Dust acoustic rogue waves of fractional-order model in dusty plasma
本站小编 Free考研考试/2022-01-02
Jun-Chao Sun1, Zong-Guo Zhang2, Huan-He Dong1, Hong-Wei Yang,1,∗1College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China 2School of Mathematics Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China
First author contact: Author to whom any correspondence should be addressed. Received:2020-05-26Revised:2020-08-18Accepted:2020-09-1Online:2020-11-12
Abstract In this paper, the fractional-order model is used to study dust acoustic rogue waves in dusty plasma. Firstly, based on control equations, the multi-scale analysis and reduced perturbation method are used to derive the (3+1)-dimensional modified Kadomtsev-Petviashvili (MKP) equation. Secondly, using the semi-inverse method and the fractional variation principle, the (3+1)-dimensional time-fractional modified Kadomtsev-Petviashvili (TF-MKP) equation is derived. Then, the Riemann-Liouville fractional derivative is used to study the symmetric property and conservation laws of the (3+1)-dimensional TF-MKP equation. Finally, the exact solution of the (3+1)-dimensional TF-MKP equation is obtained by using fractional order transformations and the definition and properties of Bell polynomials. Based on the obtained solution, we analyze and discuss dust acoustic rogue waves in dusty plasma. Keywords：time-fractional modified Kadomtsev-Petviashvili equation;conservation laws;dust acoustic rogue waves
PDF (1068KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Jun-Chao Sun, Zong-Guo Zhang, Huan-He Dong, Hong-Wei Yang. Dust acoustic rogue waves of fractional-order model in dusty plasma. Communications in Theoretical Physics, 2020, 72(12): 125001- doi:10.1088/1572-9494/abb7d7
1. Introduction
Dusty plasmas [1, 2] are complex plasmas composed of electrons, ions, neutral gases, and charged dust particles. Dust particles are an important part of dusty plasmas. The size of dust particles ranges from nanometer to micron to millimeter. Dust particles [3, 4] are charged, each with thousands of elementary charges, but their charge-mass ratio is many orders of magnitude smaller than that of ions. In addition to the action of gravity, the motion of dust particles is mainly governed by electromagnetic force. The interaction between dust particles and electron ions; the interaction between dust particles and the interaction between the entire system and external factors increase the complexity of the system. These properties make dusty plasmas different from ordinary plasmas and present many new physical phenomena, which arouse the interest of scholars.
Dusty plasmas are widespread in nuclear fusion reactors, the Earth's ionosphere, planetary rings and tails of comets. In the early 1980s, from the photos of Saturn rings returned from the Voyager 2 spacecraft to Saturn, it was found that radial spoke-like material in the ring B of Saturn rotates around the outside of the ring B. These spoke-like substances are composed of fine particles. Hill and Goertz et al [5, 6] first proposed that these particles are charged. In 1989, Selwyn et al [7] first reported dust contamination during plasma etching of conductor chips. These two major events have made the research of dusty plasmas develop rapidly. Now, dusty plasmas have developed into a more popular research field, and it has become the fastest-growing one besides Bose-Einstein condensation [8] subject.
Collective fluctuation and instability are important topics in dusty plasma physics. There are ultra-low frequency dust acoustic waves [9] at the time scale of the movement of dust particles in dusty plasmas, and this waves has the nature of acoustic waves. In 1990, Rao et al [10] predicted dust acoustic waves in theory. In 1995, Barkan [11] first confirmed the existence of dust acoustic waves in experiments. Since then, scholars have been more interested in the study of dust acoustic waves. Ghosh [12] used a (1+1)-dimensional model to study the influence of the change of the charge of the non-adiabatic dust in nonlinear dust acoustic waves on dusty plasmas. Gill [13] used the reduced perturbation method to derive a model of the (2+1)-dimensional KP equation in dusty plasmas. Saini [14] analyzed and discussed the characteristics of dust acoustic solitary waves in dusty plasma system.
In the early stage, the theoretical study of fractional calculus [15, 16] was mainly focused on the field of mathematics. However, with the development of science, the theory of fractional calculus is becoming more and more complete, so fractional calculus has attracted more and more scholars' attention. In particular, fractional differential equations abstracted from practical problems have become a research hotspot. Fractional calculus is the theory of any order differential and integral, it is unified with integer order calculus [17, 18], it is a generalization of integer order calculus. With the continuous improvement of fractional calculus theory by scholars, in some practical situations, the concept of fractional derivatives [19, 20] is more suitable for modeling than integer derivatives, which makes fractional differential equations gradually begin to be used to solve problems in various fields and become more and more practical. In the study of dust plasma, most scholars describe the propagation of waves in dusty plasmas by establishing integer-order models, which makes it necessary to establish fractional-order models to study the propagation of waves in dusty plasmas. The use of fractional differential equations to study practical problems is inseparable from the solution of fractional differential equations [21, 22]. Therefore, the solution of fractional differential equations has become an important topic. At present, based on the solution method of integer order equations [23-25], there are many methods for solving fractional differential equations. For example, $\exp (-\varphi (\xi ))$ method [26], $(G/G^{\prime} )$-expansion method [27], and the Hirota bilinear method [28], etc.
Lie symmetry analysis [3] provides an efficient and powerful tool for the determination of boundary value problems, initial value problems and conservation laws of differential equations. Conservation laws [3] are of great significance in analyzing the integrability, internal properties of differential equations, and proving the existence and uniqueness of solutions. Lie symmetry analysis was first proposed by Norwegian mathematician Lie [29] at the end of the 19th century. Noether's theorem [30] established the relationship between conservation laws and differential equations, which led to the development of conservation laws. These methods and theories have been used for the study of integer order differential equations, but the symmetry and conservation laws of fractional differential equations have not been widely discussed. In order to promote the research on the symmetry and conservation laws of fractional differential equations, in this paper, we have established a fractional-order model and studied the symmetry and conservation laws of the (3+1)-dimensional time-fractional modified Kadomtsev-Petviashvili (TF-MKP) equation.
The concept of rogue waves [31-33] was first proposed by Draper [34] in the 1960s. The peak of rogue waves is very sharp, the height of rogue waves is very high, the duration of the rogue waves is very short, rogue waves will disappear quickly, they will suddenly appear in a certain area at a certain time. The amplitude of rogue waves will rapidly increase to the extreme value, and then the amplitude will quickly fall back and eventually disappear. It can be seen that rogue waves show a local structure in space and time. Rogue waves can gather huge energy in a short time, which makes them very powerful and destructive. At present, there is no unified definition of rogue waves, and most scholars define rogue waves from the perspective of wave height. In recent years, rogue waves have been a hot topic in the marine atmosphere, plasma, Bose-Einstein condensation and other fields. The study of phenomena of rogue waves in plasma is still in its infancy. In 2011, Bailung et al [35] first observed the phenomenon of rogue waves in plasmas in experiments. After that, scholars studied the phenomenon of rogue waves in several important plasma models [36-38]. However, the research on rogue waves in fractional models is almost blank. Therefore, in this paper, we establish a fractional-order model to study dust acoustic rogue waves in dusty plasmas.
The rest of the paper is as follows: in the second section, based on the control equations, the reduced perturbation method is used to derive the (3+1)-dimensional MKP equation. In the third section, the TF-MKP equation is derived by using the semi-inverse method and the fractional variation principle. In the forth section, the symmetry and conservation laws of the TF-MKP equation are discussed. In the fifth section, according to the definition and properties of the bell polynomials, the exact solution of the TF-MKP equation is obtained. Using the obtained solution, phenomena of dust acoustic rogue waves in dusty plasmas is studied and the effect of the time-fractional order ω on dust acoustic rogue waves is analyzed.
