An investigation of the physical dynamics of a traveling wave solution called a bright soliton

2021 ◽  
Vol 96 (12) ◽  
pp. 125251
Author(s):  
Serbay Duran
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Applied Mathematics ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Bright Soliton ◽  
Kink Wave Solution ◽  
Advantages And Disadvantages ◽  
Kink Wave ◽  
Mathematics And Physics

Abstract This study examines the 1 + 2 -dimensional Zoomeron equation, which has recently become popular in applied mathematics and physics. Bright soliton (non-topological), kink wave solution and traveling wave solutions are generated with the advantages of the generalized exponential rational function method. With the help of this method, it is aimed to produce different types of solutions for the Zoomeron equation compared to other traditional exponential function methods. The effects of parameters on the amplitude of the wave function are discussed, along with physical explanations backed by simulations. In addition, the advantages and disadvantages of the method for the 1 + 2 -dimensional Zoomeron equation are discussed.

Exact Traveling Wave Solutions and Bifurcations of the Time-Fractional Differential Equations with Applications

2019 ◽  
Vol 29 (03) ◽  
pp. 1950041 ◽  
Author(s):  
Wenjing Zhu ◽  
Yonghui Xia ◽  
Bei Zhang ◽  
Yuzhen Bai
Keyword(s):  
Differential Equations ◽  
Traveling Wave ◽  
Wave Solution ◽  
Bifurcation Theory ◽  
Traveling Wave Solutions ◽  
Kink Wave Solution ◽  
Fractional Pdes ◽  
Kink Wave ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

This paper presents a method to investigate exact traveling wave solutions and bifurcations of the nonlinear time-fractional partial differential equations with the conformable fractional derivative proposed by [Khalil et al., 2014]. The method is based on employing the bifurcation theory of planar dynamical systems proposed by [Li, 2013]. For the fractional PDEs, up till now, there is no related paper to obtain the exact solutions by applying bifurcation theory. We show how to use this method with applications to two fractional PDEs: the fractional Klein–Gordon equation and the fractional generalized Hirota–Satsuma coupled KdV system, respectively. We find the new exact solutions including periodic wave solution, kink wave solution, anti-kink wave solution and solitary wave solution (bright and dark), which are different from previous works in the literature. This approach can also be extended to other nonlinear time-fractional differential equations with the conformable fractional derivative.

scholarly journals Single peaked traveling wave solutions to a generalized μ-Novikov Equation

2020 ◽  
Vol 10 (1) ◽  
pp. 66-75
Author(s):  
Byungsoo Moon
Keyword(s):  
Linear Combination ◽  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Wave Solutions ◽  
Quadratic Nonlinearities ◽  
Novikov Equation

Abstract In this paper, we study the existence of peaked traveling wave solution of the generalized μ-Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions.

Energy-carrying wave simulation of the Lonngren-wave equation in semiconductor materials

2021 ◽  
pp. 2150213
Author(s):  
Hülya Durur
Keyword(s):  
Wave Equation ◽  
Traveling Wave ◽  
Energy Transport ◽  
Three Dimensional ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Discussion Section ◽  
Advantages And Disadvantages ◽  
Wave Solutions ◽  
Package Program

In this study, the Lonngren-wave equation, which is physically semiconductor, is taken into consideration. Traveling wave solutions of this equation are presented with generalized exponential rational function method, which is one of the mathematically powerful analytical methods. These solutions are produced in bright (non-topological) soliton and complex trigonometric-type traveling wave solutions. Three-dimensional (3D), 2D and contour graphics are presented with the help of a ready-made package program with special values given to constants in these solutions. The effect of the change in wave velocity on the traveling wave solution showing energy transport is presented with the help of simulation. It is argued that velocity is one of the important factors in wave diffraction. In the results and discussion section, the advantages and disadvantages of the method are discussed.

scholarly journals Approximate Damped Oscillatory Solutions for Generalized KdV-Burgers Equation and Their Error Estimates

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Weiguo Zhang ◽  
Xiang Li
Keyword(s):  
Dynamical Systems ◽  
Error Estimates ◽  
Traveling Wave ◽  
Wave Solution ◽  
Burgers Equation ◽  
Traveling Wave Solutions ◽  
Approximate Solutions ◽  
Traveling Wave Solution ◽  
Oscillatory Solutions ◽  
Wave Solutions

We focus on studying approximate solutions of damped oscillatory solutions of generalized KdV-Burgers equation and their error estimates. The theory of planar dynamical systems is employed to make qualitative analysis to the dynamical systems which traveling wave solutions of this equation correspond to. We investigate the relations between the behaviors of bounded traveling wave solutions and dissipation coefficient, and give two critical valuesλ1andλ2which can characterize the scale of dissipation effect, for right and left-traveling wave solution, respectively. We obtain that for the right-traveling wave solution if dissipation coefficientα≥λ1, it appears as a monotone kink profile solitary wave solution; that if0<α<λ1, it appears as a damped oscillatory solution. This is similar for the left-traveling wave solution. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, we obtain their approximate damped oscillatory solutions by undetermined coefficients method. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between exact and approximate solutions. The errors are infinitesimal decreasing in the exponential form.

