Higher-order Nonlinear Schrodinger Equation Research Articles

Construction of modulation instability analysis and optical soliton solutions of pertubed nonlinear Schrödinger dynamical equation with power law nonlinearity in non-kerr medium

Abstract In this paper the perturbed nonlinear Schrodinger equation (PNLSE) with power law nonlinearity in the medium of non-kerr is analyzed by employing Reccati equation mapping method. Perturbed nonlinear Schrodinger equation depict the effect of quantic non-linearity on propagation of the ultra-short optical pulses in the medium of non-kerr, similar to optical fibers. We achieved various types of soliton solutions, some of them are novel and does not exist previously. Graphical presentation of some obtained exact solutions are also given through assigning suitable values to the parameters that aids to understand the physical phenomenon of this models. Modulation Instability (MI) is also discussed by standard linear-stability analysis that shows about all achieved results are exact and stable. The computational exertion and achieved outputs shows that the current proposed technique is powerful and effectual. Furthermore many higher order non-linear Schrodinger equations can be solved using current method.

Study of optical soliton fibers with power law model by means of higher-order nonlinear Schrödinger dynamical system

Abstract In this paper, the higher-order nonlinear Schrodinger equation (NLSE) represents description of the propagation of short light pulses in mono-mode optical fibers. The optical solitons and solitary wave solutions of higher-order nonlinear Schrodinger equation mono-mode optical fibers are organized by apply the modified simple equation method to get the exact solutions and possibility of given this solution graphically at a given moment. This solutions help the researchers to study the physical properties of this model and its applications. There are many different types of models that we find in applied science that can be solved by this effective and reliable method. This method can be effective and successful in solving and understanding many of the problems of higher-order non-linear in the various fields of research and advanced.

Lax Pair, Improved Γ $\varGamma $ -Riccati Bäcklund Transformation and Soliton-Like Solutions to Variable-Coefficient Higher-Order Nonlinear Schrödinger Equation in Optical Fibers

In this paper, the higher-order nonlinear Schrodinger equation with variable-coefficient for the propagation of femtosecond pulses in optical fibers is analytically investigated. Lax pair is constructed via the Ablowitz-Kaup-Newell-Segur system, an infinite number of conservation laws are derived based on the Lax pair. Introducing an auxiliary function, an improved $\varGamma $ -Riccati Backlund transformation is presented, which can generate successively nonlinear superposition formula. Moreover, one and two soliton-like solutions are obtained.

Periodic attenuating oscillation between soliton interactions for higher-order variable coefficient nonlinear Schrödinger equation

According to the change in the amplitude of the oscillation, it can be divided into equal-amplitude oscillation, amplitude-reduced oscillation (attenuating oscillation) and amplitude-increasing oscillation (divergence oscillation). In this paper, the periodic attenuating oscillation of solitons for a higher-order variable coefficient nonlinear Schrodinger equation is investigated. Analytic one- and two-soliton solutions of this equation are obtained by the Hirota bilinear method. By analyzing the soliton propagation properties, we study how to choose the corresponding parameters to control the soliton propagation and periodic attenuation oscillation phenomena. Results might be of significance for the study of optical communications including soliton control, amplification, compression and interactions.

Contrast of optical activity and rogue wave propagation in chiral materials

We report the contrast of optical activity and properties of nonparaxial optical rogue waves for the higher-order nonparaxial chiral nonlinear Schrodinger (NLS) equation. The latter describes the propagation of ultrashort optical pulses in chiral optical fibers. Both first- and second-order rogue wave solutions are investigated analytically by the modified Darboux transformation (MDT) and numerically by one pseudo-spectral method. The interplay of chiral materials on rogue wave propagation is elucidated, and the results can be applied on optical communication and in others physical systems.

Vector rogue waves, rogue wave-to-soliton conversions and modulation instability of the higher-order matrix nonlinear Schrödinger equation

Using the $4\times 4$ Lax pair and Darboux-dressing transformation, we study the existence and properties of vector rogue waves of the higher-order matrix nonlinear Schrodinger equation. Our analytical results show at least three important features that are associated to rogue-wave formations and dynamics. Firstly, we show that vector rogue waves can be converted to vector solitons on mixed backgrounds. To be specific, one component of the vector soliton propagates on the constant background, while the other one propagates on the zero background. Such novel characteristics (vector solitons on mixed backgrounds) arise from higher-order effects. Secondly, the link between baseband modulation instability and rogue waves is displayed by showing the gain function. Thirdly, with the continuous-wave and mixed backgrounds, we produce a family of rational solutions for the purpose of describing the rogue waves. Bright-bright and dark-bright rogue waves with single hump and double humps are presented. We observe that the dark rogue wave could turn into the double-hump bright rogue wave. It is found that the bright rogue wave on the continuous-wave background can split up, giving birth to multiple rogue waves, while two humps of bright rogue wave on the zero background merge, giving birth to a bright rogue wave with one hump.

