Abundant breather and semi-analytical investigation: On high-frequency waves’ dynamics in the relaxation medium

2021 ◽  
pp. 2150372
Author(s):  
Mostafa M. A. Khater
Keyword(s):  
Analytical Solutions ◽  
High Frequency ◽  
Evolution Equations ◽  
Dynamical Behavior ◽  
Analytical Investigation ◽  
Nonlinear Evolution Equations ◽  
Research Papers ◽  
Ostrovsky Equation ◽  
High Frequency Waves ◽  
Published Research

This paper investigates the high-frequency waves’ dynamical behavior in the relaxation medium through two recent analytical schemes. This study depends on the Vakhnenko–Parkes (VP) equation that has been reduced from the well-known Ostrovsky equation. The modified Khater (MKhat) and the extended simplest equation (ESE) methods are used to handle the considered model. As a result, many novel solitary wave solutions have been obtained to construct the initial and boundary conditions. These conditions allow employing the variational iteration (VI) method to study the semi-analytical solutions of the considered model. The accuracy of solutions is explained along with showing the matching between analytical and semi-analytical solutions and comparing our obtained solutions with the previous results that have been obtained in published research papers. Moreover, the high-frequency waves’ behavior relaxation medium is illustrated through some distinct sketches. The methods’ performance shows their effectiveness, direct, easy, and consequential for studying many nonlinear evolution equations.

scholarly journals Investigation of Dark and Bright Soliton Solutions of Some Nonlinear Evolution Equations

2018 ◽  
Vol 22 ◽  
pp. 01056 ◽  
Author(s):  
Seyma Tuluce Demiray ◽  
Hasan Bulut
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Bright Soliton ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method ◽  
Ostrovsky Equation

In this paper, generalized Kudryashov method (GKM) is used to find the exact solutions of (1+1) dimensional nonlinear Ostrovsky equation and (4+1) dimensional Fokas equation. Firstly, we get dark and bright soliton solutions of these equations using GKM. Then, we remark the results we found using this method.

Lie symmetry analysis and dynamical structures of soliton solutions for the (2 + 1)-dimensional modified CBS equation

2020 ◽  
Vol 34 (25) ◽  
pp. 2050221
Author(s):  
S. Kumar ◽  
D. Kumar
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Surface Condition ◽  
Dynamical Behavior ◽  
Lie Symmetry ◽  
Nonlinear Evolution Equations ◽  
Optimal System ◽  
Explicit Solutions ◽  
Nonlinear Odes ◽  
Symmetry Reductions

In this present article, the new [Formula: see text]-dimensional modified Calogero-Bogoyavlenskii-Schiff (mCBS) equation is studied. Using the Lie group of transformation method, all of the vector fields, commutation table, invariant surface condition, Lie symmetry reductions, infinitesimal generators and explicit solutions are constructed. As we all know, an optimal system contains constructively important information about the various types of exact solutions and it also offers clear understandings into the exact solutions and its features. The symmetry reductions of [Formula: see text]-dimensional mCBS equation is derived from an optimal system of one-dimensional subalgebra of the Lie invariance algebra. Then, the mCBS equation can further be reduced into a number of nonlinear ODEs. The generated explicit solutions have different wave structures of solitons and they are analyzed graphically and physically in order to exhibit their dynamical behavior through 3D, 2D-shapes and respective contour plots. All the produced solutions are definitely new and totally different from the earlier study of the Manukure and Zhou (Int. J. Mod. Phys. B 33, (2019)). Some of these solutions are demonstrated by the means of solitary wave profiles like traveling wave, multi-solitons, doubly solitons, parabolic waves and singular soliton. The calculations show that this Lie symmetry method is highly powerful, productive and useful to study analytically other nonlinear evolution equations in acoustics physics, plasma physics, fluid dynamics, mathematical biology, mathematical physics and many other related fields of physical sciences.

scholarly journals On Highly Dimensional Elastic and Nonelastic Interaction between Internal Waves in Straight and Varying Cross-Section Channels

2020 ◽  
Vol 2020 ◽  
pp. 1-9 ◽  
Author(s):  
Mostafa M. A. Khater ◽  
Qiang Zheng ◽  
Haiyong Qin ◽  
Raghda A. M. Attia
Keyword(s):  
Internal Waves ◽  
Dynamical Behavior ◽  
Manuscript Studies ◽  
Research Papers ◽  
Derivative Operator ◽  
Dynamical Characteristics ◽  
Extension Model ◽  
Spatial Dimensions ◽  
Kp Equations ◽  
Published Research

This manuscript studies the computational solutions of the highly dimensional elastic and nonelastic interaction between internal waves through the fractional nonlinear (4 + 1)-dimensional Fokas equation. This equation is considered as the extension model of the two-dimensional Davey–Stewartson (DS) and Kadomtsev–Petviashvili (KP) equations to a four spatial dimensions equation with time domain. The modified Khater method is employed along the Atangana–Baleanu (AB) derivative operator to construct many novel explicit wave solutions. These solutions explain more physical and dynamical behavior of that kind of the interaction. Moreover, 2D, 3D, contour, and stream plots are demonstrated to explain the detailed dynamical characteristics of these solutions. The novelty of our paper is shown by comparing our results with those obtained in previous published research papers.

