Bright–dark soliton solutions for the (2+1)-dimensional variable-coefficient coupled nonlinear Schrödinger system in a graded-index waveguide

2017 ◽  
Vol 31 (10) ◽  
pp. 1750100
Author(s):  
Yu-Qiang Yuan ◽  
Bo Tian ◽  
Xi-Yang Xie ◽  
Jun Chai ◽  
Lei Liu
Keyword(s):  
Soliton Solutions ◽  
Original System ◽  
Bound State ◽  
Dark Soliton ◽  
Elastic Collision ◽  
Variable Coefficients ◽  
Nonlinear Schrödinger ◽  
Graded Index ◽  
The One ◽  
Gain Loss

Under investigation in this paper is the (2[Formula: see text]+[Formula: see text]1)-dimensional coupled nonlinear Schrödinger (NLS) system with variable coefficients, which describes the propagation of an optical beam inside the two-dimensional graded-index waveguide amplifier with the polarization effects. Through a similarity transformation, we convert that system into a set of the integrable defocusing (1[Formula: see text]+[Formula: see text]1)-dimensional coupled NLS equations, and subsequently construct the bright–dark soliton solutions for the original system which are converted from the ones of the latter set. With the graphic analysis, we discuss the soliton propagation and collision with r(t), which is related to the nonlinear, profile and gain/loss coefficients. When r(t) is a constant, one soliton propagates with the amplitude, width and velocity unvaried, while velocity and width of the one soliton can be affected, and two solitons possess the elastic collision; When r(t) is a linear function, velocity and width of the one soliton varies with t increasing, and collision of the two solitons is altered. Besides, bound-state solitons are seen.

Dark-soliton collisions for a coupled AB system in the geophysical fluids or nonlinear optics

2018 ◽  
Vol 32 (04) ◽  
pp. 1850039 ◽  
Author(s):  
Xi-Yang Xie ◽  
Gao-Qing Meng
Keyword(s):  
Nonlinear Optics ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Elastic Collision ◽  
Mean Flow ◽  
Short Pulses ◽  
Dark Solitons ◽  
Parameter Measuring ◽  
The One ◽  
Geophysical Fluids

Under investigation in this paper is a coupled AB system, which describes the marginally unstable baroclinic wave packets in the geophysical fluids or ultra-short pulses in nonlinear optics. As the dark solitons are more resistant against various perturbations than the bright ones, we aim to investigate the dark solitons in the geophysical fluids or nonlinear optics. Dark one- and two-soliton solutions for such a system are derived based on the bilinear forms and propagations of the one solitons and collisions between the two solitons are graphically illustrated and analyzed. Further, influences of the coefficients [Formula: see text] and [Formula: see text] on the solitons are discussed, where [Formula: see text] is a parameter measuring the state of the basic flow and [Formula: see text] is the group velocity. The dark-one solitons with invariant shapes and amplitudes are viewed, and elastic collisions between the dark-two solitons are observed. Also, elastic collision between the bright and dark solitons is viewed, and the dark soliton is found to possess two peaks. [Formula: see text] is found to affect the widths of the dark-one solitons and the travelling directions of the dark-two solitons. It is shown that [Formula: see text] cannot influence shapes of [Formula: see text] and [Formula: see text], but affect the plane of the one soliton for [Formula: see text], and the two-peak dark soliton for [Formula: see text] changes to the single-peak one with the value of [Formula: see text] decreasing, where [Formula: see text] and [Formula: see text] are the packets of short waves and [Formula: see text] is the mean flow.

