scholarly journals The local wave phenomenon in the quintic nonlinear Schrödinger equation by numerical methods

2021 ◽  
Author(s):  
Yaning Tang ◽  
Zaijun Liang ◽  
Wenxian Xie
Keyword(s):  
Optical Fibers ◽  
Analytical Solutions ◽  
Initial Conditions ◽  
Fourier Method ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Light Pulses ◽  
Propagation Of Light ◽  
Wide Range ◽  
Local Wave

Abstract The nonlinear Schrodinger hierarchy has a wide range of applications in modeling the propagation of light pulses in optical fibers. In this paper, we focus on the integrable nonlinear Schrodinger (NLS) equation with quintic terms, which play a prominent role when the pulse duration is very short. First, we investigate the spectral signatures of the spatial Lax pair with distinct analytical solutions and their periodized wavetrains by Fourier oscillatory method. Then, we numerically simulate the wave evolution of the quintic NLS equation from different initial conditions through the symmetrical split-step Fourier method. We find many localized high-peak structures whose profiles are very similar to the analytical solutions, and we analyze the formation of rouge waves (RWs) in different cases. These results may be helpful to understand the excitation of nonlinear waves in some nonlinear fields, such as optical fibers, oceanography and so on.

Analytical simulations of the Fokas system; extension (2 + 1)-dimensional nonlinear Schrödinger equation

2021 ◽  
Author(s):  
Mostafa M. A. Khater
Keyword(s):  
Optical Fibers ◽  
Pulse Propagation ◽  
Dynamical Behavior ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Contour Plots ◽  
System Extension ◽  
Inverse Spectral Method ◽  
Nonlinear Pulse Propagation ◽  
The Stability

This paper studies novel analytical solutions of the extended [Formula: see text]-dimensional nonlinear Schrödinger (NLS) equation which is also known with [Formula: see text]-dimensional complex Fokas ([Formula: see text]D–CF) system. Fokas derived this system in 1994 by using the inverse spectral method. This model is considered as an icon model for nonlinear pulse propagation in monomode optical fibers. Many novel computational solutions are constructed through two recent analytical schemes (Ansatz and Projective Riccati expansion (PRE) methods). These solutions are represented through sketches in 2D, 3D, and contour plots to demonstrate the dynamical behavior of pulse propagation in breather, rogue, periodic, lump, and solitary characteristics. The stability property of the obtained solutions is examined based on the Hamiltonian system’s properties. The obtained solutions are checked by putting them back into the original equation through Mathematica 12 software.

Optical solitons in (n + 1) dimensions with Kerr and power law nonlinearities

2017 ◽  
Vol 31 (15) ◽  
pp. 1750186 ◽  
Author(s):  
Muhammad Younis
Keyword(s):  
Optical Fibers ◽  
Power Law ◽  
Schrodinger Equation ◽  
Optical Solitons ◽  
Nonlinear Schrodinger Equation ◽  
Bright Soliton ◽  
Light Pulses ◽  
Propagation Of Light ◽  
First Time ◽  
Expansion Scheme

The paper studies the dynamics of optical solitons in [Formula: see text]-dimensional nonlinear Schrödinger equation with Kerr and power law nonlinearities that describe the propagation of light pulses in optical fibers. First time the dark and singular optical solitons are extracted in [Formula: see text] dimensions. The [Formula: see text]-expansion scheme is used to analyze these solutions. Additionally, the constraint conditions for the existence of the solutions are also listed. However, the scheme fails to retrieve the bright soliton.

Optical dark and singular solitons of generalized nonlinear Schrödinger’s equation with anti-cubic law of nonlinearity

2019 ◽  
Vol 33 (13) ◽  
pp. 1950158 ◽  
Author(s):  
Nauman Raza ◽  
Asad Zubair
Keyword(s):  
Optical Fibers ◽  
Soliton Solutions ◽  
Algebraic Method ◽  
Rational Solutions ◽  
Schrödinger’S Equation ◽  
Light Pulses ◽  
Propagation Of Light ◽  
Schrödinger's Equation ◽  
Nonlinear Schrodinger’S Equation ◽  
Nonlinear Schrödinger’S Equation

This work is devoted to scrutinize new optical soliton solutions to the spatially temporal [Formula: see text]-dimensional nonlinear Schrödinger’s equation (NLSE) with anti-cubic nonlinearity. Two different versatile integration architectures are used to extract these solitons. Extended direct algebraic method (EDAM) is utilized to pluck out optical, dark and singular soliton solutions, whereas generalized Kudryashov method (GKM) provides rational solutions. The fetched results are new and useful for the propagation of light pulses in optical fibers in [Formula: see text]-dimensions. For the existence of these solitons, constraint conditions are also listed.

scholarly journals Analytical construction of the projectile motion trajectory in midair

MOMENTO ◽  
2021 ◽  
pp. 79-96
Author(s):  
Peter Chudinov ◽  
Vladimir Eltyshev ◽  
Yuri Barykin
Keyword(s):  
Analytical Solutions ◽  
Initial Conditions ◽  
Resistance Force ◽  
Tennis Ball ◽  
Projectile Motion ◽  
Classic Problem ◽  
Analytical Construction ◽  
Air Resistance ◽  
Wide Range ◽  
Large Period

A classic problem of the motion of a projectile thrown at an angle to the horizon is studied. Air resistance force is taken into account with the use of the quadratic resistance law. An analytic approach is mainly applied for the investigation. Equations of the projectile motion are solved analytically for an arbitrarily large period of time. The constructed analytical solutions are universal, that is, they can be used for any initial conditions of throwing. As a limit case of motion, the vertical asymptote formula is obtained.  The value of the vertical asymptote is calculated directly from the initial conditions of motion. There is no need to study the problem numerically. The found analytical solutions are highly accurate over a wide range of parameters. The motion of a baseball, a tennis ball, and a shuttlecock of badminton are presented as examples.