2. Basic model equations
We consider the dust acoustic rogue waves in a dusty plasma system consisting of dust fluid and superthermal electrons as well as ions.
In order to facilitate the study of problems, we make the following assumptions: The dusty plasma system studied is not affected by external magnetic field, that is, it is unmagnetized. The dust particles in the dusty plasma system have no collision effect, that is, they are collisionless. Dust acoustic solitary waves propagates in the direction of x, but there are weak high-order lateral disturbances in the directions of y and z.
The dusty plasma studied in this paper consisting of dust fluid, ions and electrons obeying $\kappa \mbox{-} $ type particle distributions. Charge neutrality at equilibrium reads: ${n}_{e0}={{sZ}}_{d0}{n}_{d0}+{n}_{i0}$, where ne0, nd0, and ni0 are the equilibrium value of the number density of electrons, dust particles and ions, respectively. Zd0 is the unperturbed number of charges on dust particles. The nonlinear dynamics of dust acoustic solitary waves in such a dusty plasma system can be described by the following set of normalized equations:$\begin{eqnarray}\left\{\begin{array}{l}\tfrac{\partial {n}_{d}}{\partial t}+\tfrac{\partial ({n}_{d}{u}_{d})}{\partial x}+\tfrac{\partial ({n}_{d}{v}_{d})}{\partial y}+\tfrac{\partial ({n}_{d}{w}_{d})}{\partial z}=0,\\ \tfrac{\partial {u}_{d}}{\partial t}+{u}_{d}\tfrac{\partial {u}_{d}}{\partial x}+{v}_{d}\tfrac{\partial {u}_{d}}{\partial y}+{w}_{d}\tfrac{\partial {u}_{d}}{\partial z}=-s\tfrac{\partial \phi }{\partial x},\\ \tfrac{\partial {v}_{d}}{\partial t}+{u}_{d}\tfrac{\partial {v}_{d}}{\partial x}+{v}_{d}\tfrac{\partial {v}_{d}}{\partial y}+{w}_{d}\tfrac{\partial {v}_{d}}{\partial z}=-s\tfrac{\partial \phi }{\partial y},\\ \tfrac{\partial {w}_{d}}{\partial t}+{u}_{d}\tfrac{\partial {w}_{d}}{\partial x}+{v}_{d}\tfrac{\partial {w}_{d}}{\partial y}+{w}_{d}\tfrac{\partial {w}_{d}}{\partial z}=-s\tfrac{\partial \phi }{\partial z},\\ \tfrac{{\partial }^{2}\phi }{\partial {x}^{2}}+\tfrac{{\partial }^{2}\phi }{\partial {y}^{2}}+\tfrac{{\partial }^{2}\phi }{\partial {z}^{2}}=-s({n}_{d}-1)+{n}_{e}-{n}_{i},\end{array}\right.\end{eqnarray}$where $s=+1(-1)$ for positive (negative) dust. The number density of the dust particles nd is normalized by nd0. The electrostatic potential φ is normalized by $\tfrac{{K}_{{\rm{B}}}{T}_{\mathrm{eff}}}{e}$, where kB is the Boltzmann constant, ${T}_{\mathrm{eff}}$ is the effective temperature and e is the electronic charge. ud, vd and wd are the velocities of the dust flow along x, y and z directions, respectively and normalized by the dust acoustic speed ${C}_{d}={\left(\tfrac{{K}_{{\rm{B}}}{Z}_{d0}{T}_{\mathrm{eff}}}{{m}_{d}}\right)}^{\tfrac{1}{2}}$, where md is the mass of the dust particles. The spatial coordinates (x, y, z) are normalized by the effective Debye length ${\lambda }_{d}={\left(\tfrac{{K}_{{\rm{B}}}{T}_{\mathrm{eff}}}{4\pi {n}_{d0}{Z}_{d0}{e}^{2}}\right)}^{\tfrac{1}{2}}$ and the time t is normalized by the inverse dust plasma frequency ${\omega }_{{pi}}^{-1}={\left(\tfrac{{m}_{i}}{4\pi {n}_{d0}{Z}_{d0}^{2}{e}^{2}}\right)}^{\tfrac{1}{2}}$.
The normalized number densities of electrons and ions obeying kappa distribution are expressed as$\begin{eqnarray}\begin{array}{rcl}{n}_{{e}} & = & {\left(1-\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{{e}}}\displaystyle \frac{\phi }{{\kappa }_{{e}}-\tfrac{3}{2}}\right)}^{-{\kappa }_{{e}}+\displaystyle \frac{1}{2}},\\ {n}_{{i}} & = & {\left(1-\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{{i}}}\displaystyle \frac{\phi }{{\kappa }_{{i}}-\tfrac{3}{2}}\right)}^{-{\kappa }_{{i}}+\displaystyle \frac{1}{2}},\end{array}\end{eqnarray}$where Te and Ti are the electron and ion plasma temperatures respectively.
Equation (1) is a complex system of nonlinear fluid mechanics equations. In order to study the propagation of dust acoustic rouge waves, we study the dimensionless form of the control equations by using the reductive perturbation method. The complex nonlinear equations are simplified to a differential equation, and the main nonlinear part of the original equations is reserved. First, the independent variables are expanded as follows:$\begin{eqnarray}\xi =\varepsilon (x-\lambda t),\,\eta ={\varepsilon }^{2}y,\,\zeta ={\varepsilon }^{2}z,\,\tau ={\varepsilon }^{3}t,\end{eqnarray}$according to equation (6), we have$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial }{\partial x} & = & \varepsilon \displaystyle \frac{\partial }{\partial \xi },\,\displaystyle \frac{\partial }{\partial y}={\varepsilon }^{2}\displaystyle \frac{\partial }{\partial \eta },\\ \displaystyle \frac{\partial }{\partial z} & = & {\varepsilon }^{2}\displaystyle \frac{\partial }{\partial \zeta },\,\displaystyle \frac{\partial }{\partial t}={\varepsilon }^{3}\displaystyle \frac{\partial }{\partial \tau }-\varepsilon \lambda \displaystyle \frac{\partial }{\partial \xi }.\end{array}\end{eqnarray}$
The dependent variables nd, ud, vd, wd and φ are expanded as follows:$\begin{eqnarray}\left\{\begin{array}{l}{n}_{d}=1+\varepsilon {n}_{{d}_{1}}+{\varepsilon }^{2}{n}_{{d}_{2}}+{\varepsilon }^{3}{n}_{{d}_{3}}+\cdots ,\\ {u}_{d}=\varepsilon {u}_{{d}_{1}}+{\varepsilon }^{2}{u}_{{d}_{2}}+{\varepsilon }^{3}{u}_{{d}_{3}}+\cdots ,\\ {v}_{d}={\varepsilon }^{2}{v}_{{d}_{1}}+{\varepsilon }^{3}{v}_{{d}_{2}}+\cdots ,\\ {w}_{d}={\varepsilon }^{2}{w}_{{d}_{1}}+{\varepsilon }^{3}{w}_{{d}_{2}}+\cdots ,\\ \phi =\varepsilon {\phi }_{1}+{\varepsilon }^{2}{\phi }_{2}+{\varepsilon }^{3}{\phi }_{2}+\cdots .\end{array}\right.\end{eqnarray}$
According to equation (10), we have$\begin{eqnarray}{n}_{{d}_{1}}=\displaystyle \frac{{u}_{{d}_{1}}}{\lambda },\,{u}_{{d}_{1}}=s\displaystyle \frac{{\phi }_{1}}{\lambda },\,{C}_{1}=\displaystyle \frac{1}{{\lambda }^{2}}.\end{eqnarray}$
In the second section, we derived the (3+1)-dimensional MKP equation, which is an integer-order equation. With the development of scientific research, fractional calculus began to shift from pure mathematical formulas to applications in various fields, especially in the field of physics. Phenomena in acoustics can be well described by fractional differential equations. In order to deeply and comprehensively study the nonlinear propagation of dust acoustic waves, in this section, we use the semi-inverse method and the fractional variational principle to derive the (3+1)-dimensional TF-MKP equation.