Dark–dark and bright–dark solitons for a (2+1)-dimensional variable-coefficient nonlinear Schrödinger system in a graded-index waveguide

2020 ◽  
Vol 34 (17) ◽  
pp. 2050183
Author(s):  
Jie Zhang ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
Yu-Qiang Yuan ◽  
He-Yuan Tian ◽  
...  
Keyword(s):  
Variable Coefficient ◽  
Soliton Solutions ◽  
Beam Width ◽  
Dark Soliton ◽  
Nonlinear Schrödinger ◽  
Graded Index ◽  
Dark Solitons ◽  
Nonlinear Schrödinger System ◽  
Dimensional Variable ◽  
Schrödinger System

In this letter, we study a (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient nonlinear Schrödinger system, which describes an optical beam inside the two-dimensional graded-index waveguide with polarization effects. Through the Kadomtsev–Petviashvili hierarchy reduction, the [Formula: see text] dark–dark soliton and [Formula: see text] bright-dark soliton solutions in terms of the Gramian are obtained, where [Formula: see text] is a positive integer. We analyze the interaction and propagation of the dark–dark solitons graphically. With the different values of the diffraction coefficient [Formula: see text], periodic-, cubic- and parabolic-shaped dark–dark solitons are derived. With the different values of the gain/loss coefficient [Formula: see text], periodic- and arctangent-profile background waves are obtained. Moreover, we discuss the effects from the dimensionless beam width [Formula: see text], [Formula: see text] and [Formula: see text] on the solitons and background waves: Shapes of the solitons are affected by [Formula: see text] and [Formula: see text], while profiles of the background waves are affected by [Formula: see text] and [Formula: see text].

Vector Dark Solitons for a Coupled Nonlinear Schrödinger System with Variable Coefficients in an Inhomogeneous Optical Fibre

2017 ◽  
Vol 72 (8) ◽  
pp. 779-787 ◽  
Author(s):  
Lei Liu ◽  
Bo Tian ◽  
Xiao-Yu Wu ◽  
Yu-Qiang Yuan
Keyword(s):  
Optical Fibre ◽  
Bound States ◽  
Group Velocity Dispersion ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Variable Coefficients ◽  
Nonlinear Schrödinger ◽  
Dark Solitons ◽  
Nonlinear Schrödinger System ◽  
Schrödinger System

AbstractStudied in this paper are the vector dark solitons for a coupled nonlinear Schrödinger system with variable coefficients, which can be used to describe the pulse simultaneous propagation of the M-field components in an inhomogeneous optical fibre, where M is a positive integer. When M=2, under the integrable constraint, we construct the nondegenerate N-dark-dark soliton solutions in terms of the Gramian through the Kadomtsev–Petviashvili hierarchy reduction. With the help of analytic analysis, a vector one soliton with varying amplitude and velocity is studied. Interactions and bound states between the two solitons under different group velocity dispersion and amplification/absorption coefficients are presented. Moreover, we extend our analysis to any M to obtain the nondegenerate vector N-dark soliton solutions.

scholarly journals New Optical Soliton Solutions For The Variable Coefficients Nonlinear Schrodinger Equation

2021 ◽  
Author(s):  
Yongyi Gu ◽  
Najva Aminakbari
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Dark Soliton ◽  
Variable Coefficients ◽  
Nonlinear Schrodinger Equation ◽  
Optical Soliton ◽  
Nonlinear Schrödinger

Abstract This paper is appropriated to seek new optical soliton solutions of nonlinear Schrodinger equation (NLSE) with time-dependent coefficients which describes the dispersion decreasing fiber. By proposing Bernoulli (G ′/G)-expansion method, where G = G(ζ ) satisfies Bernoulli equation, some periodic wave, bright and dark soliton solutions are successfully achieved. In addition, 3D, line and contour maps graphs of the obtained results under effect of different values of coefficients are illustrated to have acceptable conception of dynamic structures.

scholarly journals The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities

2021 ◽  
Author(s):  
Rupert L. Frank ◽  
David Gontier ◽  
Mathieu Lewin
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound State ◽  
Nonlinear Schrodinger Equation ◽  
Best Constant ◽  
Nonlinear Schrödinger ◽  
The One ◽  
Bound State Case ◽  
Cubic Nonlinear Schrödinger Equation

AbstractIn this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator $$-\Delta +V(x)$$ - Δ + V ( x ) are raised to the power $$\kappa $$ κ is never given by the one-bound state case when $$\kappa >\max (0,2-d/2)$$ κ > max ( 0 , 2 - d / 2 ) in space dimension $$d\ge 1$$ d ≥ 1 . When in addition $$\kappa \ge 1$$ κ ≥ 1 we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. https://doi.org/10.1007/s00205-021-01634-7). In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.