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.

Generation and control of multiple solitons under the influence of parameters

In this paper, the analytic three-soliton solution for a high-order nonlinear Schrodinger equation is obtained by the Hirota’s bilinear method. The transmission characteristics of three solitons are discussed. By selecting relevant parameters, soliton interactions are presented, and the method of generating new solitons is suggested. The influences of corresponding parameters on soliton transmission and interactions are analyzed. Results of this paper are helpful for enriching the soliton theory and studying the signal routing system.

Structure of optical soliton solutions for the generalized higher-order nonlinear Schrödinger equation with light-wave promulgation in an optical fiber

In this work, we investigate unprecedented optical closed form of solutions for the generalized higher-order nonlinear Schrodinger equation. This vital model in the optical fiber. Schrodinger equation describes the promulgation of the light-wave in an optical fiber. To achieve the desired objective of the search we apply a generalized Kudryashov method. This method is considered as one of the recent methods that developed in last decades. In this discussion, we discuss what differentiates this method from other methods in the field of finding precise solutions to non-linear partial differential equations and that debate takes place using two axes. The first is comparing the same way with the other ways ahead of this area while the second axis is comparing the solutions obtained using this method with solutions found using other methods and how these solutions converge and how much able to be utilized in that modern method (Generalized Kudryashov method).

Dispersive optical soliton solutions of the higher-order nonlinear Schrödinger dynamical equation via two different methods and its applications

In this paper, we apply two methods which are the arbitrary nonlinear parameters and the exponential rational function method to construct many new exact solutions of the higher-order nonlinear partial differential equations, namely, the higher-order nonlinear Schrodinger (HNLS) equation. The solutions obtained by the current methods are generalized periodic solutions. The shape of the solutions can be well controlled by adjusting the parameters of the system. Optical soliton solutions obtained can be used to transport information in the telecommunication domain. It also comes from this work that the behavior of this HNLS equation may be easily studied by means of the phase plane plot which is the best tool to predict some solutions.

Some types of dark soliton interactions in inhomogeneous optical fibers

Dark solitons are the subject of intense theoretical and experimental studies in nonlinear optics due to their unique characteristics compared with bright solitons. In this paper, the variable coefficient high-order nonlinear Schrodinger equation in the inhomogeneous optical fiber is investigated. Via the Hirota bilinear method and symbolic computation, the analytic dark two-soliton solutions are obtained. With the suitable choices of functions and coefficients for the obtained dark two-soliton solutions, some new phenomena are presented for the first time. The influences on phases and amplitudes of soliton interactions are detailed analyzed. Moreover, sets of double-triangle structures and methods of changing the propagation direction of dark solitons are introduced. Finally, by choosing suitable functions of the fourth-order dispersion parameter, the arch-structure and M-structure interactions are revealed. Results may be potentially useful in designing all-optical switches and optical fibers.

Control of interaction between femtosecond dark solitons in inhomogeneous optical fibers

In this paper, we present exact femtosecond one- and two-dark soliton solutions for a variable-coefficient higher-order nonlinear Schrodinger equation via modified Hirota method. The propagation and interaction of femtosecond dark solitons are investigated in inhomogeneous fiber systems. Elastic collision, bound oscillation and parallel propagation can be achieved in both Gaussian distributed parameter system and exponentially periodic distributed parameter system by choosing the appropriate distributed parameters and soliton parameters. The results may be beneficial to the realization of interaction control of femtosecond dark solitons in communication systems.

Stabilization of the higher order nonlinear Schrödinger equation with constant coefficients

We study the internal stabilization of the higher order nonlinear Schrodinger equation with constant coefficients. Combining multiplier techniques, a fixed point argument and nonlinear interpolation theory, we can obtain the well-posedness. Then, applying compactness arguments and a unique continuation property, we prove that the solution of the higher-order nonlinear Schrodinger equation with a damping term decays exponentially.

