Bifurcations and Exact Solutions for a Class of MKdV Equations with the Conformable Fractional Derivative via Dynamical System Method

2020 ◽  
Vol 30 (01) ◽  
pp. 2050004 ◽  
Author(s):  
Jianli Liang ◽  
Longkun Tang ◽  
Yonghui Xia ◽  
Yi Zhang
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Leibniz Rule ◽  
Wave Transformation ◽  
Conformable Fractional Derivative ◽  
Simplest Equation Method

In 2014, Khalil et al. [2014] proposed the conformable fractional derivative, which obeys chain rule and the Leibniz rule. In this paper, motivated by the monograph of Jibin Li [Li, 2013], we study the exact traveling wave solutions for a class of third-order MKdV equations with the conformable fractional derivative. Our approach is based on the bifurcation theory of planar dynamical systems, which is much different from the simplest equation method proposed in [Chen & Jiang, 2018]. By employing the traveling wave transformation [Formula: see text] [Formula: see text], we reduce the PDE to an ODE which depends on the fractional order [Formula: see text], then the analysis depends on the order [Formula: see text]. Moreover, as [Formula: see text], the exact solutions are consistent with the integer PDE. However, in all the existing papers, the reduced ODE is independent of the fractional order [Formula: see text]. It is believed that this method can be applicable to solve the other nonlinear differential equations with the conformable fractional derivative.

Explicit, periodic and dispersive soliton solutions to the conformable time-fractional Wu–Zhang system

2021 ◽  
pp. 2150417
Author(s):  
Kalim U. Tariq ◽  
Mostafa M. A. Khater ◽  
Muhammad Younis
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Conformable Fractional Derivative ◽  
Physical Behavior ◽  
Wave Solutions

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified.

scholarly journals Exact Solutions and Dynamic Properties of Perturbed Nonlinear Schrodinger Equation Arised From Nano Optical Fibers

2021 ◽  
Author(s):  
Baoshu Xin ◽  
Shuangqing Chen
Keyword(s):  
Exact Solutions ◽  
Optical Fibers ◽  
Traveling Wave ◽  
Dynamic Properties ◽  
Equilibrium Points ◽  
Main Idea ◽  
Traveling Wave Solutions ◽  
Wave Transformation ◽  
Polynomial Method ◽  
Wave System

The main idea of this paper is to investigate the exact solutions and dynamic properties in optical nanofibers, which is modeled by space-time fractional perturbed nonlinear schr\"odinger equation involving Kerr law nonlinearity with conformal fractional derivative. Firstly, by the complex fractional traveling wave transformation, the traveling wave system of the original equation is obtained, then a conserved quantity, namely the Hamiltonian is constructed, and the qualitative analysis of this system is conducted via this quantity by classifying the equilibrium points. Moreover, the prior estimate of the existence of the soliton and periodic solution is established via the bifurcation method. Furthermore, all exact traveling wave solutions are constructed to illustrate our results explicitly by the complete discrimination system for polynomial method.

scholarly journals Exact Solutions and Dynamic Properties of Perturbed Nonlinear Schrodinger Equation Arised From Nano Optical Fibers

2021 ◽  
Author(s):  
Baoshu Xin
Keyword(s):  
Exact Solutions ◽  
Optical Fibers ◽  
Traveling Wave ◽  
Dynamic Properties ◽  
Equilibrium Points ◽  
Main Idea ◽  
Traveling Wave Solutions ◽  
Wave Transformation ◽  
Polynomial Method ◽  
Wave System

The main idea of this paper is to investigate the exact solutions and dynamic properties in optical nanofibers, which is modeled by space-time fractional perturbed nonlinear schr\"odinger equation involving Kerr law nonlinearity with conformal fractional derivative. Firstly, by the complex fractional traveling wave transformation, the traveling wave system of the original equation is obtained, then a conserved quantity, namely the Hamiltonian is constructed, and the qualitative analysis of this system is conducted via this quantity by classifying the equilibrium points. Moreover, the prior estimate of the existence of the soliton and periodic solution is established via the bifurcation method. Furthermore, all exact traveling wave solutions are constructed to illustrate our results explicitly by the complete discrimination system for polynomial method.

scholarly journals Explicit Exact Solutions of Space-time Fractional Drinfel’d-Sokolov-Wilson Equations

2021 ◽  
Vol 2068 (1) ◽  
pp. 012005
Author(s):  
Hongkua Lin
Keyword(s):  
Partial Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Rational Function ◽  
Expansion Method ◽  
Space Time ◽  
Wave Transformation ◽  
Conformable Fractional Derivative ◽  
Fractional Partial Differential Equations ◽  
Function Type