CONSTRUCTING FAMILIES OF EXACT SOLUTIONS TO A (2+1)-DIMENSIONAL CUBIC NONLINEAR SCHRÖDINGER EQUATION

2004 ◽  
Vol 15 (05) ◽  
pp. 741-751 ◽  
Author(s):  
BIAO LI ◽  
HONGQING ZHANG
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Analytical Solutions ◽  
Schrodinger Equation ◽  
Riccati Equations ◽  
Trigonometric Function ◽  
Nls Equation ◽  
Exact Analytical Solutions ◽  
Nonlinear Schrödinger ◽  
Cubic Nonlinear Schrödinger Equation

The projective Riccati equations method is extended to find some novel exact solutions of a (2+1)-dimensional cubic nonlinear Schrödinger (NLS) equation. Applying the extended method and symbolic computation, six families of exact analytical solutions for this NLS equation are reported, which include some new and more general exact soliton-like solutions, trigonometric function forms solutions and rational forms solutions.

Benchmarking of the Split-Step Fourier Method on Solving a Soliton Propagation Equation in a Nonlinear Optical Medium

2020 ◽  
Vol 12 (2) ◽  
pp. 105-112
Author(s):  
Ahmad Ripai ◽  
Zulfi Abdullah ◽  
Mahdhivan Syafwan ◽  
Wahyu Hidayat
Keyword(s):  
Optical Fiber ◽  
Nonlinear Optical ◽  
Fourier Method ◽  
Optical Solitons ◽  
Propagation Equation ◽  
Dispersion Parameters ◽  
Nls Equation ◽  
Soliton Propagation ◽  
Optical Medium ◽  
Nonlinear Schrödinger

Benchmarking of the numerical split-step Fourier method in solving a soliton propagation equation in a nonlinear optical medium is considered. This study is carried out by comparing the solutions calculated by numerics with those obtained by analytics. In particular, the soliton propagation equation used as the object of observation is the nonlinear Schrödinger (NLS) equation, which describes optical solitons in optical fiber. By using the split-step Fourier method, we show that the split-step Fourier method is accurate. We also confirm that the nonlinear and dispersion parameters of the optical fiber influence the soliton propagation.

scholarly journals Simulation of Pulse Propagation in Optical Fibers

2016 ◽  
Vol 64 ◽  
pp. 159-170
Author(s):  
P.C.T. Munaweera ◽  
K.A.I.L. Wijewardena Gamalath
Keyword(s):  
Optical Fibers ◽  
Pulse Propagation ◽  
Group Velocity Dispersion ◽  
Nonlinear Coefficient ◽  
Input Pulse ◽  
Fourier Method ◽  
Nonlinear Effects ◽  
Input Power ◽  
Light Pulses ◽  
Order Dispersion

A theoretical model was developed for light pulses propagating in optical fibers by considering the nonlinear effects, the self-phase modulation and group velocity dispersion effects. The split step Fourier method was used to generate soliton pulses in a fiber composed of a glass core surrounded by a cladding layer. Gaussian and hyperbolic secant input pulses were used for the simulation. By varying the initial chirp, input power and nonlinear coefficient for an input Gaussian pulse at wavelength of λ =1.55μm with initial pulse width 125ps for second order dispersion β2=−20 ps2km-1, nonlinear parameter γ=3W-1kg-1and initial chirp C=−0.25 two near soliton pulses were generated for input powers P = 0.54mW and P = 0.64mW and a perfect soliton for the hyperbolic secant input pulse.

scholarly journals NUMERICAL SIMULATIONS OF THE ENERGY-SUPERCRITICAL NONLINEAR SCHRÖDINGER EQUATION

2010 ◽  
Vol 07 (02) ◽  
pp. 279-296 ◽  
Author(s):  
J. COLLIANDER ◽  
G. SIMPSON ◽  
C. SULEM
Keyword(s):  
Numerical Simulations ◽  
Wave Equations ◽  
Initial Conditions ◽  
A Priori ◽  
Sobolev Norm ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Long Time ◽  
Supercritical Nonlinearity ◽  
Homogeneous Norm

We present numerical simulations of the defocusing nonlinear Schrödinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig–Merle and Kilip–Visan who considered some energy supercritical wave equations and proved that if the solution is a priori bounded in the critical Sobolev space (i.e. the space whose homogeneous norm is invariant under the scaling leaving the equation invariant), then it exists for all time and scatters. In this paper, we numerically investigate the boundedness of the H2-critical Sobolev norm for solutions of the NLS equation in dimension five with quintic nonlinearity. We find that for a class of initial conditions, this norm remains bounded, the solution exists for long time, and scatters.

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