The Riemann-Liouville fractional derivative is defined as [16]$\begin{eqnarray}{D}_{\tau }^{\omega }f(\tau )=\left\{\begin{array}{l}\displaystyle \frac{1}{{\rm{\Gamma }}(n-\omega )}\displaystyle \frac{{{\rm{d}}}^{n}}{{\rm{d}}{\tau }^{n}}{\displaystyle \int }_{0}^{\tau }{\left(\tau -T\right)}^{n-\omega -1}f(T){\rm{d}}T,\qquad n-1\lt \alpha \lt n,\\ \displaystyle \frac{{\partial }^{n}f(t)}{\partial {\tau }^{n}},\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \omega =n.\end{array}\right.\end{eqnarray}$
Some properties of the Riemann-Liouville derivative are given as follows:$\begin{eqnarray}\begin{array}{l}{D}_{\tau }^{\omega }(f(\tau )g(\tau ))=g(\tau ){D}_{\tau }^{\omega }f(\tau )+f(\tau ){D}_{\tau }^{\omega }g(\tau ),\\ {D}_{\tau }^{\omega }f[g(\tau )]=f{{\prime} }_{g}[g(\tau )]{D}_{\tau }^{\omega }g(\tau )={f}_{g}^{\omega }f[g(\tau )]{\left(g^{\prime} (\tau )\right)}^{\omega }.\end{array}\end{eqnarray}$
Equation (15) can be rewritten as follows:$\begin{eqnarray}{\phi }_{\tau }+{a}_{1}{\phi }^{2}{\phi }_{\xi }+{a}_{2}{\phi }_{\xi \xi \xi }+{a}_{3}{D}^{-1}({\phi }_{\eta \eta }+{\phi }_{\zeta \zeta })=0,\end{eqnarray}$where ${D}^{-1}\phi =\int \phi {\rm{d}}\xi $.
Introducing the potential function ψ(ξ, η, ζ, τ) such that φ(ξ, η, ζ, τ)=ψξ(ξ, η, ζ, τ), and the potential equation of equation (3.3) is as follows:$\begin{eqnarray}{\psi }_{\xi \tau }+{a}_{1}{\psi }_{\xi }^{2}{\psi }_{\xi \xi }+{a}_{2}{\psi }_{\xi \xi \xi \xi }+{a}_{3}({\psi }_{\eta \eta }+{\phi }_{\zeta \zeta })=0.\end{eqnarray}$
The Lagrangian of equation (19) can be defined by using the semi-inverse method [13], the functional of equation (20) can be represented by$\begin{eqnarray}\begin{array}{rcl}J(\psi ) & = & {\displaystyle \int }_{X}{\rm{d}}\xi {\displaystyle \int }_{Y}{\rm{d}}\eta {\displaystyle \int }_{Z}{\rm{d}}\zeta {\displaystyle \int }_{T}{\rm{d}}\tau [\psi ({b}_{1}{\psi }_{\xi \tau }+{b}_{2}{a}_{1}{\psi }_{\xi }^{2}{\psi }_{\xi \xi }\\ & & +{b}_{3}{a}_{2}{\psi }_{\xi \xi \xi \xi }+{b}_{4}{a}_{3}{\psi }_{\eta \eta }+{b}_{5}{a}_{3}{\psi }_{\zeta \zeta })],\end{array}\end{eqnarray}$where bi (i=1, 2, …, 4) are Lagrangian multipliers which can be determined later.
By applying the variation of equation (22) and using the optimal conditions of variation to integrate each term by parts, the following relation can be obtained:$\begin{eqnarray}\begin{array}{l}L(\xi ,\eta ,\zeta ,\tau ,{\psi }_{\xi },{\psi }_{\eta },,{\psi }_{\zeta },{\psi }_{\tau },{\psi }_{\xi \xi })\\ \quad =\,\displaystyle \frac{\partial F}{\partial \psi }-\displaystyle \frac{\partial }{\partial \tau }\left(\displaystyle \frac{\partial F}{\partial {\psi }_{\tau }}\right)-\displaystyle \frac{\partial }{\partial \xi }\left(\displaystyle \frac{\partial F}{\partial {\psi }_{\xi }}\right)+\displaystyle \frac{{\partial }^{2}}{\partial {\xi }^{2}}\left(\displaystyle \frac{\partial F}{\partial {\psi }_{\xi \xi }}\right)\\ \qquad -\,\displaystyle \frac{\partial }{\partial \eta }\left(\displaystyle \frac{\partial F}{\partial {\psi }_{\eta }}\right)-\displaystyle \frac{\partial }{\partial \zeta }\left(\displaystyle \frac{\partial F}{\partial {\psi }_{\zeta }}\right)\\ \quad =\,2{b}_{1}{\psi }_{\xi \tau }+4{b}_{2}{a}_{1}{\psi }_{\xi }^{2}{\psi }_{\xi \xi }+2{b}_{3}{a}_{2}{\psi }_{\xi \xi \xi \xi }\\ \quad +\,2{b}_{4}{a}_{3}{\psi }_{\eta \eta }+2{b}_{5}{a}_{3}{\psi }_{\zeta \zeta }=0.\end{array}\end{eqnarray}$
Obviously, equation (23) is equivalent to equation (20), so the Lagrangian multiplier bi(i=1-5) can be determined as follows:$\begin{eqnarray}{b}_{1}=\displaystyle \frac{1}{2},\,{b}_{2}=\displaystyle \frac{1}{4},\,{b}_{3}=\displaystyle \frac{1}{2},\,{b}_{4}=\displaystyle \frac{1}{2},\,{b}_{5}=\displaystyle \frac{1}{2}.\end{eqnarray}$
According to equations (22) and (24), we obtain the Lagrangian form of (3+1)-dimensional MKP equation$\begin{eqnarray}\begin{array}{l}L({\psi }_{\xi },{\psi }_{\tau },{\psi }_{\eta },{\psi }_{\zeta },{\psi }_{\xi \xi })\\ =\,-\displaystyle \frac{1}{2}{\psi }_{\xi }{\psi }_{\tau }-\displaystyle \frac{1}{12}{a}_{1}{\psi }_{\xi }^{4}+\displaystyle \frac{1}{2}{a}_{2}{\psi }_{\xi \xi }^{2}-\displaystyle \frac{1}{2}{a}_{3}{\psi }_{\eta }^{2}-\displaystyle \frac{1}{2}{a}_{3}{\psi }_{\zeta }^{2}.\end{array}\end{eqnarray}$
Similarly, the Lagrangian form of the TF-MKP equation can be written as$\begin{eqnarray}\begin{array}{l}{ \mathcal L }({D}_{\tau }^{\omega }\psi ,{\psi }_{\tau },{\psi }_{\eta },{\psi }_{\zeta },{\psi }_{\xi \xi })\\ =\,-\displaystyle \frac{1}{2}{\psi }_{\xi }{D}_{\tau }^{\omega }\psi -\displaystyle \frac{1}{12}{a}_{1}{\psi }_{\xi }^{4}+\displaystyle \frac{1}{2}{a}_{2}{\psi }_{\xi \xi }^{2}-\displaystyle \frac{1}{2}{a}_{3}{\psi }_{\eta }^{2}-\displaystyle \frac{1}{2}{a}_{3}{\psi }_{\zeta }^{2}.\end{array}\end{eqnarray}$
