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.

scholarly journals Integrability and Multisoliton Solutions of the Reverse Space or/and Time Nonlocal Fokas-Lenells (FL) Equation

2021 ◽  
Author(s):  
Wen-Xin Zhang ◽  
Yaqing Liu
Keyword(s):  
Optical Fibers ◽  
Soliton Solutions ◽  
Elastic Interaction ◽  
Space Time ◽  
Inelastic Interaction ◽  
Hirota Bilinear Method ◽  
Reverse Time ◽  
Bilinear Method ◽  
Hirota Bilinear ◽  
Multisoliton Solutions

Abstract This paper studies reverse space or/and time nonlocal Fokas-Lenells (FL) equation, which describes the propagation of nonlinear light pulses in monomode optical fibers when certain higher-order nonlinear effects are considered, by Hirota bilinear method. Firstly, variable transformations from reverse space nonlocal FL equation to reverse time and reverse space-time nonlocal FL equations are constructed. Secondly, the one-, two- and three-soliton solutions of the reverse space nonlocal FL equation are derived through Hirota bilinear method, and the soliton solutions of reverse time and reverse space-time nonlocal FL equations are given through variable transformations. Dynamical behaviors of the multisoliton solutions are discussed in detail by analyzing their wave structures. Thirdly, asymptotic analysis of two- and three-soliton solutions of reverse space nonlocal FL equation is used to investigated the elastic interaction and inelastic interaction. At last, the Lax integrability and conservation laws of three types of nonlocal FL equations is studied. The results obtained in this paper possess new properties that different from the ones for FL equation, which are useful in exploring novel physical phenomena of nonlocal systems in nonlinear media.

Y-shaped soliton solutions for the (2+1)-dimensional bidirectional Sawada–Kotera equation

2021 ◽  
Author(s):  
Shuxin Yang ◽  
Zhao Zhang ◽  
Biao Li
Keyword(s):  
Soliton Solutions ◽  
Complex Interaction ◽  
High Order ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Dynamic Behaviors ◽  
Interaction Solutions ◽  
Localized Waves ◽  
Hirota Bilinear ◽  
Interaction Phenomenon

On the basis of the Hirota bilinear method, resonance Y-shaped soliton and its interaction with other localized waves of (2+1)-dimensional bidirectional Sawada–Kotera equation are derived by introducing the constraint conditions. These types of mixed soliton solutions exhibit complex interaction phenomenon between the resonance Y-shaped solitons and line waves, breather waves, and high-order lump waves. The dynamic behaviors of the interaction solutions are analyzed and illustrated.

MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (25) ◽  
pp. 5003-5015 ◽  
Author(s):  
XING LÜ ◽  
TAO GENG ◽  
CHENG ZHANG ◽  
HONG-WU ZHU ◽  
XIANG-HUA MENG ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Symbolic Computation ◽  
Bilinear Form ◽  
Soliton Solutions ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Truncated Painlevé Expansion ◽  
Hirota Bilinear ◽  
Bilinear Equations

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painlevé expansion and Hirota bilinear method. Firstly, based on the truncation of the Painlevé series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painlevé expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

Multiple-soliton solutions for coupled KdV and coupled KP systems

2009 ◽  
Vol 87 (12) ◽  
pp. 1227-1232 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Multiple Soliton Solutions ◽  
Two Systems ◽  
Kp Equations ◽  
Hirota Bilinear ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

In this work we study two systems of coupled KdV and coupled KP equations. The Hirota bilinear method is applied to show that these two systems are completely integrable. Multiple-soliton solutions and multiple singular-soliton solutions are derived for each system. The resonance phenomenon is examined as well.

scholarly journals Solitary wave solutions and integrability for generalized nonlocal complex modified Korteweg-de Vries (cmKdV) equations

2021 ◽  
Vol 6 (10) ◽  
pp. 11046-11075
Author(s):  
Wen-Xin Zhang ◽  
◽  
Yaqing Liu
Keyword(s):  
Soliton Solutions ◽  
Solitary Wave Solutions ◽  
Hirota Bilinear Method ◽  
Reverse Time ◽  
Local Equation ◽  
Bilinear Method ◽  
Dynamical Behaviors ◽  
Wave Solutions ◽  
Nonlocal Equations ◽  
Hirota Bilinear

<abstract><p>In this paper, the reverse space cmKdV equation, the reverse time cmKdV equation and the reverse space-time cmKdV equation are constructed and each of three types diverse soliton solutions is derived based on the Hirota bilinear method. The Lax integrability of three types of nonlocal equations is studied from local equation by using variable transformations. Based on exact solution formulae of one- and two-soliton solutions of three types of nonlocal cmKdV equation, some figures are used to describe the soliton solutions. According to the dynamical behaviors, it can be found that these solutions possess novel properties which are different from the ones of classical cmKdV equation.</p></abstract>

