is governed by a generalized inhomogeneous higher order nonlinear Schrodinger equation. We demonstrate the magnetization reversal through the flipping of a soliton in the ferromagnetic medium by solving numerically the two coupled evolution equations for the velocity and amplitude of the soliton using the fourth order Runge-Kutta method. We propose a new approach for inducing the flipping behaviour of a soliton in the presence of an inhomogeneity by tuning the parameter associated with the Dzyaloshinskii-Moriya interaction, which causes the soliton to move with constant velocity and amplitude along the spin lattice.

Radiationless Higher-Order Embedded Solitons

The soliton solutions of a higher-order nonlinear Schrödinger equation including nonlinearities of orders \(2b+1\) and \(4b+1\), as well as second- and fouth-order dispersive terms, are obtained. It is shown that these solitons may be (or may not be) embedded, depending on the coefficients of the equation. The radiationless character of the embedded solitons is explained by analyzing the Fourier transform of the equation. The stability of these solitons is studied analytically and numerically. In the analytical approach, we combined the Vakhitov–Kolokolov stability criterion with the standard variational method. This combined technique predicts that the standard solitons are unstable and the embedded ones might be stable. The numerical results confirm these predictions, and show that the embedded and standard solitons respond very differently to perturbations.

The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions

The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation, the generalized nonlinear Schrodinger (GNLS) equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.

Dark Bound Solitons and Soliton Chains for the Higher-Order Nonlinear Schrödinger Equation

Dark bound solitons and soliton chains without interactions are investigated for the higher-order nonlinear Schrodinger (HNLS) equation, which can model the propagation of the femtosecond optical pulse under some physical situations in nonlinear fiber optics. Via the modulation of parameters for the analytic solutions, different types of dark bound solitons and soliton chains can be derived for the HNLS equation. In addition, stabilities of those structures are checked through numerical simulations. Our discussions are expected to be helpful in interpreting those new structures, and applied to the long-distance transmission of the femtosecond pulses in optical fibers.

New Solutions for the Higher-Order Nonlinear Schrödinger Equation Using Integral Methods

In this paper, we use two integral methods, the first integral method and the direct integral method to study a higher-order nonlinear Schrodinger equation (NLSE). The application of the first integral method yield trigonometric func- tion solutions and solitary wave solutions. Using the direct integration lead to shock wave solution and Jacobi elliptic func- tion solutions. The direct integral method is more concise and direct than the first integral method.

Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrödinger equation

We have studied the modulational instability (MI) of the higher-order nonlinear Schrodinger (HNLS) equation with non-Kerr nonlinearities in an optical context and presented an analytical expression for MI gain to show the effects of non-Kerr nonlinearities and higher-order dispersions on MI gain spectra. In our study, we demonstrate that MI can exist not only for the anomalous group-velocity dispersion (GVD) regime, but also in the normal GVD regime in the HNLS equation in the presence of non-Kerr quintic nonlinearities. The non-Kerr quintic nonlinear effect reduces the maximum value of the MI gain and bandwidth and plays a sensitive role over the Kerr nonlinearity, which leads to continuous wave breaking into a number of stable wave trains of ultrashort optical pulses that can be used to generate the stable supercontinuum white-light coherent sources.

Computer simulation of the propagation of extremely short femtosecond pulse in single-mode fiber with shifted dispersion characteristics

We present the results of theoretical study of generation of spectral supercontinuum by an extremely short femtosecond laser pulse with the central wavelength corresponding to zero dispersion, propagating in a single-mode fiber with shifted dispersion characteristics. High-order nonlinear Schrodinger equation (HONSE) is considered with higher-order terms taken into account. Numerical integration of HONSE is performed by the method of lines. Time dependence of instantaneous frequencies of the femtosecond pulse is obtained. The evolution of the pulse spectrum is calculated and the medium length dependence of the spectrum width is obtained. For study of the time-frequency dynamics of extremely short femtosecond laser pulse propagating in single-mode optical fiber with allowance for higher order terms, the Wigner transformation (WT) was used.

