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.

scholarly journals Bright-Dark Mixed N-Soliton Solution of Two-Dimensional Multicomponent Maccari System

2017 ◽  
Vol 72 (8) ◽  
pp. 745-755 ◽  
Author(s):  
Zhong Han ◽  
Yong Chen
Keyword(s):  
Soliton Solution ◽  
Reduction Method ◽  
Soliton Solutions ◽  
Two Dimensional ◽  
Kp Hierarchy ◽  
Dark Solitons ◽  
Long Wave ◽  
Special Cases ◽  
Kp Hierarchy Reduction Method ◽  
Kp Hierarchy Reduction

AbstractBased on the KP hierarchy reduction method, we construct the general bright-dark mixed N-soliton solution of the two-dimensional (2D) (M+1)-component Maccari system comprised of M-component short waves (SWs) and one-component long wave (LW) with all possible combinations of nonlinearities. We firstly consider two types of mixed N-soliton solutions (two-bright-one-dark and one-bright-two-dark solitons in SW components) to the (3+1)-component Maccari system in detail. Then by extending our analysis to the (M+1)-component Maccari system, its general m-bright-(M–m)-dark mixed N-soliton solution is obtained. The formula obtained also contains the general all-bright and all-dark N-soliton solutions as special cases. For the two-bright-one-dark mixed soliton solution of the (3+1)-component Maccari system, it can be shown that solioff excitation and solioff interaction take place in the two SW components supporting bright solitons, whereas the SW component supporting dark solitons and the LW component possess V-type solitary and interaction.

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.

Discrete Solitons and Bäcklund Transformation for the Coupled Ablowitz–Ladik Equations

2017 ◽  
Vol 72 (10) ◽  
pp. 963-972
Author(s):  
Xiao-Yu Wu ◽  
Bo Tian ◽  
Lei Liu ◽  
Yan Sun
Keyword(s):  
Lattice Spacing ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Bound State ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Bright Solitons ◽  
Dark Solitons ◽  
Spacing Parameter ◽  
Binary Bell Polynomials

AbstractUnder investigation in this paper are the coupled Ablowitz–Ladik equations, which are linked to the optical fibres, waveguide arrays, and optical lattices. Binary Bell polynomials are applied to construct the bilinear forms and bilinear Bäcklund transformation. Bright/dark one- and two-soliton solutions are also obtained. Asymptotic analysis indicates that the interactions between the bright/dark two solitons are elastic. Amplitudes and velocities of the bright solitons increase as the value of the lattice spacing increases. Increasing value of the lattice spacing can lead to the increase of both the bright solitons’ amplitudes and velocities, and the decrease of the velocities of the dark solitons. The lattice spacing parameter has no effect on the amplitudes of the dark solitons. Overtaking interaction between the unidirectional bright two solitons and a bound state of the two equal-velocity solitons is presented. Overtaking interaction between the unidirectional dark two solitons and the two parallel dark solitons is also plotted.

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.

scholarly journals A Novel Dynamic of Localized Solitary Waves for the Hirota-Maccari System

2021 ◽  
Author(s):  
Pei Xia ◽  
Yi Zhang ◽  
Heyan Zhang ◽  
Yindong Zhuang
Keyword(s):  
Solitary Waves ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Dark Solitons ◽  
Kp Hierarchy Reduction Method ◽  
Waves Propagation ◽  
Short Time ◽  
Physical Phenomena ◽  
Localized Waves ◽  
Kp Hierarchy Reduction

Abstract This paper investigates a particular family of semi-rational solutions in determinant form by using the KP hierarchy reduction method, which describe resonant collisions among lumps or resemble line rogue waves and dark solitons in the Hirota-Maccari system. Due to the resonant collisions, the line resemble rogue waves are generated and attenuated in the background of dark solitons with line profiles of finite length, it takes a short time for the lumps to appear from and disappear into the dark solitons background. These novel dynamic of localized solitary waves may be help to understand some physical phenomena of nonlinear localized waves propagation in many physical settings.

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.

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