scholarly journals On Soliton Solutions of Perturbed Boussinesq and KdV-Caudery-Dodd-Gibbon Equations

Coatings ◽  
2021 ◽  
Vol 11 (11) ◽  
pp. 1429
Author(s):  
Muhammad Imran Asjad ◽  
Hamood Ur Rehman ◽  
Zunaira Ishfaq ◽  
Jan Awrejcewicz ◽  
Ali Akgül ◽  
...  
Keyword(s):  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Hyperbolic Function ◽  
Nonlinear Phenomena ◽  
3D Graphics ◽  
New Exact Solutions ◽  
Hyperbolic Function Method ◽  
The Common ◽  
2D And 3D ◽  
Exact Soliton Solutions

Nonlinear science is a fundamental science frontier that includes research in the common properties of nonlinear phenomena. This article is devoted for the study of new extended hyperbolic function method (EHFM) to attain the exact soliton solutions of the perturbed Boussinesq equation (PBE) and KdV–Caudery–Dodd–Gibbon (KdV-CDG) equation. We can claim that these solutions are new and are not previously presented in the literature. In addition, 2d and 3d graphics are drawn to exhibit the physical behavior of obtained new exact solutions.

scholarly journals The conformable space–time fractional Fokas–Lenells equation and its optical soliton solutions based on three analytical schemes

2020 ◽  
pp. 2150004
Author(s):  
Asim Zafar ◽  
Muhammad Raheel ◽  
Ahmet Bekir ◽  
Waseem Razzaq
Keyword(s):  
Optical Fibers ◽  
Function Method ◽  
Pulse Propagation ◽  
Soliton Solutions ◽  
Space Time ◽  
Hyperbolic Function ◽  
Nonlinear Wave Propagation ◽  
Different Types ◽  
Hyperbolic Function Method ◽  
Exact Soliton Solutions

This paper is about the study of space–time fractional Fokas–Lenells equation that describes nonlinear wave propagation in optical fibers. Three prominent schemes are employed for extracting different types of exact soliton solutions. In particular, the [Formula: see text] function method, the hyperbolic function method and the simplest Riccati equation scheme are investigated for the said model. As a sequel, a series of soliton solutions are obtained and verified through MATHEMATICA. The obtained solutions are significant additions in some specific fields of physics and engineering. Furthermore, the 3D graphical descriptions are left to analyze the pulse propagation for the reader.

New optical soliton solutions via two distinctive schemes for the DNA Peyrard–Bishop equation in fractal order

2021 ◽  
pp. 2150444
Author(s):  
Loubna Ouahid ◽  
M. A. Abdou ◽  
S. Owyed ◽  
Sachin Kumar
Keyword(s):  
Evolution Equations ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Optical Soliton ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Closed Form Solutions ◽  
Special Solutions ◽  
Engineering Physics ◽  
Exact Soliton Solutions

The deoxyribonucleic acid (DNA) dynamical equation, which emerges from the oscillator chain known as the Peyrard–Bishop (PB) model for abundant optical soliton solutions, is presented, along with a novel fractional derivative operator. The Kudryashov expansion method and the extended hyperbolic function (HF) method are used to construct novel abundant exact soliton solutions, including light, dark, and other special solutions that can be directly evaluated. These newly formed soliton solutions acquired here lead one to ask whether the analytical approach could be extended to deal with other nonlinear evolution equations with fractional space–time derivatives arising in engineering physics and nonlinear sciences. It is noted that the newly proposed methods’ performance is most reliable and efficient, and they will be used to construct new generalized expressions of exact closed-form solutions for any other NPDEs of fractional order.

Exact Soliton Solutions To An Averaged Dispersion-Managed Fiber System Equation

2004 ◽  
Vol 59 (12) ◽  
pp. 919-926
Author(s):  
Biao Li
Keyword(s):  
Evolution Equations ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Fiber System ◽  
System Equation ◽  
Generalized Riccati Equation ◽  
Hyperbolic Function Solutions ◽  
Hyperbolic Function Method

By introducing a set of ordinary differential equations which possess q-deformed hyperbolic function solutions, and a new ansatz, a method is developed for constructing a series of exact analytical solutions of some nonlinear evolution equations. The proposed method is more powerful than various tanh methods, the secq-tanhq-method, generalized hyperbolic-function method, generalized Riccati equation expansion method, generalized projective Riccati equations method and other sophisticated methods. As an application of the method, an averaged dispersion-managed (DM) fiber system equation, which governs the dynamics of the core of the DM soliton, is chosen to illustrate the method. With the help of symbolic computation, rich new soliton solutions are obtained. From these solutions, some previously known solutions obtained by some authors can be recovered by means of some suitable choices of the arbitrary functions and arbitrary constants. Further, the soliton propagation and solitons interaction scenario are discussed and simulated by computer.

scholarly journals An application of the MEFM to the modified Boussinesq equation

2018 ◽  
Vol 9 (1) ◽  
pp. 11-17
Author(s):  
Tolga Akturk
Keyword(s):  
Function Method ◽  
Boussinesq Equation ◽  
Travelling Wave ◽  
3D Graphics ◽  
Travelling Wave Solutions ◽  
Trigonometric Functions ◽  
Wolfram Mathematica ◽  
Modified Boussinesq Equation ◽  
Mathematica Software ◽  
2D And 3D

In this paper, some travelling wave solutions of the Modified Boussinesq (MBQ) equation are obtained by using the modified expansion function method (MEFM). When the obtained solutions are commented, trigonometric functions including hyperbolic features are obtained. The 2D and 3D graphics of the solutions have been investigated by selecting appropriate parameters. All the obtained solutions provide the MBQ equation. In this work, all mathematical calculations are done with Wolfram Mathematica software. 

