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

scholarly journals ASYMPTOTIC STABILITY OF NONLINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL

2005 ◽  
Vol 17 (10) ◽  
pp. 1143-1207 ◽  
Author(s):  
ZHOU GANG ◽  
I. M. SIGAL
Keyword(s):  
Asymptotic Stability ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Initial Conditions ◽  
Nonlinear Schrodinger Equation ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Generalized Nonlinear Schrödinger Equation

We prove asymptotic stability of trapped solitons in the generalized nonlinear Schrödinger equation with a potential in dimension 1 and for even potential and even initial conditions.

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.

The current-modified nonlinear Schrödinger equation

1999 ◽  
Vol 399 ◽  
pp. 335-353 ◽  
Author(s):  
J. R. STOCKER ◽  
D. H. PEREGRINE
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Scale ◽  
Numerical Solutions ◽  
Numerical Data ◽  
Surface Current ◽  
Nonlinear Schrodinger Equation ◽  
Formal Approach ◽  
Nonlinear Schrödinger

By comparison with both experimental and numerical data, Dysthe's (1979) O(ε4) modified nonlinear Schrödinger; equation has been shown to model the evolution of a slowly varying wavetrain well (here ε is the wave steepness). In this work, we extend the equation to include a prescribed, large-scale, O(ε2) surface current which varies about a mean value. As an introduction, a heuristic derivation of the O(ε3) current-modified equation, used by Bakhanov et al. (1996), is given, before a more formal approach is used to derive the O(ε4) equation. Numerical solutions of the new equations are compared in one horizontal dimension with those from a fully nonlinear solver for velocity potential in the specific case of a sinusoidal surface current, such as may be due to an underlying internal wave. The comparisons are encouraging, especially for the O(ε4) equation.

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