Abundant excited optical breathers for a nonlinear Schrödinger equation with variable dispersion and nonlinearity terms in inhomogenous fiber optics

Optik ◽  
2020 ◽  
Vol 201 ◽  
pp. 162821 ◽  
Author(s):  
Yu-Lan Ma
Keyword(s):  
Schrödinger Equation ◽  
Fiber Optics ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Variable Dispersion

scholarly journals On the Analytical and Numerical Solutions of the One-Dimensional Nonlinear Schrodinger Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Neveen G. A. Farag ◽  
Ahmed H. Eltanboly ◽  
M. S. EL-Azab ◽  
S. S. A. Obayya
Keyword(s):  
Schrödinger Equation ◽  
Fiber Optics ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Real Life ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Numerical Approaches

In this paper, four compelling numerical approaches, namely, the split-step Fourier transform (SSFT), Fourier pseudospectral method (FPSM), Crank-Nicolson method (CNM), and Hopscotch method (HSM), are exhaustively presented for solving the 1D nonlinear Schrodinger equation (NLSE). The significance of this equation is referred to its notable contribution in modeling wave propagation in a plethora of crucial real-life applications such as the fiber optics field. Although exact solutions can be obtained to solve this equation, these solutions are extremely insufficient because of their limitations to only a unique structure under some limited initial conditions. Therefore, seeking high-performance numerical techniques to manipulate this well-known equation is our fundamental purpose in this study. In this regard, extensive comparisons of the proposed numerical approaches, against the exact solution, are conducted to investigate the benefits of each of them along with their drawbacks, targeting a broad range of temporal and spatial values. Based on the obtained numerical simulations via MATLAB, we extrapolated that the SSFT invariably exhibits the topmost robust potentiality for solving this equation. However, the other suggested schemes are substantiated to be consistently accurate, but they might generate higher errors or even consume more processing time under certain conditions.

Numerical approaches for solving the nonlinear Schrödinger equation in the nonlinear fiber optics formalism

2020 ◽  
Author(s):  
Hugo Enrique Ibarra- Villalón ◽  
Olivier Pottiez ◽  
Armando Gómez-Vieyra ◽  
Jesús Pablo Lauterio-Cruz ◽  
Yazmin Esmeralda Bracamontes- Rodriguez
Keyword(s):  
Schrödinger Equation ◽  
Fiber Optics ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Fiber Optics ◽  
Nonlinear Schrödinger ◽  
Nonlinear Fiber ◽  
Numerical Approaches

Seesaw drift of bright solitons of the nonlinear Schrödinger equation with a periodic potential

2016 ◽  
Vol 25 (03) ◽  
pp. 1650038 ◽  
Author(s):  
Camilo J. Castro ◽  
Deterlino Urzagasti
Keyword(s):  
External Field ◽  
Schrödinger Equation ◽  
Fiber Optics ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Periodic Potential ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Analytical Approximations

Soliton solutions are investigated employing the nonlinear Schrödinger equation (NLSE) with an additional term corresponding to an external periodic field. In particular, we use this equation to describe the behavior of solitons in fiber optics in the case of anomalous dispersion. Employing the framework of variational analysis and analytical approximations, single peaked soliton solutions are derived, which exhibit variations of the solitonic parameters due to the effect of the periodic potential and a harmonic oscillator motion of the soliton center, when the frequency of the external field is small, whereas high values of the frequency of the external field produce static solitons. Finally, a variational-numerical analysis was developed and compared with a purely numerical model.

scholarly journals Some results on discrete eigenvalues for the Stochastic Nonlinear Schrödinger Equation in fiber optics

2018 ◽  
Vol 9 (1) ◽  
pp. 87-103
Author(s):  
Laura Prati ◽  
Luigi Barletti
Keyword(s):  
Schrödinger Equation ◽  
Fiber Optics ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Signal To Noise Ratio ◽  
Additive White Gaussian Noise ◽  
Nonlinear Schrodinger Equation ◽  
Order Perturbation ◽  
Perturbation Approach ◽  
Nonlinear Schrödinger

Abstract We study a stochastic Nonlinear Schrödinger Equation (NLSE), with additive white Gaussian noise, by means of the Nonlinear Fourier Transform (NFT). In particular, we focus on the propagation of discrete eigenvalues along a focusing fiber. Since the stochastic NLSE is not exactly integrable by means of the NFT, then we use a perturbation approach, where we assume that the signal-to-noise ratio is high. The zeroth-order perturbation leads to the deterministic NLSE while the first-order perturbation allows to describe the statistics of the discrete eigenvalues. This is important to understand the properties of the channel for recently devised optical transmission techniques, where the information is encoded in the nonlinear Fourier spectrum.

scholarly journals Transformation of Eigenvalues of the Zakharov–Shabat Problem under the Effect of Soliton Collision

