scholarly journals Solitons, Rogues and Interaction Behaviors of Third-Order Nonlinear Schrodinger Equation

2022 ◽  
Author(s):  
Ren Bo ◽  
Shi Kai-Zhong ◽  
Shou-Feng Shen ◽  
Wang Guo-Fang ◽  
Peng Jun-Da ◽  
...  
Keyword(s):  
Bilinear Form ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Ultrashort Pulses ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Femtosecond Regime ◽  
Wave Limit

Abstract In this paper, we investigate the third-order nonlinear Schr\"{o}dinger equation which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The multi-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detail calculations.PACS numbers: 02.30.Ik, 05.45.Yv.

scholarly journals Rogue Wave Solutions and Generalized Darboux Transformation for an Inhomogeneous Fifth-Order Nonlinear Schrödinger Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
N. Song ◽  
W. Zhang ◽  
P. Wang ◽  
Y. K. Xue
Keyword(s):  
Darboux Transformation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Wave Solutions ◽  
Generalized Darboux Transformation ◽  
Rogue Wave Solutions ◽  
Fifth Order

The rogue wave solutions are discussed for an inhomogeneous fifth-order nonlinear Schrödinger equation, which describes the dynamics of a site-dependent Heisenberg ferromagnetic spin chain. Using the Darboux matrix, the generalized Darboux transformation is constructed and a recursive formula is derived. Based on the transformation, the first-order to the third-order rogue wave solutions are obtained. Then, the nonlinear dynamics of the first-order to the third-order rogue waves are studied on the basis of some free parameters. Several new structures of the rogue waves are found using numerical simulation. The conclusions will be a supportive tool to study the rogue waves better.

Breathers and multiple rogue waves solutions of the (3+1)-dimensional Jimbo–Miwa equation

2021 ◽  
pp. 2150183
Author(s):  
Hong-Yi Zhang ◽  
Yu-Feng Zhang
Keyword(s):  
Dynamic Characteristics ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Wave Solutions ◽  
Hirota Bilinear

In this paper, we construct the breathers of the (3+1)-dimensional Jimbo–Miwa (JM) equation by means of the Hirota bilinear method, then based on the Hirota bilinear method with a new ansatz form, the multiple rogue wave solutions are constructed. Here, we discuss the general breathers, first-order rogue waves, the second-order rogue waves and the third-order rogue waves. Then we draw the 3- and 2-dimensional plots to illustrate the dynamic characteristics of breathers and multiple rogue waves. These interesting results will help us better reveal (3+1)-dimensional JM equation evolution mechanism.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

Solitons and Rogue Waves for a Higher-Order Nonlinear Schrödinger–Maxwell–Bloch System in an Erbium-Doped Fiber

2015 ◽  
Vol 70 (11) ◽  
pp. 935-948 ◽  
Author(s):  
Chuan-Qi Su ◽  
Yi-Tian Gao ◽  
Long Xue ◽  
Xin Yu
Keyword(s):  
Rogue Wave ◽  
Rogue Waves ◽  
Dispersion Parameter ◽  
Frequency Detuning ◽  
Third Order ◽  
First Order ◽  
Third Order Dispersion ◽  
Erbium Doped ◽  
Order Dispersion ◽  
Extant Population

AbstractUnder investigation in this article is a higher-order nonlinear Schrödinger–Maxwell–Bloch (HNLS-MB) system for the optical pulse propagation in an erbium-doped fiber. Lax pair, Darboux transformation (DT), and generalised DT for the HNLS-MB system are constructed. Soliton solutions and rogue wave solutions are derived based on the DT and generalised DT, respectively. Properties of the solitons and rogue waves are graphically presented. The third-order dispersion parameter, fourth-order dispersion parameter, and frequency detuning all influence the characteristic lines and velocities of the solitons. The frequency detuning also affects the amplitudes of solitons. The separating function has no effect on the properties of the first-order rogue waves, except for the locations where the first-order rogue waves appear. The third-order dispersion parameter affects the propagation directions and shapes of the rogue waves. The frequency detuning influences the rogue-wave types of the module for the measure of polarization of resonant medium and the extant population inversion. The fourth-order dispersion parameter impacts the rogue-wave interaction range and also has an effect on the rogue-wave type of the extant population inversion. The value of separating function affects the spatial-temporal separation of constituting elementary rogue waves for the second-order and third-order rogue waves. The second-order and third-order rogue waves can exhibit the triangular and pentagon patterns under different choices of separating functions.

Higher order rogue waves for the(3 + 1)-dimensional Jimbo–Miwa equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammed K. Elboree
Keyword(s):  
Symbolic Computation ◽  
Fourth Order ◽  
Bilinear Form ◽  
Rogue Waves ◽  
Higher Order ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

Abstract Based on the Hirota bilinear form for the (3 + 1)-dimensional Jimbo–Miwa equation, we constructed the first-order, second-order, third-order and fourth-order rogue waves for this equation using the symbolic computation approach. Also some properties of the higher-order rogue waves and their interaction are explained by some figures via some special choices of the parameters.