The (2+1) and (3+1)-Dimensional CBS Equations: Multiple Soliton Solutions and Multiple Singular Soliton Solutions

2010 ◽  
Vol 65 (3) ◽  
pp. 173-181 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Necessary Conditions ◽  
Soliton Solutions ◽  
Complete Integrability ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Multiple Soliton Solutions ◽  
Hirota Bilinear ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

In this work, the generalized (2+1) and (3+1)-dimensional Calogero-Bogoyavlenskii-Schiff equations are studied. We employ the Cole-Hopf transformation and the Hirota bilinear method to derive multiple-soliton solutions and multiple singular soliton solutions for these equations. The necessary conditions for complete integrability of each equation are derived

New Breather and Multiple-Wave Soliton Dynamics for Generalized Vakhnenko-Parkes Equation with Variable Coefficients

2021 ◽  
Author(s):  
Bang-Qing Li
Keyword(s):  
High Frequency ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Time Dependent ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Multiple Wave ◽  
Frequency Wave ◽  
Soliton Dynamics ◽  
Hirota Bilinear

Abstract In investigation is the generalized Vakhnenko--Parkes equation (GVPE) with time-dependent coefficients. GVPE is a new nonlinear model connecting to high-frequency wave propagation in relaxing media with variable perturbations. An extended Hirota bilinear method is proposed to construct soliton, breather and multiple-wave soliton solutions. The soliton solutions can degenerate into existing single soliton solutions. The breather and multiple-wave soliton solutions are first obtained. By utilizing the two free functions involved in the solutions, the dynamics of some novel excited breathers and multiple-wave solitons are demonstrated.

Constructing Quasi-Periodic Wave Solutions of Differential-Difference Equation by Hirota Bilinear Method

2016 ◽  
Vol 71 (12) ◽  
pp. 1159-1165
Author(s):  
Qi Wang
Keyword(s):  
Difference Equation ◽  
Wave Solution ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Differential Difference Equation ◽  
Hirota Bilinear

AbstractIn the present paper, based on the Riemann theta function, the Hirota bilinear method is extended to directly construct a kind of quasi-periodic wave solution of a new integrable differential-difference equation. The asymptotic property of the quasi-periodic wave solution is analyzed in detail. It will be shown that quasi-periodic wave solution converge to the soliton solutions under certain conditions and small amplitude limit.

Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hajar F. Ismael ◽  
Aly Seadawy ◽  
Hasan Bulut
Keyword(s):  
Soliton Solutions ◽  
Higher Order ◽  
Contour Plot ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Density Plot ◽  
Breather Solution ◽  
Wave Patterns ◽  
Hirota Bilinear

Abstract In this research, we explore the dynamics of Caudrey–Dodd–Gibbon–Sawada–Kotera equations in (1 + 1)-dimension, such as N-soliton, and breather solutions. First, a logarithmic variable transform based on the Hirota bilinear method is defined, and then one, two, three and N-soliton solutions are constructed. A breather solution to the equation is also retrieved via N-soliton solutions. All the solutions that have been obtained are novel and plugged into the equation to guarantee their existence. 2-D, 3-D, contour plot and density plot are also presented.

Two–Soliton Solutions Of The Kadomtsevpetviashvili Equation

2012 ◽  
Author(s):  
Wei King Tiong ◽  
Chee Tiong Ong ◽  
Mukheta Isa
Keyword(s):  
Analytic Solution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Hirota Bilinear Method ◽  
Kp Equation ◽  
Bilinear Method ◽  
Traditional Group ◽  
Soliton Interactions ◽  
De Vries ◽  
Hirota Bilinear

Beberapa keputusan tentang penjanaan penyelesaian soliton oleh persamaan Kadomtsev–Petviashvili akan dibincangkan dalam kertas ini. Kaedah teori kumpulan mampu memberikan penyelesaian secara analitik kerana persamaan KP mempunyai ketakterhinggaan banyaknya hukum keabadian. Dengan kaedah Bilinear Hirota, ditunjukkan melalui simulasi berkomputer bagaimana penyelesaian dua soliton persamaan KP mampu menghasilkan strukturstruktur “triad”, kuadruplet dan struktur tak beresonan dalam interaksi soliton. Kata kunci: Soliton, kaedah Bilinear Hirota, persamaan Kortewegde Vries dan Kadomtsev- Petviashvili Several findings on soliton solutions generated by the Kadomtsev–Petviashvili (KP) equation were discussed in this paper. This equation is a two dimensional of the Korteweg–de Vries (KdV) equation. Traditional group–theoretical approach can generate analytic solution of solitons because KP equation has infinitely many conservation laws. By using Hirota Bilinear method, we show via computer simulation how two solitons solution of KP equation produces triad, quadruplet and a non–resonance structures in soliton interactions. Key words: Soliton, Hirota Bilinear method, Korteweg-de Vries and Kadomtsev-Petviashvili equations

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