Darboux transformation for the Zn-Hirota systems

2019 ◽  
Vol 33 (21) ◽  
pp. 1950246 ◽  
Author(s):  
Lulu Geng ◽  
Chuanzhong Li
Keyword(s):  
Lie Algebra ◽  
Optical Fibers ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Taylor Series Expansion ◽  
Nls Equation ◽  
Hirota Equation ◽  
Strongly Coupled ◽  
Commutative Subalgebra ◽  
Algebraic Reductions

Hirota equation is a modified nonlinear Schrödinger (NLS) equation, which takes into account higher order dispersion and delay correction of cubic nonlinearity. The propagation of the waves in the ocean is described, and the optical fiber can be regarded as a more accurate approximation than the NLS equation. Using the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text], we construct the general [Formula: see text]-Hirota systems. Considering the potential applications of two-mode nonlinear waves in nonlinear optical fibers, including its Lax pairs, we use the algebraic reductions of the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text]. Then, we construct Darboux transformation of the strongly coupled Hirota equation, which implies the new solutions of [Formula: see text] generated from the known solution [Formula: see text]. The new solutions [Formula: see text] furnish soliton solutions and breather solutions of the strongly coupled Hirota equation. Furthermore, using Taylor series expansion of the breather solutions, the rogue waves of the strongly coupled Hirota equation can be given demonstrably. It is obvious that different images can be obtained by choosing different parameters.

A weakly coupled Hirota equation and its rogue waves

2019 ◽  
Vol 34 (22) ◽  
pp. 1950179 ◽  
Author(s):  
Huijuan Zhou ◽  
Chuanzhong Li
Keyword(s):  
Optical Fibers ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Taylor Series Expansion ◽  
Rogue Waves ◽  
Nls Equation ◽  
Hirota Equation ◽  
Weakly Coupled ◽  
Parameter Values ◽  
Higher Order Dispersion

The Hirota equation, a modified nonlinear Schrödinger (NLS) equation, takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity. Its wave propagation is like in the ocean and optical fibers can be viewed as an approximation which is more accurate than the NLS equation. By considering the potential application of two mode nonlinear waves in nonlinear fibers under a certain case, we use the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text] and [Formula: see text] to define a weakly coupled Hirota equation (called Frobenius Hirota equation) including its Lax pair, in this paper. Afterwards, Darboux transformation of the Frobenius Hirota equation is constructed. The Darboux transformation implies the new solutions of ([Formula: see text], [Formula: see text]) generated from the known solution ([Formula: see text], [Formula: see text]). The new solutions ([Formula: see text], [Formula: see text]) provide soliton solutions, breather solutions of the Frobenius Hirota equation. Further, rogue waves of the Frobenius Hirota equation are given explicitly by a Taylor series expansion of the breather solutions. In particular, by choosing different parameter values for the rogue waves, we can get different images.

Dispersive optical solitons for the Schrödinger–Hirota equation in optical fibers

2020 ◽  
pp. 2150060
Author(s):  
Wen-Tao Huang ◽  
Cheng-Cheng Zhou ◽  
Xing Lü ◽  
Jian-Ping Wang
Keyword(s):  
Asymptotic Behavior ◽  
Optical Fibers ◽  
Modulation Instability ◽  
Soliton Solutions ◽  
Optical Solitons ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Hirota Equation ◽  
The One ◽  
Hirota Bilinear

Under investigation in this paper is the dynamics of dispersive optical solitons modeled via the Schrödinger–Hirota equation. The modulation instability of solutions is firstly studied in the presence of a small perturbation. With symbolic computation, the one-, two-, and three-soliton solutions are obtained through the Hirota bilinear method. The propagation and interaction of the solitons are simulated, and it is found the collision is elastic and the solitons enjoy the particle-like interaction properties. In the end, the asymptotic behavior is analyzed for the three-soliton solutions.

SOLITON PROPAGATION IN NONUNIFORM OPTICAL FIBERS

2003 ◽  
Vol 12 (03) ◽  
pp. 341-348 ◽  
Author(s):  
YAN XIAO ◽  
ZHIYONG XU ◽  
LU LI ◽  
ZHONGHAO LI ◽  
GUOSHENG ZHOU
Keyword(s):  
Optical Fibers ◽  
Soliton Solution ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Transmission System ◽  
Soliton Propagation ◽  
Soliton Transmission System ◽  
Soliton Transmission

In this paper, we construct the Lax pair for a soliton transmission system in nonuniform optical fibers and give N-soliton solution using the Darboux transformation. The explicit one-soliton and two-soliton solutions are presented. Further, we discuss the interaction scenario between two neighboring solitons and the effect of the inhomogeneities of the fiber (z0) on the interaction between two neighboring solitons.

scholarly journals Exact optical solitons of the perturbed nonlinear Schrödinger–Hirota equation with Kerr law nonlinearity in nonlinear fiber optics

