Generalized Darboux transformation and asymptotic analysis on the degenerate dark-bright solitons for a coupled nonlinear Schrödinger system

2021 ◽  
Author(s):  
He-yuan Tian ◽  
Bo Tian ◽  
Yan Sun ◽  
Su-Su Chen
Keyword(s):  
Asymptotic Analysis ◽  
Darboux Transformation ◽  
Coupled System ◽  
Bound State ◽  
Bright Soliton ◽  
Spectral Parameters ◽  
Nonlinear Superposition ◽  
Bright Solitons ◽  
Generalized Darboux Transformation ◽  
The Given

Abstract In this paper, our work is based on a coupled nonlinear Schr ̈odinger system in a two-mode nonlinear fiber. A (N,m)-generalized Darboux transformation is constructed to derive the Nth-order solutions, where the positive integers N and m denote the numbers of iterative times and of distinct spectral parameters, respectively. Based on the Nth-order solutions and the given steps to perform the asymptotic analysis, it is found that a degenerate dark-bright soliton is the nonlinear superposition of several asymptotic dark-bright solitons possessing the same profile. For those asymptotic dark-bright solitons, their velocities are z-dependent except that one of those velocities could become z-independent under the certain condition, where z denotes the evolution dimension. Those asymptotic dark-bright solitons are reflected during the interaction. When a degenerate dark-bright soliton interacts with a nondegenerate/degenerate dark-bright soliton, the interaction is elastic, and the asymptotic bound-state dark-bright soliton with z-dependent or z-independent velocity could take place under certain conditions. Our study extends the investigation on the degenerate solitons from the bright soliton case for the scalar equations to the dark-bright soliton case for a coupled system.

scholarly journals Darboux transformation, bright and dark-bright solitons of an N-coupled high-order nonlinear Schrödinger system in an optical fiber

2021 ◽  
Author(s):  
Xi-Hu Wu ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Cui-Cui Ding ◽  
Fei-Yan Liu ◽  
...  
Keyword(s):  
Optical Fiber ◽  
Asymptotic Analysis ◽  
Darboux Transformation ◽  
Lax Pair ◽  
Second Order ◽  
High Order ◽  
First Order ◽  
Nonlinear Schrödinger ◽  
Bright Solitons ◽  
Schrödinger System

Abstract In this paper, an N -coupled high-order nonlinear Schrödinger system, which describes the properties of the ultrashort optical pulses in an optical fiber, is investigated with the help of Darboux transformation (DT) method and asymptotic analysis. Starting from the given (2 N +1)th-order Lax pair, we construct a new form of DT (complex eigenfunctions of Lax pair involved) to derive the formulas of the n th-iterated solutions, where n and N are the positive integers. On the zero background, the first- and second-order solitons are obtained and analysed through the asymptotic analysis. Multi-parameter adjustment is proceeded since there are 3 N +4 real parameters in the second-order solitons. We find that under certain conditions each of the two interaction patterns (elastic, inelastic) holds in the second-order soliton. On the plane wave background, the first-order bright and dark-bright solitons are obtained. Soliton velocities, amplitudes, widths and characteristic lines of the first-order bright and dark-bright solitons are presented and analysed.

scholarly journals N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 733
Author(s):  
Yu-Shan Bai ◽  
Peng-Xiang Su ◽  
Wen-Xiu Ma
Keyword(s):  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Lax Pairs ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
The Given

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

Localised Nonlinear Waves in the Three-Component Coupled Hirota Equations

2017 ◽  
Vol 72 (11) ◽  
pp. 1053-1070 ◽  
Author(s):  
Tao Xu ◽  
Yong Chen
Keyword(s):  
Darboux Transformation ◽  
Dark Soliton ◽  
Lax Pair ◽  
Higher Order ◽  
Bright Soliton ◽  
Order Effects ◽  
Vector Solution ◽  
Hybrid Solutions ◽  
Three Components ◽  
Hirota Equations

