Lax pair and soliton solutions for the generalized nonlinear Schrödinger equation in inhomogeneous optical fiber media

2011 ◽  
Author(s):  
Jingru Qu
Keyword(s):  
Optical Fiber ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Dinger Equation

scholarly journals Soliton solutions for variable coefficient nonlinear Schr?dinger equation for optical fiber and their application

2006 ◽  
Vol 55 (8) ◽  
pp. 3805
Author(s):  
Zong Feng-De ◽  
Dai Chao-Qing ◽  
Yang Qin ◽  
Zhang Jie-Fang
Keyword(s):  
Optical Fiber ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Dinger Equation

Dark solitons for a variable-coefficient Kundu–Eckhaus equation in an inhomogeneous optical fiber with the relevant Lax pair and binary Darboux transformations

2019 ◽  
Vol 33 (01) ◽  
pp. 1850418 ◽  
Author(s):  
Ze Zhang ◽  
Bo Tian ◽  
Han-Peng Chai ◽  
Hui-Min Yin ◽  
Chen-Rong Zhang
Keyword(s):  
Optical Fiber ◽  
Group Velocity ◽  
Variable Coefficient ◽  
Velocity Dispersion ◽  
Group Velocity Dispersion ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Darboux Transformations ◽  
Dark Solitons ◽  
The One

In this paper, we study a Kundu–Eckhaus equation with variable coefficients, which describes the ultra-short optical pulses in an inhomogeneous optical fiber. We construct the Lax pair under certain variable-coefficient constraints. With the gauge transformation, one/N-fold binary Darboux transformations and limit forms of the one-fold binary Darboux transformation are obtained. Based on such transformations, one/N-dark (N = 2,3, [Formula: see text]) soliton solutions under those constraints are derived. Linear, periodic and parabolic dark solitons are presented, and numerical simulations are used to investigate the influence of the group velocity dispersion on the structures of the one-dark solitons. Based on the two-dark soliton solutions under certain variable-coefficient constraints, we also discuss the influence of the group velocity dispersion on the structures of the two-dark solitons. Head-on and overtaking collisions between the two linear, parabolic and cubic-type dark solitons are presented.

Noncommutative negative order AKNS equation and its soliton solutions

2018 ◽  
Vol 33 (35) ◽  
pp. 1850209 ◽  
Author(s):  
H. Wajahat A. Riaz ◽  
Mahmood ul Hassan
Keyword(s):  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Curvature Condition ◽  
Integrable Structure ◽  
Zero Curvature ◽  
Negative Order ◽  
Akns Equation

A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.

Riemann–Hilbert approach and N-soliton solutions for the Chen–Lee–Liu equation

2019 ◽  
Vol 33 (02) ◽  
pp. 1950002 ◽  
Author(s):  
Ming-Jun Xu ◽  
Tie-Cheng Xia ◽  
Bei-Bei Hu
Keyword(s):  
Soliton Solution ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Potential Matrix ◽  
Riemann Hilbert Problem

In this paper, we construct the Riemann–Hilbert problem to the Lax pair of Chen–Lee–Liu (CLL) equation. As far as we know, many researchers have studied various equations with Riemann–Hilbert method before, but no one compared the N-soliton solutions calculated by different symmetries of potential matrix. Using different symmetries of potential matrix, we get two N-soliton solution formulae of the CLL equation. The interesting thing is that we find the equivalence of these two N-soliton solutions.

Bilinear forms and soliton solutions for a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation in an optical fiber

2020 ◽  
Vol 34 (30) ◽  
pp. 2050336
Author(s):  
Dong Wang ◽  
Yi-Tian Gao ◽  
Jing-Jing Su ◽  
Cui-Cui Ding
Keyword(s):  
Schrödinger Equation ◽  
Optical Fiber ◽  
Nonlinear Schrödinger Equation ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Nonlinear Schrodinger Equation ◽  
Bilinear Forms ◽  
Dimensional Variable

In this paper, under investigation is a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation, which is introduced to the study of an optical fiber, where [Formula: see text] is the temporal variable, variable coefficients [Formula: see text] and [Formula: see text] are related to the group velocity dispersion, [Formula: see text] and [Formula: see text] represent the Kerr nonlinearity and linear term, respectively. Via the Hirota bilinear method, bilinear forms are obtained, and bright one-, two-, three- and N-soliton solutions as well as dark one- and two-soliton solutions are derived, where [Formula: see text] is a positive integer. Velocities and amplitudes of the bright/dark one solitons are obtained via the characteristic-line equations. With the graphical analysis, we investigate the influence of the variable coefficients on the propagation and interaction of the solitons. It is found that [Formula: see text] can only affect the phase shifts of the solitons, while [Formula: see text], [Formula: see text] and [Formula: see text] determine the amplitudes and velocities of the bright/dark solitons.

Dark solitons, Lax pair and infinitely-many conservation laws for a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in the inhomogeneous Heisenberg ferromagnetic spin chain

2017 ◽  
Vol 31 (03) ◽  
pp. 1750013 ◽  
Author(s):  
Xue-Hui Zhao ◽  
Bo Tian ◽  
De-Yin Liu ◽  
Xiao-Yu Wu ◽  
Jun Chai ◽  
...  
Keyword(s):  
Conservation Laws ◽  
Spin Chain ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Nonlinear Schrodinger Equation ◽  
Dark Solitons ◽  
Heisenberg Ferromagnetic Spin Chain ◽  
Dimensional Variable

Under investigation in this paper is a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain. Lax pair and infinitely-many conservation laws are derived, indicating the existence of the multi-soliton solutions for such an equation. Via the Hirota method with an auxiliary function, bilinear forms, dark one-, two- and three-soliton solutions are derived. Propagation and interactions for the dark solitons are illustrated graphically: Velocity of the solitons is linearly related to the coefficients of the second- and fourth-order dispersion terms, while amplitude of the solitons does not depend on them. Interactions between the two solitons are shown to be elastic, while those among the three solitons are pairwise elastic.

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