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.

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.

scholarly journals Bright--Dark Solitons in the Space-Shifted Nonlocal Coupled Nonlinear Schrodinger Equation

2021 ◽  
Author(s):  
Ping Ren ◽  
Jiguang Rao
Keyword(s):  
Soliton Solutions ◽  
Bound State ◽  
Dark Soliton ◽  
Collision Dynamics ◽  
Double Pole ◽  
Dark Solitons ◽  
Kp Hierarchy Reduction Method ◽  
Wave Limit ◽  
04.20 Jb ◽  
Kp Hierarchy Reduction

Abstract Multiple bright-dark soliton solutions in terms of determinants for the space-shifted nonlocal coupled nonlinear Schro¨dinger (CNLS) equation are constructed by using the bilinear (Kadomtsev-Petviashvili) KP hierarchy reduction method. It is found that the bright-dark two-soliton only occur elastic collisions. Upon their amplitudes, the bright two solitons only admit one pattern whose amplitude are equal, and the dark two solitons have three different non-degenerated patterns and two different degenerated patterns. The bright-dark four-soliton is the superposition of the two-soliton pairs and can generated bound-state solitons. The multiple double-pole bright-dark soliton solutions are generated through the long wave limit of the obtained bright-dark soliton solutions, and their collision dynamics are also investigated.PACS 02.30.Jr · 03.75.Lm · 04.20.Jb · 05.45.Yv

scholarly journals Multi-dark soliton solutions for the (2+1)-dimensional multi-component Maccari system

2019 ◽  
Vol 33 (31) ◽  
pp. 1950390
Author(s):  
Tao Xu ◽  
Yong Chen ◽  
Zhijun Qiao
Keyword(s):  
Bound States ◽  
Arbitrary Order ◽  
Soliton Solutions ◽  
Bound State ◽  
Dark Soliton ◽  
Stationary Case ◽  
Kp Hierarchy ◽  
Short Waves ◽  
Dark Solitons ◽  
Two Component

Based on reduction of the KP hierarchy, the general multi-dark soliton solutions in Gram type determinant forms for the (2[Formula: see text]+[Formula: see text]1)-dimensional multi-component Maccari system are constructed. Especially, the two component coupled Maccari system comprising of two component short waves and single-component long waves are discussed in detail. Besides, the dynamics of one and two dark-dark solitons are analyzed. It is shown that the collisions of two dark-dark solitons are elastic by asymptotic analysis. Additionally, the two dark-dark solitons bound states are studied through two different cases (stationary and moving cases). The bound states can exist up to arbitrary order in the stationary case, however, only two-soliton bound state exists in the moving case. Besides, the oblique stationary bound state can be generated for all possible combinations of nonlinearity coefficients consisting of positive, negative and mixed cases. Nevertheless, the parallel stationary and the moving bound states are only possible when nonlinearity coefficients take opposite signs.

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.

N-dark–dark solitons for the coupled higher-order nonlinear Schrödinger equations in optical fibers

2017 ◽  
Vol 31 (33) ◽  
pp. 1750305 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Yue Wang
Keyword(s):  
Optical Fibers ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Higher Order ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Dark Solitons ◽  
Nonlinear Schrodinger Equations

In this paper, we construct the binary Darboux transformation on the coupled higher-order dispersive nonlinear Schrödinger equations in optical fibers. We present the N-fold iterative transformation in terms of the determinants. By the limit technique, we derive the N-dark–dark soliton solutions from the non-vanishing background. Based on the obtained solutions, we find that the collision mechanisms of dark vector solitons exhibit the standard elastic collisions in both two components.

scholarly journals Perturbations of dark solitons

2011 ◽  
Vol 467 (2133) ◽  
pp. 2597-2621 ◽  
Author(s):  
M. J. Ablowitz ◽  
S. D. Nixon ◽  
T. P. Horikis ◽  
D. J. Frantzeskakis
Keyword(s):  
Outer Region ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Nonlinear Damping ◽  
Background Intensity ◽  
Nls Equation ◽  
The Core ◽  
Dark Solitons ◽  
Direct Perturbation ◽  
Inner Region

A direct perturbation method for approximating dark soliton solutions of the nonlinear Schrödinger (NLS) equation under the influence of perturbations is presented. The problem is broken into an inner region, where the core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton that propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated, including linear and nonlinear damping type perturbations.

Dark soliton solutions of the cubic-quintic complex Ginzburg–Landau equation with high-order terms and potential barriers in normal-dispersion fiber lasers

2021 ◽  
Author(s):  
Marco A. Viscarra ◽  
Deterlino Urzagasti
Keyword(s):  
Optical Fibers ◽  
Landau Equation ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Two Dimensions ◽  
Ginzburg Landau ◽  
Normal Dispersion ◽  
Homogeneous Solutions ◽  
Dark Solitons ◽  
Ginzburg Landau Equation

In this paper, we numerically study dark solitons in normal-dispersion optical fibers described by the cubic-quintic complex Ginzburg–Landau equation. The effects of the third-order dispersion, self-steepening, stimulated Raman dispersion, and external potentials are also considered. The existence, chaotic content and interactions of these objects are analyzed, as well as the tunneling through a potential barrier and the formation of dark breathers aside from dark solitons in two dimensions and their mutual interactions as well as with periodic potentials. Furthermore, the homogeneous solutions of the model and the conditions for their stability are also analytically obtained.

Dynamics and elastic interactions of the discrete multi-dark soliton solutions for the Kaup–Newell lattice equation

2018 ◽  
Vol 32 (07) ◽  
pp. 1850085 ◽  
Author(s):  
Nan Liu ◽  
Xiao-Yong Wen
Keyword(s):  
Soliton Solutions ◽  
Elastic Interaction ◽  
Dark Soliton ◽  
Lattice Equation ◽  
Small Noise ◽  
Elastic Interactions ◽  
Dark Solitons ◽  
Dynamical Behaviors ◽  
Integrable Discretization ◽  
Interaction Phenomena

Under consideration in this paper is the Kaup–Newell (KN) lattice equation which is an integrable discretization of the KN equation. Infinitely, many conservation laws and discrete N-fold Darboux transformation (DT) for this system are constructed and established based on its Lax representation. Via the resulting N-fold DT, the discrete multi-dark soliton solutions in terms of determinants are derived from non-vanishing background. Propagation and elastic interaction structures of such solitons are shown graphically. Overtaking interaction phenomena between/among the two, three and four solitons are discussed. Numerical simulations are used to explore their dynamical behaviors of such multi-dark solitons. Numerical results show that their evolutions are stable against a small noise. Results in this paper might be helpful for understanding the propagation of nonlinear Alfvén waves in plasmas.

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