Surface wave behavior and refraction simulation on the ocean for the fractional Ostrovsky–Benjamin–Bona–Mahony equation

2021 ◽  
pp. 2150477
Author(s):  
Serbay Duran ◽  
Asif Yokuş ◽  
Hülya Durur
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Stationary Wave ◽  
Traveling Wave Solution ◽  
Incoming Wave ◽  
Topological Soliton ◽  
Advantages And Disadvantages ◽  
Wave Event ◽  
Ostrovsky Equation ◽  
General Graphs

In this study, we have taken into account the time-fractional Ostrovsky–Benjamin–Bona–Mahony equation, which is a synthesis of the time-fractional Ostrovsky equation and time-fractional Benjamin–Bona–Mahony equations and contains both mathematical and physical properties. Traveling wave solutions are produced by using the Ostrovsky–Benjamin–Bona–Mahony equation that physically sheds light on the incoming wave event on the ocean surface, using the sub-equation and Bernoulli sub-equation function methods. These solutions are presented in hyperbolic, trigonometric, singular and dark (topological) soliton types. With the help of special values given to the coefficients in the solitons obtained, it is associated with the solutions in the literature and it is observed that the solitons produced in this study are more general. Graphs representing the stationary wave at any given moment are presented. The advantages and disadvantages as well as the similarities and differences of the method are discussed in detail. Also, the behavior of the wave and its refraction according to the velocity variable, which is a physically important factor of the traveling wave solution, is analyzed and supported by simulation.

scholarly journals Traveling wave solutions for a class of reaction-diffusion system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bingyi Wang ◽  
Yang Zhang
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Traveling Wave Solution ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
One Dimensional ◽  
Wave Solutions

AbstractIn this paper we investigate the existence of traveling wave for a one-dimensional reaction diffusion system. We show that this system has a unique translation traveling wave solution.

scholarly journals Traveling waves in cooperative predation: relaxation of sublinearity

2021 ◽  
pp. 1-10
Author(s):  
Srijana Ghimire ◽  
Xiang-Sheng Wang
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Wave Speed ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Positive Equilibrium ◽  
Open Problems ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Wave Solutions

In this paper, we investigate traveling wave solutions of a diffusive predator-prey model which takes into consideration hunting cooperation. Sublinearity condition is violated for the function of cooperative predation. When the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. On the other hand, if the wave speed exceeds this critical value, we prove the existence of a positive traveling wave solution connecting the predator-free equilibrium to the unique positive equilibrium under a technical assumption of weak cooperative predation. The key idea of the proof contains two major steps: (i) we construct a suitable pentahedron and find inside it a trajectory connecting the predator-free equilibrium; and (ii) we construct a suitable Lyapunov function and use LaSalle invariance principle to prove that the trajectory also connects the positive equilibrium. In the end of this paper, we propose five open problems related to traveling wave solutions in cooperative predation.

On the Gaussian traveling wave solution to a special kind of Schrödinger equation with logarithmic nonlinearity

2021 ◽  
Author(s):  
Yue Kai ◽  
Zhixiang Yin
Keyword(s):  
Schrödinger Equation ◽  
Traveling Wave ◽  
Wave Solution ◽  
Schrodinger Equation ◽  
Traveling Wave Solutions ◽  
Special Kind ◽  
Traveling Wave Solution ◽  
Complete Analysis ◽  
Wave Solutions ◽  
Dispersion Terms

We present the complete analysis of traveling wave solutions to a special kind of nonlinear Schrödinger equation with logarithmic nonlinearity, and obtain all traveling wave solutions. As a result, we prove this equation does not have any Gaussian traveling wave solution. However, by modifying this equation into another form, we can actually obtain a Gaussian traveling wave solution, which verifies the conclusion that existing Gaussian traveling solution requires two restrictions: (1) balance between the dispersion terms and logarithmic nonlinearity; and (2) balance of the parameters.

scholarly journals Simulation and refraction event of complex hyperbolic type solitary wave  in plasma and obtical fiber for the perturbed Chen-Lee-Liu equation

2021 ◽  
Author(s):  
Asıf Yokuş ◽  
Hülya Durur ◽  
Serbay Duran
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Hyperbolic Type ◽  
Traveling Wave Solution ◽  
Algebraic Equations ◽  
Discussion Section ◽  
Advantages And Disadvantages ◽  
Special Values ◽  
Analytic Methods

Abstract In this presented article, modified 1/G'-expansion and modified Kudryashov methods are applied to generate traveling wave solutions of perturbed Chen-Lee-Liu (CLL) equation. The similar and different aspects of the solutions produced by both analytic methods are discussed in the results and discussion section. By giving special values to the constants in the solutions obtained by analytical methods, 2D, 3D and contour graphics representing the shape of the standing wave at any time are presented. Additionally, the advantages and disadvantages of the two analytic methods are discussed and presented in the results and discussion section. Also, a solitary wave is produced by giving special values ​​to the parameters in the hyperbolic type complex traveling wave solution. Simulations are created for different values ​​of the frequency and velocity propagation parameters of the solitary wave. The values ​​of these parameters are calculated for the breakage event physically. A computer package program is used for operations such as solving complex operations, drawing graphics and systems of algebraic equations.

scholarly journals Some Traveling Wave Solutions of a Hyperbolic Model for Chemotaxis in 1-D

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Qin Zhang ◽  
Wen-yan Chen
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Space Dimension ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Hyperbolic Model ◽  
Independent Variables ◽  
Wave Solutions ◽  
Tanh Method ◽  
Dependent Variables

The traveling wave solution of a hyperbolic model for chemotaxis in one space dimension is studied in this paper. By using some transformations of dependent variables and independent variables, we apply the tanh method and improved tanh method to the model, from which some traveling wave solutions in explicit form are presented.

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