2020 ◽  
Vol 20 (4) ◽  
pp. 248-257
Author(s):  
Andrey I. Konyukhov ◽  
Keyword(s):  
Schrödinger Equation ◽  
Spectral Problem ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Dispersion Coefficient ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Complex Eigenvalues ◽  
All Optical ◽  
Variable Dispersion

Background and Objectives: The Zakharov–Shabat spectral problem allows to find soliton solutions of the nonlinear Schrodinger equation. Solving the Zakharov–Shabat problem gives both a discrete set of eigenvalues λj and a continuous one. Each discrete eigenvalue corresponds to an individual soliton with the real part Re(λj) providing the soliton velocity and the imaginary part Im(λj) determining the soliton amplitude. Solitons can be used in optical communication lines to compensate both non-linearity and dispersion. However, a direct use of solitons in return-to-zero signal encoding is inhibited. The interaction between solitions leads to the loss of transmitted data. The problem of soliton interaction can be solved using eigenvalues. The latter do not change when the solitons obey the nonlinear Schrodinger equation. Eigenvalue communication was realized recently using electronic signal processing. To increase the transmission speed the all-optical method for controlling eigenvalues should be developed. The presented research is useful to develop optical methods for the transformation of the eigenvalues. The purpose of the current paper is twofold. First, we intend to clarify the issue of whether the dispersion perturbation can not only split a bound soliton state but join solitons into a short oscillating period breather. The second goal of the paper is to describe the complicated dynamics and mutual interaction of complex eigenvalues of the Zakharov–Shabat spectral problem. Materials and Methods: Pulse propagation in single-mode optical fibers with a variable core diameter can be described using the nonlinear Schrödinger equation (NLSE) which coefficients depends on the evolution coordinate. The NLSE with the variable dispersion coefficient was considered. The dispersion coefficient was described using a hyperbolic tangent function. The NLSE and the Zakharov– Shabat spectral problem were solved using the split-step method and the layer-peeling method, respectively. Results: The results of numerical analysis of the modification of soliton pulses under the effect of variable dispersion coefficient are presented. The main attention is paid to the process of transformation of eigenvalues of the Zakharov–Shabat problem. Collision of two in-phase solitons, which are characterized by two complex eigenvalues is considered. When the coefficients of the nonlinear Schrodinger equation change, the collision of the solitons becomes inelastic. The inelastic collision is characterized by the change of the eigenvalues. It is shown that the variation of the coefficients of the NLSE allows to control both real and imaginary parts of the eigenvalues. Two scenarios for the change of the eigenvalues were identified. The first scenario is characterized by preserving the zero real part of the eigenvalues. The second one is characterized by the equality of their imaginary parts. The transformation of eigenvalues is most effective at the distance where the field spectrum possesses a two-lobe shape. Variation of the NLSE coefficient can introduce splitting or joining of colliding soliton pulses. Conclusion: The presented results show that the eigenvalues can be changed only with a small variation of the NLSE coefficients. On the one hand, a change in the eigenvalues under the effect of inelastic soliton collision is an undesirable effect since the inelastic collision of solitons will lead to unaccounted modulation in soliton optical communication links. On the other hand, the dependence of the eigenvalues on the parameters of the colliding solitons allows to modulate the eigenvalues using all-fiber optical devices. Currently, the modulation of the eigenvalues is organized using electronic devices. Therefore, the transmission of information is limited to nanosecond pulses. For picosecond pulse communication, the development of all-optical modulation methods is required. The presented results will be useful in the development of methods for controlling optical solitons and soliton states of the Bose–Einstein condensate.

scholarly journals Chirped Periodic and Solitary Waves for Improved Perturbed Nonlinear Schrödinger Equation with Cubic Quadratic Nonlinearity

2021 ◽  
Vol 5 (4) ◽  
pp. 234
Author(s):  
Aly R. Seadawy ◽  
Syed T. R. Rizvi ◽  
Saad Althobaiti
Keyword(s):  
Schrödinger Equation ◽  
Fiber Optics ◽  
Nonlinear Schrödinger Equation ◽  
Solitary Waves ◽  
Schrodinger Equation ◽  
Elliptic Functions ◽  
Nonlinear Schrodinger Equation ◽  
Quadratic Nonlinearity ◽  
Periodic Waves ◽  
Nonlinear Schrödinger

In this paper, we study the improved perturbed nonlinear Schrödinger equation with cubic quadratic nonlinearity (IPNLSE-CQN) to describe the propagation properties of nonlinear periodic waves (PW) in fiber optics. We obtain the chirped periodic waves (CPW) with some Jacobi elliptic functions (JEF) and also obtain some solitary waves (SW) such as dark, bright, hyperbolic, singular and periodic solitons. The nonlinear chirp associated with each of these optical solitons was observed to be dependent on the pulse intensity. The graphical behavior of these waves will also be displayed.

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