Bright, dark optical and other solitons to the generalized higher-order NLSE in optical fibers

In this paper, we study the generalized higher-order nonlinear Schrodinger equation analytically. We use two integral schemes for conducting this study. Dark, bright, combined dark–bright optical, singular soliton, soliton-like and trigonometric function solutions are successfully constructed. We give the constraint conditions for the existence of valid solutions. The 2D, 3D and the contour graphs for the dark and bright solitons are plotted.

Uniqueness and symmetry of ground states for higher-order equations

We establish uniqueness and radial symmetry of ground states for higher-order nonlinear Schrodinger and Hartree equations whose higher-order differentials have small coefficients. As an application, we obtain error estimates for higher-order approximations to the pseudo-relativistic ground state. Our proof adapts the strategy of Lenzmann (Anal PDE 2:1–27, 2009) using local uniqueness near the limit of ground states in a variational problem. However, in order to bypass difficulties from lack of symmetrization tools for higher-order differential operators, we employ the contraction mapping argument in our earlier work (Choi et al. 2017. arXiv:1705.09068 ) to construct radially symmetric real-valued solutions, as well as improving local uniqueness near the limit.

Averaging Principle for the Higher Order Nonlinear Schrödinger Equation with a Random Fast Oscillation

This work concerns the problem associated with averaging principle for a higher order nonlinear Schrodinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction–diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrodinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrodinger equation with a modified coefficient.

Integrability and solitons for the higher-order nonlinear Schrödinger equation with space-dependent coefficients in an optical fiber

Under investigation in this paper is a higher-order nonlinear Schrodinger equation with space-dependent coefficients, related to an optical fiber. Based on the self-similarity transformation and Hirota method, related to the integrability, the N-th-order bright and dark soliton solutions are derived under certain constraints. It is revealed that the velocities and trajectories of the solitons are both affected by the coefficient of the sixth-order dispersion term while the amplitudes of the solitons are determined by the gain function. Amplitudes increase when the gain function is positive and decrease when the gain function is negative. Furthermore, we find that the intensities of dark solitons are presented as a superposition of the solitons and stationary waves.

Amplification, reshaping, fission and annihilation of optical solitons in dispersion-decreasing fiber

Amplification, reshaping, fission and annihilation of optical solitons can be applied in fiber lasers, all-optical switching devices and optical communications. In this paper, for the variable coefficient high-order nonlinear Schrodinger equation, analytic two- and three-soliton solutions are derived by the Hirota bilinear method. Optical solitons propagation in the dispersion-decreasing fibers is investigated theoretically. The influence of corresponding parameters is discussed based on obtained solutions. By choosing properly parameters, optical solitons are amplified and reshaped stably in a long distance. Besides, the number of amplified solitons can be chosen as required. Moreover, a novel phenomenon that three solitons can split into four solitons or merge into two solitons has been proposed. Results may be helpful to realize the amplification, reshaping, fission and annihilation of solitons, and will be valuable to the applications of optical amplifier, all-optical switching and optical self-routing.

基于首次积分法求解非线性薛定谔方程的精确解

基于首次积分法求解非线性薛定谔方程的精确解

Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions

In this paper, we modify with an appropriate analytical technique, the characteristics of the optical fiber through the modification of the coefficients of the highly nonlinear partial differential equation, which initially governs the dynamics of the propagation in such a wave guide. The procedure consists to assign arbitrary coefficients to the various terms of the established nonlinear partial differential equation, such as the one that embodies the propagation dynamics in a strongly nonlinear optical fiber and subsequently establishing the constraint equations linking these coefficients and thus the analys is makes it possible to enumerate the criteria for which obtaining the desired solutions is possible. These coefficients are like indicators which characterize the various modifications made in this medium of transmission. The nonlinear evolution equation that served as mathematical model for this study is the higher-order nonlinear Schrodinger equation which better describes the propagation of an ultrafast pulse in an optical fiber. The use of the Bogning-Djeumen Tchaho-Kofane method enabled not only to establish the constraint relations, but also the solitary wave solutions and plane wave solutions. We want through the results obtained in this article to give the specialists of the manufacture of transmission media such as optical fiber, to consider the modification of the properties of this wave guide during manufacture, depending on the type of signal that one wants to propagate in this case notably the solitary wave.