Numerical simulation of propagation of a femtosecond optical soliton in a single-mode fiber with allowance for the imaginary part of Raman response

We consider the effect of Raman inertial response of a medium on the stability of a first-order femtosecond soliton. Numerical solution to the high-order nonlinear Schrodinger equation, with the complex Raman term, describing propagation of a femtosecond optical soliton in a single-mode fiber, is obtained. It is shown that a soliton solution of the high-order nonlinear Schrodinger equation exists under certain conditions imposed on the equation coefficients. These conditions lead to limitations on the wavelength, fiber type, and the highest energy. Results of numerical solutions are in agreement with available experimental data.

Slow-light solitons in three-level atomic systems modified by a microwave field

We study the modified effect of slow-light soliton in a resonant three-level atomic system via electromagnetically induced transparency (EIT) by utilizing a microwave field. We derive a high-order nonlinear Schrodinger equation by using a perturbation method of multiple-scales, and calculate the modification of soliton velocity and frequency shift. We find that in the presence of the microwave field an obvious decrease of propagating velocity of soliton can be realized, which provides an effective method to slow down optical solitons in EIT systems. We also find that the down shift of oscillating frequency of soliton in such system can be largely suppressed by the microwave field.

Conservation laws, soliton solutions and modulational instability for the higher-order dispersive nonlinear Schrödinger equation

In this paper, analytically investigated is a higher-order dispersive nonlinear Schrodinger equation. Based on the linear eigenvalue problem associated with this equation, the integrability is identified by admitting an infinite number of conservation laws. By using the Darboux transformation method, the explicit multi-soliton solutions are generated in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.

The auxiliary elliptic-like equation and the exp-function method

The auxiliary equation method is very useful for finding the exact solutions of the nonlinear evolution equations. In this paper, a new idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the auxiliary elliptic-like equation are derived using exp-function method, and then the exact solutions of the nonlinear evolution equations are derived with the aid of auxiliary elliptic-like equation. As examples, the RKL models, the high-order nonlinear Schrodinger equation, the Hamilton amplitude equation, the generalized Hirota-Satsuma coupled KdV system and the generalized ZK-BBM equation are investigated and the exact solutions are presented using this method.

Solitary wave solutions of higher-order NLSE with Raman and self-steepening effect in a cubic–quintic–septic medium

We find bright and dark solitary wave solutions for the higher-order nonlinear Schrodinger equation with cubic–quintic–septic terms adopting the ansatz solution of Li et al. [Li Z, Li L, Tian H, Zhou G. Phys. New types of solitary wave solutions for the higher-order nonlinear Schrödinger equation. Phys Rev Lett 2000;84(18):4096–99.] which may describe propagation of pulses upto the order of 10fs in a non-Kerr media. In this context, we have taken into account both the Raman and the self-steepening effect. All the solitary wave parameters and the parametric conditions for the solitary wave solutions are worked out.

High-order nonlinear Schrödinger equation and superluminal optical solitons in room-temperature active-Raman-gain media

We make a detailed study on the dynamics of gain-assisted superluminal optical solitons in a three-state active-Raman-gain medium at room temperature. Using a method of multiple-scales we derive a high-order nonlinear Schrodinger equation with correction terms contributed from differential gain, nonlinear dispersion, delay in nonlinear refractive index, and third-order dispersion of the system. We show that for a long pulse with realistic physical parameters the high-order correction terms are small and can be taken as perturbations. However, for a shorter pulse these higher-order correction terms are significant and hence must be treated on equal footing as the terms in the nonlinear Schrodinger equation. We provide exact soliton solutions of the higher-order nonlinear Schrodinger equation and demonstrate that such solitons have still superluminal propagating velocity and can be generated at very low light intensity.

Ultra-short pulse propagation in birefringent fibers—the projection operator method

We examine the propagation of ultra-short optical light pulses in dispersion-managed birefringent fiber transmission systems, in which the pulse dynamics is governed by the coupled higher-order nonlinear Schrodinger equations with higher-order linear and nonlinear optical effects. We derive the equations of motion in terms of pulse parameters such as amplitude, temporal position, width, chirp, frequency and phase, using a projection operator method, and we obtain the spatial dynamical behavior of picosecond and femtosecond pulse parameters. From our detailed analysis, we show that the stimulated Raman scattering has a strong impact on the pulse dynamics.