Therefore, we obtain the functional of the TF-MKP equation as following form:$\begin{eqnarray}\begin{array}{rcl}{J}_{{ \mathcal L }}(\psi ) & = & {\int }_{X}{\rm{d}}\xi {\int }_{Y}{\rm{d}}\eta {\int }_{Z}{\rm{d}}\zeta \\ & & \times {\int }_{T}{\left({\rm{d}}\tau \right)}^{\omega }{ \mathcal L }({D}_{\xi }^{\alpha }\psi ,{\psi }_{\tau },{\psi }_{\eta },{\psi }_{\zeta },{\psi }_{\xi \xi }).\end{array}\end{eqnarray}$
According to the Agrawal's method [2], the variation of equation (22) relative to $\psi (\xi ,\eta ,\zeta ,\tau )$ results in$\begin{eqnarray}\begin{array}{rcl}\delta {J}_{{ \mathcal L }}(\psi ) & = & {\displaystyle \int }_{X}{\rm{d}}\xi {\displaystyle \int }_{Y}{\rm{d}}\eta {\displaystyle \int }_{Z}{\rm{d}}\zeta {\displaystyle \int }_{T}{\left({\rm{d}}\tau \right)}^{\omega }\left[\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {D}_{\tau }^{\omega }\psi }\right)\delta {D}_{\tau }^{\omega }\psi \right.\\ & & +\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {\psi }_{\xi }}\right)\delta {\psi }_{\xi }+\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {\psi }_{\xi \xi }}\right)\delta {\psi }_{\xi \xi }\\ & & \left.+\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {\psi }_{\eta }}\right)\delta {\psi }_{\eta }+\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {\psi }_{\zeta }}\right)\delta {\psi }_{\zeta }\right].\end{array}\end{eqnarray}$
Optimizing the variation of the functional ${J}_{{ \mathcal L }}(\psi )$, i.e. $\delta {J}_{{ \mathcal L }}(\psi )=0$, the Euler-Lagrange equation of the TF-MKP equation is as follows:$\begin{eqnarray}\begin{array}{l}-{D}_{\tau }^{\omega }\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {D}_{\tau }^{\omega }\psi }\right)-\displaystyle \frac{\partial }{\partial \xi }\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {\psi }_{\xi }}\right)+\displaystyle \frac{{\partial }^{2}}{\partial {\xi }^{2}}\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {\psi }_{\xi \xi }}\right)\\ -\,\displaystyle \frac{\partial }{\partial \eta }\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {\psi }_{\eta }}\right)-\displaystyle \frac{\partial }{\partial \zeta }\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {\psi }_{\zeta }}\right)=0.\end{array}\end{eqnarray}$
Substituting equation (26) into (31), we have$\begin{eqnarray}{D}_{\tau }^{\omega }{\psi }_{\xi }+{a}_{1}{\left({\psi }_{\xi }\right)}^{2}{\psi }_{\xi \xi }+{a}_{2}{\psi }_{\xi \xi \xi \xi }+{a}_{3}{\psi }_{\eta \eta }+{a}_{3}{\psi }_{\zeta \zeta }=0,\end{eqnarray}$then, taking the derivative of of equation (32), we obtain$\begin{eqnarray}{\left({D}_{\tau }^{\omega }{\psi }_{\xi }+{a}_{1}{\left({\psi }_{\xi }\right)}^{2}{\psi }_{\xi \xi }+{a}_{2}{\psi }_{\xi \xi \xi \xi }\right)}_{\xi }+{a}_{3}{\left({\psi }_{\eta \eta }+{\psi }_{\zeta \zeta }\right)}_{\xi }=0.\end{eqnarray}$
According to the potential function ψξ(ξ, η, ζ, τ)=φ(ξ, η, ζ, τ), we obtain the TF-MKP equation as follows:$\begin{eqnarray}{\left({D}_{\tau }^{\omega }\phi +{a}_{1}{\phi }^{2}{\phi }_{\xi }+{a}_{2}{\phi }_{\xi \xi \xi }\right)}_{\xi }+{a}_{3}({\psi }_{\eta \eta }+{\psi }_{\zeta \zeta })=0.\end{eqnarray}$
4. Conservation laws of time-fractional MKP equation
Conservation laws play an important role in exploring the integrability, uniqueness and intrinsic properties of differential equations. In the past, most scholars have studied conservation laws of differential equations of integer order. In this section, we discuss the symmetry and conservation laws of TF-MKP equations.
4.1. Lie symmetry analysis
The generalized Leibnitz rule is defined as [39]$\begin{eqnarray}{D}_{t}^{\omega }(f(t)g(t))=\displaystyle \sum _{n=0}^{\infty }\left(\begin{array}{c}\omega \\ n\end{array}\right){D}_{t}^{\omega -n}f(t){D}_{t}^{n}g(t),\qquad \omega \gt 0,\end{eqnarray}$where$\begin{eqnarray}\left(\begin{array}{c}\omega \\ n\end{array}\right)=\displaystyle \frac{{\left(-1\right)}^{n-1}\omega {\rm{\Gamma }}(n-\omega )}{{\rm{\Gamma }}(1-\omega ){\rm{\Gamma }}(n+1)}.\end{eqnarray}$
The (3+1)-dimensional TF-MKP equation can be rewritten as follows:$\begin{eqnarray}{D}_{\tau }^{\omega }\phi +{a}_{1}{\phi }^{2}{\phi }_{\xi }+{a}_{2}{\phi }_{\xi \xi \xi }+{a}_{3}{D}^{-1}({\phi }_{\eta \eta }+{\phi }_{\zeta \zeta })=0.\end{eqnarray}$
Equation (37) is converted into the following form of fractional nonlinear partial differential equation:$\begin{eqnarray}F(\xi ,\eta ,\zeta ,\tau ,\phi ,{\phi }_{\xi },{\phi }_{\xi \xi \xi },{\phi }_{\eta \eta },{\phi }_{\zeta \zeta },{D}_{\tau }^{\omega }\phi )=0,\,\,\omega \gt 0.\end{eqnarray}$