Abstract The space-time fractional Drinfel’d-Sokolov-Wilson equations (DSWEs) has attracted many researchers’ attention in recent years. In this study, combining the (G’/G,1/G)-expansion method and a fractional wave transformation, we derive abundant explicit exact solutions of the DSWEs with the conformable fractional derivative. All of the resulting solutions include triangle, hyperbolic and rational function type. It shows this technique is effective and reliable to find exact solutions of other similar nonlinear fractional partial differential equations (NFPDEs).

scholarly journals Exact Single Traveling Wave Solutions for Generalized Fractional Gardner Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Zhao Li ◽  
Tianyong Han ◽  
Chun Huang
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave Transformation ◽  
Wave Solutions ◽  
System Method ◽  
Order Polynomial ◽  
Complete Discrimination System ◽  
Fourth Order Polynomial

In this paper, the classification of all single traveling wave solutions to generalized fractional Gardner equations is presented by utilizing the complete discrimination system method. Under the fractional traveling wave transformation, generalized fractional Gardner equations can be reduced to an ordinary differential equations. All possible exact traveling wave solutions are given through the complete discrimination system of the fourth-order polynomial. Moreover, graphical representations of different kinds of the exact solutions reveal that the method is of significance for searching the exact solutions to generalized fractional Gardner equations.

scholarly journals Explicit Exact Solutions of the (2+1)-Dimensional Integro-Differential Jaulent–Miodek Evolution Equation Using the Reliable Methods

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Supaporn Kaewta ◽  
Sekson Sirisubtawee ◽  
Nattawut Khansai
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave Transformation ◽  
Jacobi Elliptic Function ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Wave Solutions

In this article, we utilize the G′/G2-expansion method and the Jacobi elliptic equation method to analytically solve the (2 + 1)-dimensional integro-differential Jaulent–Miodek equation for exact solutions. The equation is shortly called the Jaulent–Miodek equation, which was first derived by Jaulent and Miodek and associated with energy-dependent Schrödinger potentials (Jaulent and Miodek, 1976; Jaulent, 1976). The equation is converted into a fourth order partial differential equation using a transformation. After applying a traveling wave transformation to the resulting partial differential equation, we obtain an ordinary differential equation which is the main equation to which the both schemes are applied. As a first step, the two methods give us distinguish systems of algebraic equations. The first method provides exact traveling wave solutions including the logarithmic function solutions of trigonometric functions, hyperbolic functions, and polynomial functions. The second approach provides the Jacobi elliptic function solutions depending upon their modulus values. Some of the obtained solutions are graphically characterized by the distinct physical structures such as singular periodic traveling wave solutions and peakons. A comparison between our results and the ones obtained from the previous literature is given. Obtaining the exact solutions of the equation shows the simplicity, efficiency, and reliability of the used methods, which can be applied to other nonlinear partial differential equations taking place in mathematical physics.

The wave patterns to the time-space fractional Casimir equation for the Ito system

2021 ◽  
pp. 2150396
Author(s):  
Damin Cao ◽  
Wei Xu ◽  
Fajiang He
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Soliton Solution ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Graphical Representations ◽  
Wave Solutions ◽  
Time Space ◽  
Wave Patterns

In this paper, the time-space fractional Casimir equation for the Ito system with conformal fractional derivative is taken into consideration and the corresponding traveling wave solutions are given and the effects of the fractional order to the peakon soliton solution are also discussed and analyzed. In addition, some graphical representations are also provided to show the properties of the solution directly.

scholarly journals Symmetry properties and exact solutions of the time fractional Kolmogorov-Petrovskii-Piskunov equation

2019 ◽  
Vol 65 (5 Sept-Oct) ◽  
pp. 529 ◽  
Author(s):  
M. S. Hasheim ◽  
M. Inc ◽  
M. Bayram
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Lie Symmetry ◽  
Conformable Fractional Derivative ◽  
Symmetry Approach ◽  
Simplest Equation Method ◽  
Symmetry Properties ◽  
Lie Symmetry Approach

In this paper, the time fractional Kolmogorov-Petrovskii-Piskunov (FKP) equation is analyzed by means of Lie symmetry approach. The FKP is reduced to ordinary differential equation of fractional order via the attained point symmetries. Moreover, the simplest equation method is used in construct the exact solutions of underlying equation with recently introduced conformable fractional derivative.

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