Solitons and rogue waves for a nonlinear system in the geophysical fluid

2016 ◽  
Vol 30 (35) ◽  
pp. 1650412 ◽  
Author(s):  
Xi-Yang Xie ◽  
Bo Tian ◽  
Lei Liu ◽  
Xiao-Yu Wu ◽  
Yan Jiang
Keyword(s):  
Nonlinear System ◽  
Wave Packet ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Second Order ◽  
Basic Flow ◽  
Mean Flow ◽  
First Order ◽  
Derived Properties

In this paper, we investigate a nonlinear system, which describes the marginally unstable baroclinic wave packets in the geophysical fluid. Based on the symbolic computation and Hirota method, bright one- and two-soliton solutions for such a system are derived. Propagation and collisions of the solitons are graphically shown and discussed with [Formula: see text], which reflects the collision between the wave packet and mean flow, [Formula: see text], which measures the state of the basic flow, and group velocity [Formula: see text]. [Formula: see text] is observed to affect the amplitudes of the solitons, and [Formula: see text] can influence the solitons’ traveling directions. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions are derived. Properties of the first- and second-order rogue waves are graphically presented and analyzed: The first-order rogue waves are shown in the figures. [Formula: see text] has no effects on A, which is the amplitude of the wave packet, but with the increase of [Formula: see text], amplitude of B, which is a quantity measuring the correction of the basic flow, decreases. When [Formula: see text] is chosen differently, A and B do not keep their shapes invariant. With the value of [Formula: see text] increasing, amplitudes of A and B become larger. The second-order rogue wave is presented, from which we observe that with [Formula: see text] increasing, amplitude of B decreases, but [Formula: see text] has no effects on A. Collision features of A and B alter with the value of [Formula: see text] changing. When we make the value of [Formula: see text] larger, amplitudes of A and B increase.

Hybrid soliton solutions in the (2+1)-dimensional nonlinear Schrödinger equation

2017 ◽  
Vol 31 (32) ◽  
pp. 1750298 ◽  
Author(s):  
Meidan Chen ◽  
Biao Li
Keyword(s):  
Rogue Wave ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Nls Equation ◽  
Bilinear Method ◽  
Nonlinear Schrödinger ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit

Rational solutions and hybrid solutions from N-solitons are obtained by using the bilinear method and a long wave limit method. Line rogue waves and lumps in the (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Schrödinger (NLS) equation are derived from two-solitons. Then from three-solitons, hybrid solutions between kink soliton with breathers, periodic line waves and lumps are derived. Interestingly, after the collision, the breathers are kept invariant, but the amplitudes of the periodic line waves and lumps change greatly. For the four-solitons, the solutions describe as breathers with breathers, line rogue waves or lumps. After the collision, breathers and lumps are kept invariant, but the line rogue wave has a great change.

Families of exact solutions of the generalized (3+1)-dimensional nonlinear-wave equation

2018 ◽  
Vol 32 (29) ◽  
pp. 1850359 ◽  
Author(s):  
Wenhao Liu ◽  
Yufeng Zhang
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Lump Solution ◽  
Rational Solutions ◽  
Bilinear Method ◽  
The One ◽  
Wave Limit

In this paper, the traveling wave method is employed to investigate the one-soliton solutions to two different types of bright solutions for the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear-wave equation, primarily. In the following parts, we derive the breathers and rational solutions by using the Hirota bilinear method and long-wave limit. More specifically, we discuss the lump solution and rogue wave solution, in which their trajectory will be changed by varying the corresponding coefficient or coordinate axis. On the one hand, the breathers express the form of periodic line waves in different planes, on the other hand, rogue waves are localized in time.

Intertextual Precision Expectations

2019 ◽  
pp. 107-116
Author(s):  
Karin Kukkonen
Keyword(s):  
Third Order ◽  
Structural Patterns ◽  
First Order ◽  
The Third ◽  
Order Probability

In the chapters that follow, the third-order probability design is developed. The third-order probability design revolves around how expectations about second- and first-order predictions are developed through structural patterns yielded by genre (III.1), textual gaps and shadow stories (III.2), and intertextual references to unfamiliar texts (III.3). The final chapter of the section, then, traces the tension between flexibility and constraint in probability designs.

General high-order breather, lump, and semi-rational solutions to the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation

2020 ◽  
pp. 2150057
Author(s):  
Xin-Mei Zhou ◽  
Shou-Fu Tian ◽  
Ling-Di Zhang ◽  
Tian-Tian Zhang
Keyword(s):  
Bilinear Form ◽  
Soliton Solutions ◽  
High Order ◽  
Rational Solutions ◽  
Lump Solutions ◽  
Long Wave ◽  
Wave Limit

In this work, we investigate the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation. Based on its bilinear form, the [Formula: see text]th-order breather solutions of the gBK equation are successful given by taking appropriate parameters. Furthermore, the [Formula: see text]th-order lump solutions of the gBK equation are obtained via the long-wave limit method. In addition, the semi-rational solutions are generated to reveal the interaction between lump solutions, soliton solutions, and breather solutions.

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