We assume that equation (38) is invariant under the one parameter Lie group of point transformations:$\begin{eqnarray}\left\{\begin{array}{l}\overline{\xi }=\xi +\epsilon X(\xi ,\eta ,\zeta ,\tau ,\phi )+O({\epsilon }^{2}),\\ \overline{\eta }=\eta +\epsilon Y(\xi ,\eta ,\zeta ,\tau ,\phi )+O({\epsilon }^{2}),\\ \overline{\zeta }=\zeta +\epsilon Z(\xi ,\eta ,\zeta ,\tau ,\phi )+O({\epsilon }^{2}),\\ \overline{\tau }=\tau +\epsilon T(\xi ,\eta ,\zeta ,\tau ,\phi )+O({\epsilon }^{2}),\\ \overline{\phi }=\phi +\epsilon U(\xi ,\eta ,\zeta ,\tau ,\phi )+O({\epsilon }^{2}),\\ {D}_{\tau }^{\omega }\overline{\phi }\to {D}_{\tau }^{\omega }\phi +\epsilon {U}^{\omega ,\tau }+O({\epsilon }^{2}),\\ \tfrac{\partial \overline{\phi }}{\partial \xi }\to \tfrac{\partial \phi }{\partial \xi }+\epsilon {U}^{\xi }+O({\epsilon }^{2}),\\ \tfrac{{\partial }^{3}\overline{\phi }}{\partial {\xi }^{3}}\to \tfrac{{\partial }^{3}\phi }{\partial {\xi }^{3}}+\epsilon {U}^{\xi \xi \xi }+O({\epsilon }^{2}),\\ \tfrac{{\partial }^{2}\overline{\phi }}{\partial {\eta }^{2}}\to \tfrac{{\partial }^{2}\phi }{\partial {\eta }^{2}}+\epsilon {U}^{\eta \eta }+O({\epsilon }^{2}),\\ \tfrac{{\partial }^{2}\overline{\phi }}{\partial {\zeta }^{2}}\to \tfrac{{\partial }^{2}\phi }{\partial {\zeta }^{2}}+\epsilon {U}^{\zeta \zeta }+O({\epsilon }^{2}),\end{array}\right.\end{eqnarray}$where ε≪1 is the group parameter, X, Y, Z, T are the infinitesimals of the transformations for the dependent variables, U is the infinitesimals of the transformations for the independent variables, the explicit expression of Uξ, ${U}^{\xi \xi \xi }$, ${U}^{\eta \eta }$, ${U}^{\zeta \zeta }$ are as follows:$\begin{eqnarray}\begin{array}{l}{U}^{\xi }={D}_{\xi }(U)-{\phi }_{\xi }{D}_{\xi }(X)-{\phi }_{\eta }{D}_{\xi }(Y)\\ \,-\,{\phi }_{\zeta }{D}_{\xi }(Z)-{\phi }_{\tau }{D}_{\xi }(T),\\ {U}^{\xi \xi \xi }={D}_{\xi }({U}^{\xi \xi })-{\phi }_{\xi \xi \xi }{D}_{\xi }(X)-{\phi }_{\xi \xi \eta }{D}_{\xi }(Y)\\ \,-\,{\phi }_{\xi \xi \zeta }{D}_{\xi }(Z)-{\phi }_{\xi \xi \tau }{D}_{\xi }(T),\\ {U}^{\eta \eta }={D}_{\eta }({U}^{\eta })-{\phi }_{\xi \eta }{D}_{\eta }(X)-{\phi }_{\eta \eta }{D}_{\eta }(Y)\\ \,-\,{\phi }_{\eta \zeta }{D}_{\eta }(Z)-{\phi }_{\eta \tau }{D}_{\eta }(T),\\ {U}^{\zeta \zeta }={D}_{\zeta }({U}^{\zeta })-{\phi }_{\xi \zeta }{D}_{\zeta }(X)-{\phi }_{\eta \zeta }{D}_{\zeta }(Y)\\ \,-\,{\phi }_{\zeta \zeta }{D}_{\zeta }(Z)-{\phi }_{\zeta \tau }{D}_{\zeta }(T).\end{array}\end{eqnarray}$
The chain rule for compound function [39] is defined as$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{m}f(g(t))}{{\rm{d}}{t}^{m}}=\displaystyle \sum _{k=0}^{m}\displaystyle \sum _{r=0}^{k}\left(\begin{array}{c}k\\ r\end{array}\right)\displaystyle \frac{1}{k!}[-g{\left(t\right)}^{r}]\displaystyle \frac{{{\rm{d}}}^{m}}{{\rm{d}}{t}^{m}}[{\left(g(t)\right)}^{k-r}]\displaystyle \frac{{{\rm{d}}}^{k}f(g)}{{\rm{d}}{g}^{k}}.\end{eqnarray}$
According to equations (49) and (51), we obtain$\begin{eqnarray}\begin{array}{l}{U}^{\omega ,\tau }+2{a}_{1}\phi {\phi }_{\xi }U+{a}_{1}{\phi }^{2}{U}_{\xi }+{a}_{2}{U}_{\xi \xi \xi }\\ \quad +\,{a}_{3}{D}^{-1}({U}_{\eta \eta }+{U}_{\zeta \zeta })=0.\end{array}\end{eqnarray}$
Substituting equations (39), (41) and (47) into (52), then equating coefficients of various monomials to zero, we obtain following infinitesimals:$\begin{eqnarray}\begin{array}{rcl}U & = & {c}_{1}\omega \phi ,\,X=\displaystyle \frac{{c}_{1}\omega }{3}\xi +{c}_{2},\,Y=\displaystyle \frac{{c}_{1}\omega }{2}\eta +{c}_{3},\\ Z & = & \displaystyle \frac{{c}_{1}\omega }{2}\zeta +{c}_{4},\,T={c}_{1}\tau +{c}_{5},\end{array}\end{eqnarray}$where ${c}_{i},\,\,i=1,2,3,4,5$ are arbitrary constants.
Therefore, the infinitesimal symmetric Lie algebra of equation (37) is spanned by the following infinitesimal generators:$\begin{eqnarray}\left\{\begin{array}{l}{M}_{1}=\tfrac{\partial }{\partial \tau },\\ {M}_{2}=\tfrac{\partial }{\partial \xi },\\ {M}_{3}=\tfrac{\partial }{\partial \eta },\\ {M}_{4}=\tfrac{\partial }{\partial \zeta },\\ {M}_{5}=\tfrac{\omega \xi }{3}\tfrac{\partial }{\partial \xi }+\tfrac{\omega \eta }{2}\tfrac{\partial }{\partial \eta }+\tfrac{\omega \zeta }{2}\tfrac{\partial }{\partial \zeta }+\tau \tfrac{\partial }{\partial \tau }+\omega \phi \tfrac{\partial }{\partial \phi }.\end{array}\right.\end{eqnarray}$
4.2. Conservation laws
Conservation laws of equation (37) satisfies the following equation:$\begin{eqnarray}{D}_{\tau }({C}^{\tau })+{D}_{\xi }({C}^{\xi })+{D}_{\eta }({C}^{\eta })+{D}_{\zeta }({C}^{\zeta })=0,\end{eqnarray}$where Cτ, Cξ, Cη and Cζ are the conserved vectors.
A formal Lagrangian of equation (37) is given as follows:$\begin{eqnarray}\begin{array}{rcl}{\ell } & = & \rho (\xi ,\eta ,\zeta ,\tau )[{D}_{\tau }^{\omega }\phi +{a}_{1}{\phi }^{2}{\phi }_{\xi }\\ & & +{a}_{2}{\phi }_{\xi \xi \xi }+{a}_{3}{D}^{-1}({\phi }_{\eta \eta }+{\phi }_{\zeta \zeta })],\end{array}\end{eqnarray}$where ρ(ξ, η, ζ, τ) is a new dependent variable.