AbstractWe construct the Lax pair and Darboux transformation for the three-component coupled Hirota equations including higher-order effects such as third-order dispersion, self-steepening, and stimulated Raman scattering. A special vector solution of the Lax pair with 4×4 matrices for the three-component Hirota system is elaborately generated, based on this vector solution, various types of mixed higher-order localised waves are derived through the generalised Darboux transformation. Instead of considering various arrangements of the three potential functions q1, q2, and q3, here, the same combination is considered as the same type solution. The first- and second-order localised waves are mainly discussed in six mixed types: (1) the hybrid solutions degenerate to the rational ones and three components are all rogue waves; (2) two components are hybrid solutions between rogue wave (RW) and breather (RW+breather), and one component is interactional solution between RW and dark soliton (RW+dark soliton); (3) two components are RW+dark soliton, and one component is RW+bright soliton; (4) two components are RW+breather, and one component is RW+bright soliton; (5) two components are RW+dark soliton, and one component is RW+bright soliton; (6) three components are all RW+breather. Moreover, these nonlinear localised waves merge with each other by increasing the absolute values of two free parameters α, β. These results further uncover some striking dynamic structures in the multicomponent coupled system.

Bound-state solitons for a non-linear Schrödinger system with the negatively coherent coupling in a weakly birefringent fiber

2020 ◽  
Vol 34 (36) ◽  
pp. 2050423
Author(s):  
Jie Zhang ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
Chen-Rong Zhang ◽  
Xia-Xia Du ◽  
...  
Keyword(s):  
Darboux Transformation ◽  
Bound State ◽  
Optical Pulses ◽  
Coherent Coupling ◽  
Non Linear ◽  
Optical Modes ◽  
Binary Darboux Transformation ◽  
Birefringent Fiber ◽  
Schrödinger System

In this paper, we study a non-linear Schrödinger system with the negatively coherent coupling in a weakly birefringent fiber for two orthogonally polarized optical pulses. With respect to the slowly-varying envelopes of two interacting optical modes and based on the existing binary Darboux transformation, we obtain four types of the bound-state solitons: degenerate-I, degenerate-II, degenerate–non-degenerate, and non-degenerate–non-degenerate bound-state solitons. We graphically analyze the interactions between the degenerate or non-degenerate solitons and four types of the bound-state solitons. When the degenerate solitons interact with the bound-state solitons, amplitudes and widths of the degenerate solitons remain unchanged. When the non-degenerate solitons interact with the bound-state solitons, amplitudes and widths of the bound-state solitons remain unchanged.

Asymptotic analysis of a viscous fluid–thin plate interaction: Periodic flow

2014 ◽  
Vol 24 (09) ◽  
pp. 1781-1822 ◽  
Author(s):  
G. P. Panasenko ◽  
R. Stavre
Keyword(s):  
Viscous Fluid ◽  
Asymptotic Analysis ◽  
Coupled System ◽  
Thin Elastic Plate ◽  
Periodic Flow ◽  
Complete Asymptotic Expansion ◽  
Thin Channel ◽  
Elastic Wall ◽  
Leading Term ◽  
2D Elasticity

The first goal of this paper is to provide an asymptotic derivation and justification of the model studied in [Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall, J. Math. Pures Appl.85 (2006) 558–579]. We consider the coupled system "viscous fluid flow–thin elastic plate" when the thickness of the plate, ε, tends to zero, while the density and the Young's modulus of the plate material are of order ε-1and ε-3, respectively. The plate lies on the fluid which occupies a thick domain. The complete asymptotic expansion is constructed when ε tends to zero and it is proved that the leading term of the expansion satisfies the equations of [Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall, J. Math. Pures Appl.85 (2006) 558–579]. The second goal is the partial asymptotic decomposition formulation of the original problem when a part of the plate is described by a one-dimensional (1D) model while the other part is simulated by the two-dimensional (2D) elasticity equations. The appropriate junction conditions based on the previous asymptotic analysis are proposed at the interface point between the 1D and 2D equations. The error of the method is evaluated.