Reductions and Solutions of Two Types of Coupled Nonlinear Evolution Equations in Optical Fibers and Fluid Dynamics

2011 ◽  
Vol 66 (8-9) ◽  
pp. 552-558
Author(s):  
Li-Cai Liu ◽  
Bo Tian ◽  
Bo Qin ◽  
Xing Lü ◽  
Zhi-Qiang Lin ◽  
...  
Keyword(s):  
Fluid Dynamics ◽  
Raman Scattering ◽  
Optical Fibers ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Dark Soliton ◽  
Nonlinear Evolution Equations ◽  
The One ◽  
Gain Loss ◽  
Burgers Type Equations

Abstract Under investigation in this paper are the coupled nonlinear Schrödinger equations (CNLSEs) and coupled Burgers-type equations (CBEs), which are, respectively, a model for certain birefringent optical fibers Raman-scattering, Kerr and gain/loss effects, and a generalized model in fluid dynamics. Special attention should be paid to the existing claim that the solitons for the CNLSEs do not exist. Through certain dependent-variable transformations, the CNLSEs are reduced to a Manakov system and the CBEs are linearized. In that way, some new solutions of the CNLSEs and CBEs are obtained via symbolic computation. Especially the one-dark-soliton-like solutions for the CNLSEs have been found, against the existing claim.

Oblique nonlinear Alfvén waves in strongly magnetized beam plasmas. Part 2. Soliton solutions and integrability

1993 ◽  
Vol 50 (3) ◽  
pp. 457-476 ◽  
Author(s):  
Bernard Deconinck ◽  
Peter Meuris ◽  
Frank Verheest
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Nonlinear Schrodinger Equation ◽  
Mhd Waves ◽  
Displacement Current ◽  
Nonlinear Schrödinger ◽  
Derivative Nonlinear Schrödinger Equation

Oblique propagation of MHD waves in warm multi-species plasmas with anisotropic pressures and different equilibrium drifts is described by a modified vector derivative nonlinear Schrödinger equation, if charge separation in Poisson's equation and the displacement current in Ampère's law are properly taken into account. This modified equation cannot be reduced to the standard derivative nonlinear Schrödinger equation, and hence requires a new approach to solitary-wave solutions, integrability and related problems. The new equation is shown to be integrable by the use of the prolongation method, and by finding a sufficient number of conservation laws, and possesses bright and dark soliton solutions, besides possible periodic solutions.

Soliton solutions to the nonlocal non-isospectral nonlinear Schrödinger equation

2020 ◽  
Vol 34 (25) ◽  
pp. 2050219 ◽  
Author(s):  
Wei Feng ◽  
Song-Lin Zhao
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Reverse Time ◽  
Nonlinear Schrödinger ◽  
Reduced Equations ◽  
Double Wronskian ◽  
The One

In this paper we study the nonlocal reductions for the non-isospectral Ablowitz-Kaup-Newell-Segur equation. By imposing the real and complex nonlocal reductions on the non-isospectral Ablowitz-Kaup-Newell-Segur equation, we derive two types of nonlocal non-isospectral nonlinear Schrödinger equations, in which one is real nonlocal non-isospectral nonlinear Schrödinger equation and the other is complex nonlocal non-isospectral nonlinear Schrödinger equation. Of both of these two equations, there are the reverse time nonlocal type and the reverse space nonlocal type. Soliton solutions in terms of double Wronskian to the reduced equations are obtained by imposing constraint conditions on the double Wronskian solutions of the non-isospectral Ablowitz-Kaup-Newell-Segur equation. Dynamics of the one-soliton solutions are analyzed and illustrated by asymptotic analysis.

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