According to equation (56), an action integral can be defined as$\begin{eqnarray}{\int }_{0}^{t}{\int }_{{R}_{1}}{\int }_{{R}_{2}}{\int }_{{R}_{3}}{\ell }(\xi ,\eta ,\zeta ,\tau ,\rho ,\phi ,{D}_{\tau }^{\omega }\phi ,{\phi }_{\xi },{\phi }_{\xi \xi \xi },{\phi }_{\eta \eta },{\phi }_{\zeta \zeta }){\rm{d}}\xi {\rm{d}}\eta {\rm{d}}\zeta {\rm{d}}\tau .\end{eqnarray}$
The Euler-Lagrangian operator is given by$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\delta }{\delta \phi } & = & \displaystyle \frac{\partial }{\partial \phi }+{\left({D}_{\tau }^{\omega }\right)}^{* }\displaystyle \frac{\partial }{\partial {D}_{\tau }^{\omega }\phi }-{D}_{\xi }\displaystyle \frac{\partial }{\partial {\phi }_{\xi }}\\ & & -{D}_{\xi }^{3}\displaystyle \frac{\partial }{\partial {\phi }_{\xi \xi \xi }}+{D}_{\eta }^{2}\displaystyle \frac{\partial }{\partial {\phi }_{\eta \eta }}+{D}_{\zeta }^{2}\displaystyle \frac{\partial }{\partial {\phi }_{\zeta \zeta }},\end{array}\end{eqnarray}$where ${({D}_{\tau }^{\omega })}^{* }={\left(-1\right)}^{n}{I}_{t}^{n-\omega }({D}_{\tau }^{n}){=}_{\tau }^{C}{D}_{t}^{\omega }$ is the adjoint operator of Riemann-Liouville fractional derivative operator ${D}_{\tau }^{\omega }$, where ${I}_{t}^{n-\omega }$ is the right-sided fractional integral operators, and ${}_{\tau }^{C}{D}_{p}^{\omega }$ is the right-sided Caputo fractional differential operator.
Therefore, we obtain the adjoint equation $\tfrac{\delta {\ell }}{\delta \phi }=0$ of equation (37) as the Euler-Lagrange equation, and we also have$\begin{eqnarray}\begin{array}{l}\overline{X}+J({D}_{\xi }(X)+{D}_{\eta }(Y)+{D}_{\zeta }(Z)+{D}_{\tau }(T))\\ \quad =\,W\displaystyle \frac{\delta }{\delta \phi }+{D}_{\xi }({C}^{\xi })+{D}_{\eta }({C}^{\eta })+{D}_{\zeta }({C}^{\zeta })\\ \qquad +\,{D}_{\tau }({C}^{\tau }),\end{array}\end{eqnarray}$where J is the identity operator, $\tfrac{\delta }{\delta \phi }$ is the Euler-Lagrangian operator, and $\overline{X}$ is given as$\begin{eqnarray}\begin{array}{rcl}\overline{X} & = & X\displaystyle \frac{\partial }{\partial \xi }+Y\displaystyle \frac{\partial }{\partial \eta }+Z\displaystyle \frac{\partial }{\partial \zeta }+T\displaystyle \frac{\partial }{\partial \tau }+U\displaystyle \frac{\partial }{\partial \phi }\\ & & +{U}^{\omega ,\tau }\displaystyle \frac{\partial }{\partial {D}_{\tau }^{\omega }\phi }+{U}_{\xi }\displaystyle \frac{\partial }{\partial {\phi }_{\xi }}+{U}_{\xi \xi \xi }\displaystyle \frac{\partial }{\partial {\phi }_{\xi \xi \xi }}\\ & & +{U}_{\eta \eta }\displaystyle \frac{\partial }{\partial {\phi }_{\eta \eta }}+{U}_{\zeta \zeta }\displaystyle \frac{\partial }{\partial {\phi }_{\zeta \zeta }}.\end{array}\end{eqnarray}$
The Lie characteristic function W is given by$\begin{eqnarray}W=U-T{\phi }_{\tau }-X{\phi }_{\xi }-Y{\phi }_{\eta }-Z{\phi }_{\zeta }.\end{eqnarray}$
5.1. Solutions of the (3+1)-dimensional time-fractional MKP equation
In this section, we obtain the solution of the (3+1)-dimensional TF-MKP equation by using fractional transforms [40] and the definition and properties of Bell polynomials.
Let $f=f({x}_{1},{x}_{2},\cdots ,{x}_{n})$ be a ${C}^{\infty }$ function with $n$ variables, the following polynomials are called multi-dimensional Bell polynomials [41]:$\begin{eqnarray}{Y}_{{n}_{1}{x}_{1},\cdots ,{n}_{l}{x}_{l}}(f)\equiv {Y}_{{n}_{1},\cdots ,{n}_{l}}({f}_{{r}_{1}{x}_{1},\cdots ,{r}_{l}{x}_{l}})={{\rm{e}}}^{-f}{\partial }_{{x}_{1}}^{{n}_{1}}\cdots {\partial }_{{x}_{l}}^{{n}_{l}}{{\rm{e}}}^{f},\end{eqnarray}$where$\begin{eqnarray}{f}_{{r}_{1}{x}_{1},\cdots ,{r}_{l}{x}_{l}}={\partial }_{{x}_{1}}^{{n}_{1}}\cdots {\partial }_{{x}_{l}}^{{n}_{l}}f\,\,({r}_{1}=0,\cdots ,{n}_{1};\cdots ;{r}_{l}=0,\cdots ,{n}_{l}).\end{eqnarray}$
A specific example is used to explain equation (68). When f=f(x, t), the two-dimensional Bell polynomials can be written as follows:$\begin{eqnarray}\begin{array}{l}{Y}_{x}(f)={f}_{x},\,{Y}_{2x}(f)={f}_{2x}+{f}_{x}^{2},\\ {Y}_{3x}(f)={f}_{3x}+3{f}_{x}{f}_{2x}+{f}_{x}^{3},\\ {Y}_{x,t}(f)={f}_{x,t}+{f}_{x}{f}_{t},\\ {Y}_{2x,t}(f)={f}_{2x,t}+2{f}_{x,t}{f}_{x}+{f}_{x}^{2}{f}_{t}.\end{array}\end{eqnarray}$
On the basis of equation (68), the following Bell polynomial containing functions v and w are defined as the multi-dimensional binary Bell polynomials:$\begin{eqnarray}\begin{array}{l}{{ \mathcal Y }}_{{n}_{1}{x}_{1},\cdots ,{n}_{l}{x}_{l}}(v,w)\equiv {Y}_{{n}_{1},\cdots ,{n}_{l}}(f)| {f}_{{r}_{1}{x}_{1},\cdots ,{r}_{l}{x}_{l}}\\ =\,\left\{\begin{array}{l}{v}_{{r}_{1}{x}_{1},\cdots ,{r}_{l}{x}_{l}},\,\,{r}_{1}+{r}_{2}+\cdots +{r}_{l}\,{\rm{is}}\,{\rm{odd}},\\ {w}_{{r}_{1}{x}_{1},\cdots ,{r}_{l}{x}_{l}},\,\,{r}_{1}+{r}_{2}+\cdots +{r}_{l}\,{\rm{is}}\,{\rm{even}}.\end{array}\right.\end{array}\end{eqnarray}$
Some specific examples of the multi-dimensional binary Bell polynomials are expressed as follows:$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal Y }}_{x}(f) & = & {v}_{x},\,{{ \mathcal Y }}_{2x}(f)={w}_{2x}+{v}_{x}^{2},\\ {{ \mathcal Y }}_{3x}(f) & = & {v}_{3x}+3{v}_{x}{w}_{2x}+{v}_{x}^{3},\,{{ \mathcal Y }}_{x,t}(f)={w}_{x,t}+{v}_{x}{v}_{t}.\end{array}\end{eqnarray}$
The relation between ${ \mathcal Y }$-polynomial and Hirota D-operator is$\begin{eqnarray}{{ \mathcal Y }}_{{n}_{1}{x}_{1},\cdots ,{n}_{l}{x}_{l}}(v=\mathrm{ln}\displaystyle \frac{F}{G},w=\mathrm{ln}{FG})={\left({FG}\right)}^{-1}{D}_{{x}_{1}}^{{n}_{1}}\cdots {D}_{{x}_{l}}^{{n}_{l}}F\cdot G,\end{eqnarray}$where ${n}_{1}+{n}_{2}+\cdots +{n}_{l}\geqslant 1$.