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.

scholarly journals Influence of Bearing Stiffness on the Nonlinear Dynamics of a Shaft-Final Drive System

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Xu Jinli ◽  
Wan Lei ◽  
Luo Wenxin
Keyword(s):  
Coupled System ◽  
Torque Ripple ◽  
Vibration Characteristics ◽  
Vibration Response ◽  
Drive System ◽  
Lumped Mass ◽  
Driving Speed ◽  
Bearing Stiffness ◽  
Final Drive ◽  
The Given

The bearing stiffness has a considerable influence on the nonlinear coupling vibration characteristics of the shaft-final drive system. A 14-DOF nonlinear coupled vibration model was established by employing the lumped mass method so as to identify the coupling effects of the bearing stiffness to the vibration response of the shaft-final drive system. The engine’s torque ripple, the alternating load from the universal joint (U-joint), and the time-varying mesh parameters of hypoid gear of the shaft-final drive system were also considered for accurate quantitative analysis. The numerical analysis of the vibration response of the coupled system was performed and the experimental measurements were carried out for the validation test. Results show that, at the given driving speed, improving the bearing stiffness can reduce the vibration response of the given coupled system; however, when the bearing stiffness increases to a critical value, the effects of bearing stiffness on the vibration reduction become insignificant; when the driving speed changes, the resonance regions of the coupled system vary with the bearing stiffness. The results are helpful to determine the proper bearing stiffness and the optimum control strategy for the shaft-final drive system. It is hoped that the optimal shaft-final drive system can provide good vibration characteristics to achieve the energy saving and noise reduction for the vehicle application.

Rogue wave solutions for the higher-order nonlinear Schrödinger equation with variable coefficients by generalized Darboux transformation

2016 ◽  
Vol 30 (10) ◽  
pp. 1650106 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Jian Chen
Keyword(s):  
Darboux Transformation ◽  
Variable Coefficient ◽  
Rogue Wave ◽  
Variable Coefficients ◽  
Higher Order ◽  
Optical Systems ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Order Variable ◽  
Generalized Darboux Transformation

In this paper, we study a higher-order variable coefficient nonlinear Schrödinger (NLS) equation, which plays an important role in the control of the ultrashort optical pulse propagation in nonlinear optical systems. Then, we construct a generalized Darboux transformation (GDT) for the higher-order variable coefficient NLS equation. The [Formula: see text]th order rogue wave solution is obtained by the iterative rule and it can be expressed by the determinant form. As application, we calculate rogue waves (RWs) from first- to fourth-order in accordance with different kinds of parameters. In particular, the dynamical properties and spatial-temporal structures of RWs are discussed and compared with Hirota equation through some figures.

Nonautonomous solitons in terms of the double Wronskian determinant for a variable-coefficient Gross–Pitaevskii equation in the Bose–Einstein condensate

2016 ◽  
Vol 30 (09) ◽  
pp. 1650103 ◽  
Author(s):  
Chuan-Qi Su ◽  
Yi-Tian Gao ◽  
Qi-Min Wang ◽  
Jin-Wei Yang ◽  
Da-Wei Zuo
Keyword(s):  
Bound State ◽  
Einstein Condensate ◽  
Pitaevskii Equation ◽  
Bilinear Forms ◽  
Spectral Parameters ◽  
Potential Coefficient ◽  
Wronskian Determinant ◽  
Bose Einstein Condensate ◽  
Double Wronskian ◽  
Bose Einstein

Under investigation in this paper is a variable-coefficient Gross–Pitaevskii equation which describes the Bose–Einstein condensate. Lax pair, bilinear forms and bilinear Bäcklund transformation for the equation under some integrable conditions are derived. Based on the Lax pair and bilinear forms, double Wronskian solutions are constructed and verified. The [Formula: see text]th-order nonautonomous solitons in terms of the double Wronskian determinant are given. Propagation and interaction for the first- and second-order nonautonomous solitons are discussed from three cases. Amplitudes of the first- and second-order nonautonomous solitons are affected by a real parameter related to the variable coefficients, but independent of the gain-or-loss coefficient [Formula: see text] and linear external potential coefficient [Formula: see text]. For Case 1 [Formula: see text], [Formula: see text] leads to the accelerated propagation of nonautonomous solitons. Parabolic-, cubic-, exponential- and cosine-type nonautonomous solitons are exhibited due to the different choices of [Formula: see text]. For Case 2 [Formula: see text], if the real part of the spectral parameter equals 0, stationary soliton can be formed. If we take the harmonic external potential coefficient [Formula: see text] as a positive constant and let the real parts of the two spectral parameters be the same, bound-state-like structures can be formed, but there are only one attractive and two repulsive procedures. For Case 3 [[Formula: see text] and [Formula: see text] are taken as nonzero constants], head-on interaction, overtaking interaction and bound-state structure can be formed based on the signs of the two spectral parameters.

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