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 526-534 ◽  
Author(s):  
Alphonse Houwe ◽  
Souleymanou Abbagari ◽  
Gambo Betchewe ◽  
Mustafa Inc ◽  
Serge Y. Doka ◽  
...  
Keyword(s):  
Fiber Optics ◽  
Optical Fibers ◽  
Soliton Solutions ◽  
Auxiliary Equation ◽  
Coupling Coefficients ◽  
Hirota Equation ◽  
Auxiliary Equation Method ◽  
Nonlinear Schrödinger ◽  
Kerr Law Nonlinearity ◽  
Constraint Relation

AbstractThis article studies dark, bright, trigonometric and rational optical soliton solutions to the perturbed nonlinear Schrödinger–Hirota equation (PNLSHE). Hence, we have examined two cases: first, restrictions have been done to the third-order (TOD) (γ) as constraint relation, and the coupling coefficients (σ) is obtained as well as the velocity of the soliton by adopting the traveling wave hypothesis. Second, the TOD and the coupling coefficients are non-zero value, sending back to the PNLSHE, which has been studied in refs. [4,10,16] recently. By employing two relevant integration technics such as the auxiliary equation and the modified auxiliary equation method, miscellaneous optical solitary wave is obtianed, which is in agreement with the outcomes collected by the previous studies [4,16]. These results help in obtaining nonlinear optical fibers in the future.

scholarly journals Bäcklund transformations of Zn-sine-Gordon systems

2017 ◽  
Vol 31 (17) ◽  
pp. 1750189 ◽  
Author(s):  
Xueping Yang ◽  
Chuanzhong Li
Keyword(s):  
Lie Algebra ◽  
General Formula ◽  
Bäcklund Transformations ◽  
Lax Pairs ◽  
Nonlinear Superposition ◽  
Commutative Subalgebra ◽  
Backlund Transformations ◽  
Algebraic Reductions ◽  
Nonlinear Superposition Formula

In this paper, from the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text], we construct the general [Formula: see text]-sine-Gordon and [Formula: see text]-sinh-Gordon systems which contain many multi-component sine-Gordon type and sinh-Gordon type equations. Meanwhile, we give the Bäcklund transformations of the [Formula: see text]-sine-Gordon and [Formula: see text]-sinh-Gordon equations which can generate new solutions from seed solutions. To see the [Formula: see text]-systems clearly, we consider the [Formula: see text]-sine-Gordon and [Formula: see text]-sine-Gordon equations explicitly including their Bäcklund transformations, the nonlinear superposition formula and Lax pairs.

scholarly journals Three Different Methods for New Soliton Solutions of the Generalized NLS Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Anwar Ja’afar Mohamad Jawad
Keyword(s):  
Optical Fibers ◽  
Evolution Equations ◽  
Pulse Propagation ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Nls Equation ◽  
Trigonometric Function Solutions ◽  
Modified Simple Equation Method

Three different methods are applied to construct new types of solutions of nonlinear evolution equations. First, the Csch method is used to carry out the solutions; then the Extended Tanh-Coth method and the modified simple equation method are used to obtain the soliton solutions. The effectiveness of these methods is demonstrated by applications to the RKL model, the generalized derivative NLS equation. The solitary wave solutions and trigonometric function solutions are obtained. The obtained solutions are very useful in the nonlinear pulse propagation through optical fibers.

scholarly journals MULTI-LINE SOLITON SOLUTIONS FOR THE TWO-DIMENSIONAL NONLINEAR HIROTA EQUATION

2021 ◽  
Vol 2 (336) ◽  
pp. 172-178
Author(s):  
G. T. Bekova ◽  
A. A. Zhadyranova
Keyword(s):  
Darboux Transformation ◽  
Soliton Solutions ◽  
Dimensional System ◽  
Scattering Method ◽  
Nonlinear Phenomena ◽  
Two Dimensional ◽  
Solid State Physics ◽  
Hirota Equation ◽  
Line Soliton ◽  
Hirota Equations

At present, the question of studying multidimensional nonlinear integrable equations in the framework of the theory of solitons is very interesting to foreign and Kazakh scientists. Many physical phenomena that occur in nature can be described by nonlinearly integrated equations. Finding specific solutions to such equations plays an important role in studying the dynamics of phenomena occurring in various scientific and engineering fields, such as solid state physics, fluid mechanics, plasma physics and nonlinear optics. There are several methods for obtaining real and soliton, soliton-like solutions of such equations: the inverse scattering method, the Hirota’s bilinear method, Darboux transformation methods, the tanh-coth and the sine-cosine methods. In our work, we studied the two-dimensional Hirota equation, which is a modified nonlinear Schrödinger equation. The nonlinear Hirota equation is one of the integrating equations and the Hirota system is used in the field of study of optical fiber systems, physics, telecommunications and other engineering fields to describe many nonlinear phenomena. To date, the first, second, and n-order Darboux transformations have been developed for the two- dimensional system of Hirota equations, and the soliton, rogue wave solutions have been determined by various methods. In this article, we consider the two-dimensional nonlinear Hirota equations. Using the Lax pair and Darboux transformation we obtained the first and the second multi-line soliton solutions for this equation and provided graphical representation.

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