In the special case of F=G, the equation (73) can be written as$\begin{eqnarray}\begin{array}{l}{F}^{-2}{D}_{{x}_{1}}^{{n}_{1}}\cdots {D}_{{x}_{l}}^{{n}_{l}}F\cdot F\\ =\,\left\{\begin{array}{l}0,\,\,{n}_{1}+{n}_{2}+\cdots +{n}_{l}\,{\rm{is}}\,{\rm{odd}},\\ {P}_{{n}_{1}{x}_{1},\cdots ,{n}_{l}{x}_{l}}(w),\,\,{n}_{1}+{n}_{2}+\cdots +{n}_{l}\,{\rm{is}}\,{\rm{even}}.\end{array}\right.\end{array}\end{eqnarray}$
If and only if ${n}_{1}+{n}_{2}+\cdots +{n}_{l}$ is even, the P-polynomials can be defined as$\begin{eqnarray}{P}_{{n}_{1}{x}_{1},\cdots ,{n}_{l}{x}_{l}}(w=2\mathrm{ln}F)={F}^{-2}{D}_{{x}_{1}}^{{n}_{1}}\cdots {D}_{{x}_{l}}^{{n}_{l}}F\cdot F.\end{eqnarray}$
The Hirota D-operator can be expressed as a combination of P-polynomials and Y-polynomials:$\begin{eqnarray}\begin{array}{l}{\left({FG}\right)}^{-1}{D}_{{x}_{1}}^{{n}_{1}}\cdots {D}_{{x}_{l}}^{{n}_{l}}F\cdot G\\ \quad =\,{{ \mathcal Y }}_{{n}_{1}{x}_{1},\cdots ,{n}_{l}{x}_{l}}({v},w){| }_{{v}={\rm{ln}}F/G,w={\rm{ln}}{FG}}\\ \quad =\,{{ \mathcal Y }}_{{n}_{1}{x}_{1},\cdots ,{n}_{l}{x}_{l}}({v},{v}+q){| }_{{v}={\rm{ln}}F/G,q={\rm{ln}}2F}\\ \quad =\,\displaystyle \sum _{{p}_{1}=0}^{{n}_{1}}\cdots \displaystyle \sum _{{p}_{l}=0}^{{n}_{l}}\left(\begin{array}{c}{n}_{1}\\ {p}_{1}\end{array}\right)\cdots \left(\begin{array}{c}{n}_{l}\\ {p}_{l}\end{array}\right)\\ \qquad \times \,{P}_{{p}_{1}{x}_{1},\cdots ,{p}_{l}{x}_{l}}(q){Y}_{({n}_{1}-{p}_{1}){x}_{1},\cdots ,({n}_{l}-{p}_{l}){x}_{l}}({v}).\end{array}\end{eqnarray}$
In order to solve the fractional MKP equation, we introduce the following fractional transformation:$\begin{eqnarray}t=\displaystyle \frac{{p}_{1}{\tau }^{\omega }}{{\rm{\Gamma }}(1+\omega )},\end{eqnarray}$where where p1 is an arbitrary constant.
According to equation (77), we obtain$\begin{eqnarray}\displaystyle \frac{{\partial }^{\omega }\phi }{\partial {\tau }^{\omega }}={p}_{1}\displaystyle \frac{\partial \phi }{\partial t}.\end{eqnarray}$
Introducing constraints ${V}_{\eta }={{pV}}_{\xi },\,{V}_{\zeta }={{qV}}_{\xi }$, p and q are arbitrary constants. Equation (80) can be rewritten as$\begin{eqnarray}{V}_{\xi t}+{a}_{1}{V}_{\xi }^{2}{V}_{\xi \xi }+{a}_{2}{V}_{\xi \xi \xi \xi }+{a}_{3}({{pV}}_{\xi \eta }+{{qV}}_{\xi \zeta })=0.\end{eqnarray}$
Integrating ξ at both ends of equation (81) and taking constants equal to zero, we obtain$\begin{eqnarray}{V}_{t}+\displaystyle \frac{{a}_{1}}{3}{V}_{\xi }^{3}+{a}_{2}{V}_{\xi \xi \xi }+{a}_{3}({{pV}}_{\eta }+{{qV}}_{\zeta })=0.\end{eqnarray}$
When ${a}_{1}=-6{a}_{2}$, equation (82) can be expressed as$\begin{eqnarray}\begin{array}{l}{V}_{t}+{a}_{2}({V}_{\xi \xi \xi }+3{W}_{\xi \xi }{V}_{\xi }+{V}_{\xi }^{3})\\ \quad -\,3{a}_{2}({V}_{\xi }^{3}+{W}_{\xi \xi }{V}_{\xi })+{a}_{3}({p}^{2}+{q}^{2}){V}_{\xi }=0.\end{array}\end{eqnarray}$
According to constraints and equation (83), we have$\begin{eqnarray}\left\{\begin{array}{l}{V}_{\eta }={{pV}}_{\xi },\\ {V}_{\zeta }={{qV}}_{\xi },\\ {V}_{t}+{a}_{2}({V}_{\xi \xi \xi }+3{W}_{\xi \xi }{V}_{\xi }+{V}_{\xi }^{3})+{a}_{3}({p}^{2}+{q}^{2}){V}_{\xi }=0,\\ -3{a}_{2}({V}_{\xi }^{2}+{W}_{\xi \xi })=0.\end{array}\right.\end{eqnarray}$
Therefore, we obtain the forms of the Bell polynomials of equation (79) as follows:$\begin{eqnarray}\left\{\begin{array}{l}{{ \mathcal Y }}_{y}(V,W)=p{{ \mathcal Y }}_{x}(V,W),\\ {{ \mathcal Y }}_{z}(V,W)=q{{ \mathcal Y }}_{x}(V,W),\\ -3{a}_{2}{{ \mathcal Y }}_{{xx}}(V,W)=0,\\ {{ \mathcal Y }}_{t}(V,W)+{a}_{3}({p}^{2}+{q}^{2}){{ \mathcal Y }}_{x}(V,W)+{a}_{2}{{ \mathcal Y }}_{{xxx}}(V,W)=0.\end{array}\right.\end{eqnarray}$
Letting $V=\mathrm{ln}(g/f),\,W=\mathrm{ln}({gf})$, and according to the relationship between ${ \mathcal Y }$-polynomial and Hirota D-operator, we can obtain the following bilinear forms:$\begin{eqnarray}\left\{\begin{array}{l}{D}_{\eta }(g\cdot f)={{pD}}_{\xi }(g\cdot f),\\ {D}_{\zeta }(g\cdot f)={{qD}}_{\xi }(g\cdot f),\\ -3{a}_{2}{D}_{\xi }^{2}(g\cdot f)=0,\\ {D}_{t}(g\cdot f)+{a}_{3}({p}^{2}+{q}^{2}){D}_{\xi }(g\cdot f)+{a}_{2}{D}_{\xi }^{3}(g\cdot f)=0.\end{array}\right.\end{eqnarray}$
In order to obtain the solution of equation (79), we assume that f and g can be expanded into the following forms:$\begin{eqnarray}\left\{\begin{array}{l}f=1+\epsilon {f}^{(1)}+{\epsilon }^{2}{f}^{(2)}+\cdots ,\\ g=1+\epsilon {g}^{(1)}+{\epsilon }^{2}{g}^{(2)}+\cdots .\end{array}\right.\end{eqnarray}$
Substituting equation (87) into (86) and letting each power of ε be zero, we have$\begin{eqnarray}\epsilon :\left\{\begin{array}{l}{g}_{\eta }^{\left(1\right)}-{f}_{\eta }^{\left(1\right)}=p({g}_{\xi }^{\left(1\right)}-{f}_{\xi }^{\left(1\right)}),\\ {g}_{\zeta }^{\left(1\right)}-{f}_{\zeta }^{\left(1\right)}=q({g}_{\xi }^{\left(1\right)}-{f}_{\xi }^{\left(1\right)}),\\ -3{a}_{2}({g}_{\xi \xi }^{\left(1\right)}-{f}_{\xi \xi }^{\left(1\right)})=0,\\ {g}_{t}^{\left(1\right)}-{f}_{t}^{\left(1\right)}+{a}_{2}({g}_{\xi \xi \xi }^{\left(1\right)}-{f}_{\xi \xi \xi }^{\left(1\right)})+{a}_{3}({p}^{2}+{q}^{2})({g}_{\xi }^{\left(1\right)}-{f}_{\xi }^{\left(1\right)})=0.\end{array}\right.\end{eqnarray}$