Lie symmetry reductions, abound exact solutions and localized wave structures of solitons for a (2 + 1)-dimensional Bogoyavlenskii equation

2021 ◽  
pp. 2150252
Author(s):  
Sachin Kumar ◽  
Monika Niwas
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Evolution Equations ◽  
Dynamical Behavior ◽  
Lie Symmetry ◽  
Nonlinear Evolution Equations ◽  
Mathematical Methods ◽  
Closed Form Solutions ◽  
Symmetry Reductions ◽  
Engineering Sciences

By applying the two efficient mathematical methods particularly with regard to the classical Lie symmetry approach and generalized exponential rational function method, numerous exact solutions are constructed for a (2 + 1)-dimensional Bogoyavlenskii equation, which describes the interaction of Riemann wave propagation along the spatial axes. Moreover, we obtain the infinitesimals, all the possible vector fields, optimal system, and Lie symmetry reductions. The governing Bogoyavlenskii equation is converted into various nonlinear ordinary differential equations through two stages of Lie symmetry reductions. Accordingly, abundant exact closed-form solutions are obtained explicitly in terms of independent arbitrary functions, rational functions, trigonometric functions, and hyperbolic functions with arbitrary free parameters. The dynamical behavior of the resulting soliton solutions is presented through 3D-plots via numerical simulation. Eventually, single solitons, multi-solitons with oscillations, kink wave with breather-type solitons, and single lump-type solitons are obtained. The proposed mathematical techniques are effective, trustworthy, and reliable mathematical tools to work out new exact closed-form solutions of various types of nonlinear evolution equations in mathematical physics and engineering sciences.

Breather soliton and cross two-soliton solutions for the fifth order Caudrey-Dodd-Gibbon (CDG) equation

2015 ◽  
Vol 25 (3) ◽  
pp. 651-655 ◽  
Author(s):  
Hanlin Chen ◽  
Zhenhui Xu ◽  
Zhengde Dai
Keyword(s):  
Nonlinear Equations ◽  
Bilinear Form ◽  
Evolution Equations ◽  
Dynamical Behavior ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
Engineering Technology ◽  
Content Type ◽  
Fifth Order

Purpose – The purpose of this paper is to reveal dynamical behavior of nonlinear wave by searching for the new breather soliton and cross two-soliton solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG) equation. Design/methodology/approach – The authors apply bilinear form and extended homoclinic test approach to the fifth-order CDG equation. Findings – In this paper, by using bilinear form and extended homoclinic test approach, the authors obtain new breather soliton and cross two-soliton solutions of the fifth-order CDG equation. It is shown that the extended homoclinic test approach, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Research limitations/implications – The research manifests that the structures of the solution to nonlinear equations are diversified and complicated. Originality/value – The methods used in this paper can be widely applied to the research of spatial and temporal characteristics of nonlinear equations in physics and engineering technology. These methods are also conducive for people to know objective laws and grasp the essential features of the development of the world.

A Riccati–Bernoulli sub-ODE Method for Some Nonlinear Evolution Equations

2019 ◽  
Vol 20 (3-4) ◽  
pp. 303-313 ◽  
Author(s):  
S. Z. Hassan ◽  
Mahmoud A. E. Abdelrahman
Keyword(s):  
High Frequency ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Langmuir Wave ◽  
Rational Functions ◽  
Nonlinear Evolution Equations ◽  
Ode Method

AbstractThis article concerns with the construction of the analytical traveling wave solutions for the model of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave and the higher-order nonlinear Schrödinger equation by Riccati–Bernoulli sub-ODE method. We give the exact solutions for these two equations. The proposed method is effective tool to solve many other nonlinear partial differential equations. Moreover, this method can give a new infinite sequence of solutions. These solutions are expressed by hyperbolic, trigonometric and rational functions. Finally, with the aid of Matlab release 15, some graphical simulations were designed to see the behavior of these solutions.

scholarly journals New analytical solutions for conformable fractional PDEs arising in mathematical physics by exp-function method

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 647-651 ◽  
Author(s):  
Orkun Tasbozan ◽  
Yücel Çenesiz ◽  
Ali Kurt ◽  
Dumitru Baleanu
Keyword(s):  
Analytical Solutions ◽  
Function Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Vital Role ◽  
Nonlinear Evolution Equations ◽  
Physical Systems ◽  
Wave Solutions ◽  
Klein Gordon

AbstractModelling of physical systems mathematically, produces nonlinear evolution equations. Most of the physical systems in nature are intrinsically nonlinear, therefore modelling such systems mathematically leads us to nonlinear evolution equations. The analysis of the wave solutions corresponding to the nonlinear partial differential equations (NPDEs), has a vital role for studying the nonlinear physical events. This article is written with the intention of finding the wave solutions of Nizhnik-Novikov-Veselov and Klein-Gordon equations. For this purpose, the exp-function method, which is based on a series of exponential functions, is employed as a tool. This method is an useful and suitable tool to obtain the analytical solutions of a considerable number of nonlinear FDEs within a conformable derivative.

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