scholarly journals Exact Soliton Solutions for Nonlinear Perturbed Schrödinger Equations with Nonlinear Optical Media

2020 ◽  
Vol 10 (24) ◽  
pp. 8929
Author(s):  
Khaled A. Gepreel
Keyword(s):  
Optical Fibers ◽  
Power Law ◽  
Soliton Solutions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Optical Media ◽  
Kerr Law ◽  
Exact Soliton Solutions

The nonlinear perturbed Schrödinger equations (NPSEs) with nonlinear terms as Kerr law, power law, quadratic-cubic law, and dual-power law nonlinearity media play an important role in optical fibers. In this article we implement the rational solitary wave method to study the NPSEs when nonlinear terms take some different forms. Additionally, we use the q-deformed hyperbolic function and q-deformed trigonometric function methods to study the exact solutions to NPSEs. Different kind of soliton solutions are obtained such as bright, dark, and singular periodic solutions to the NPSEs.

Approximate symmetry and solutions of the nonlinear Klein–Gordon equation with a small parameter

2017 ◽  
Vol 14 (03) ◽  
pp. 1750046
Author(s):  
Mohammad Rahimian ◽  
Megerdich Toomanian ◽  
Mehdi Nadjafikhah
Keyword(s):  
Small Parameter ◽  
Analytical Solutions ◽  
Function Method ◽  
Gordon Equation ◽  
Hyperbolic Function ◽  
Approximate Symmetry ◽  
Nonlinear Phenomena ◽  
Hyperbolic Function Method ◽  
Klein Gordon ◽  
Klein Gordon Equation

In this paper, the Lie approximate symmetry analysis is applied to investigate new solutions of the nonlinear Klein–Gordon equation with a small parameter. The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. The hyperbolic function method and Riccati equation method are employed to solve some of the obtained reduced ordinary differential equations. We construct new analytical solutions with a small parameter which is effectively obtained by the proposed method.

Deeper properties of the nonlinear Phi-four and Gross-Pitaevskii equations arising mathematical physics

2021 ◽  
Author(s):  
Li Yan ◽  
Ajay Kumar ◽  
Juan Luis García Guirao ◽  
Haci Mehmet Baskonus ◽  
Wei Gao
Keyword(s):  
Modulation Instability ◽  
Soliton Solutions ◽  
Mathematical Physics ◽  
Dark Soliton ◽  
Periodic Wave ◽  
Hyperbolic Function ◽  
Periodic Wave Solutions ◽  
Package Program ◽  
Hyperbolic Function Solutions ◽  
2D And 3D

In this paper, the rational sine–cosine and rational sinh–cosh methods are applied in extracting some properties of nonlinear Phi-four and Gross–Pitaevskii equations. The singular periodic wave solutions, dark soliton solutions and hyperbolic function solutions are reported. The solitary waves are observed from the traveling waves under the values of the parameters. Modulation instability analysis is also observed in various simulations. We also plot to observe the wave distributions of parameters of stability in 2D and 3D visuals via package program.

GENERALIZED RICCATI EQUATION EXPANSION METHOD AND ITS APPLICATION TO THE (2+1)-DIMENSIONAL BOUSSINESQ EQUATION

2003 ◽  
Vol 14 (04) ◽  
pp. 471-482 ◽  
Author(s):  
YONG CHEN ◽  
BIAO LI ◽  
HONGQING ZHANG
Keyword(s):  
Riccati Equation ◽  
Function Method ◽  
Boussinesq Equation ◽  
Nonlinear Differential Equations ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Generalized Riccati Equation ◽  
Hyperbolic Function Method ◽  
Formal Solutions ◽  
Singular Soliton

Based on the computerized symbolic system Maple and a Riccati equation, a new Riccati equation expansion method for constructing nontraveling wave and coefficient functions' soliton-like solutions is presented by a new general ansätz. The proposed method is more powerful than most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method, and generalized hyperbolic-function method. By using the method, we not only successfully recovered the previously known formal solutions but could also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the (2+1)-dimensional Boussinesq equation and obtain rich new families of the exact solutions, including the nontraveling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, and triangular functions solutions.

scholarly journals A comparison of analytical solutions of nonlinear complex generalized Zakharov dynamical system for various definitions of the differential operator

2022 ◽  
Vol 30 (1) ◽  
pp. 335-361
Author(s):  
Melih Cinar ◽  
◽  
Ismail Onder ◽  
Aydin Secer ◽  
Mustafa Bayram ◽  
...  
Keyword(s):  
Dynamical System ◽  
Differential Operator ◽  
Exact Solutions ◽  
3D Graphics ◽  
Langmuir Waves ◽  
New Exact Solutions ◽  
Different Types ◽  
Pde Systems ◽  
2D And 3D ◽  
Linear Pdes

<abstract><p>This paper considers deriving new exact solutions of a nonlinear complex generalized Zakharov dynamical system for two different definitions of derivative operators called conformable and $ M- $ truncated. The system models the spread of the Langmuir waves in ionized plasma. The extended rational $ sine-cosine $ and $ sinh-cosh $ methods are used to solve the considered system. The paper also includes a comparison between the solutions of the models containing separately conformable and $ M- $ truncated derivatives. The solutions are compared in the $ 2D $ and $ 3D $ graphics. All computations and representations of the solutions are fulfilled with the help of Mathematica 12. The methods are efficient and easily computable, so they can be applied to get exact solutions of non-linear PDEs (or PDE systems) with the different types of derivatives.</p></abstract>

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