Substituting equation (89) into (88), we have$\begin{eqnarray}\left\{\begin{array}{l}{l}_{1}={{pk}}_{1},\,{m}_{1}={{qk}}_{1},\,{h}_{1}=-{h}_{2},\\ {n}_{1}=-{a}_{2}{k}_{1}^{3}-{a}_{3}({p}^{2}+{q}^{2}){k}_{1}.\end{array}\right.\end{eqnarray}$
Therefore, we obtain the single soliton solution of the (3+1)-dimensional TF-MKP equation as follows:$\begin{eqnarray}\phi ={\left[\mathrm{ln}\left(\displaystyle \frac{1+{g}^{(1)}}{1+{f}^{(1)}}\right)\right]}_{x}=\displaystyle \frac{2{k}_{1}{h}_{2}\exp ({\theta }_{1})}{[1+{h}_{1}\exp ({\theta }_{1})][1+{h}_{2}\exp ({\theta }_{1})]},\end{eqnarray}$where ${\theta }_{1}={k}_{1}\xi +{l}_{1}\eta +{m}_{1}\zeta +{n}_{1}\tfrac{{\tau }^{\omega }}{{\rm{\Gamma }}(1+\omega )}+{h}_{3}$.
5.2. Dust acoustic rogue waves
Rogue waves have the characteristics of extremely high wave height, sharp peaks, and strong destructive power. Rogue waves have large peaks, but they do not necessarily have corresponding obvious troughs. The surfaces of rogue waves show strong nonlinearity. The existence time of the rogue waves are very short, and they will disappear quickly. The rogue waves are localized in both time and space, and they will gather huge energy in a short time and a certain space. In this section, based on the obtained solution of the TF-MKP equation, we study the dust acoustic rogue waves in dusty plasma.
By choosing appropriate parameters, we show dust acoustic rogue waves in figure 1. As can be seen from figure 1(a), when time $\tau =0$, two dust acoustic rogue waves appeared in a certain space $(\xi ,\eta )$, they show a local structure in space and time. Figures 1(b) and (c) are the planform and density contour map of figure 1(a) respectively, and through these two figures, we can more intuitively observe that dust acoustic rogue waves have large peaks, and the large amplitude of dust acoustic rogue waves indicates that they have strong nonlinear effect and strong energy.
Figure 1.
New window|Download| PPT slide Figure 1.Evolution plots of equation (91) by choosing k1=2.4, p=0.1, h1=0.45, h3=5, ω=1, τ=0, ζ=0.
According to figure 1, we infer that dust acoustic rogue waves will also appear in space (ξ, ζ) at a certain time. Therefore, by choosing appropriate parameters, we show dust acoustic rogue waves in figure 2. Comparing with figure 1, two strange waves also appear in figure 2, figures 2(b) and (c) are the planform and density contour map of figure 2(a) respectively, they are similar to figures 1(b) and (c), which confirms that when time τ=0, dust acoustic rogue waves also appear in the space (ξ, ζ).
Figure 2.
New window|Download| PPT slide Figure 2.Evolution plots of equation (91) by choosing k1=1.4, p=1.6, q=0.1, λ=1.9, h1=1, h3=1, ω=1, τ=0, η=0.
By choosing appropriate parameters, we obtained figure 3 of the change of dust acoustic rogue wave with time, where figures 3(b) and (c) are the planform and density contour map of figure 3(a), respectively. We can clearly observe that the duration of dust acoustic rogue wave is very short, and the waveform changes with time. When the value of τ increases from −1 to 1, the dust acoustic rogue wave quickly reach a high peak and disappear quickly, and the energy concentration and transfer of the strange waves are completed in a short time.
Figure 3.
New window|Download| PPT slide Figure 3.Evolution plots of equation (91) by choosing k1=1.5, p=1.5, q=0.5, λ=1.9, h1=1, h3=1, ω=1, η=0, ζ=0.
Expanding the range of time (τ) and space (ξ), we obtain the above figures 4(a)-(d), where figure 4(b) is the top view of figures 4(a), and 3(c), (d) are the side views. As shown in figures 4(a) and (b), different time and space can appear different dust acoustic rogue waves, dust acoustic rogue waves are localized in both the and time.
Figure 4.
New window|Download| PPT slide Figure 4.Evolution plots of equation (91) by choosing k1=1.5, p=1.5, q=0.5, λ=1.9, h1=1, h3=1, ω=1, η=0, ζ=0.
It can be seen from figure 4(c) that when the time τ increases from −5 to 5, compared with the previous dust acoustic rogue wave, the peak of the latter dust acoustic rogue wave gradually becomes smaller and the trough gradually becomes deeper. Eventually the big trough appears and rogue waves disappear. In figure 4(d), as the value of ξ increases, the same phenomenon occurs with the generated dust acoustic rogue waves.
By choosing appropriate parameters, we discuss the effect of time-fractional order ω on dust acoustic rogue waves. When ω=1, dust acoustic rogue waves in dusty plasmas are shown in figures 5(a) and (b) and it can be seen that figures 5(a) and (b) are consistent with figures 3(a) and 4(a), respectively. When ω=0.2, 0.4, 0.6, 0.8 and 0.9, evolution plots of equation (91) are shown in figures 6, 7 and 8. We find that figures 7(a)-(f) are quite different from figure 5 of integer order, and only figures 8(a) and (b) are similar to figures 5(a) and (b). Since ω=0.8 is close to 1, we infer that when the value of the time-fractional order ω is closer to 1, that is, when it is closer to the integer order, dust acoustic rogue waves will appear in the model of fractional dusty plasma. We find that figures 6(a) and (b) are similar to figures 5(a) and (b), which is consistent with our inference.
Figure 5.
New window|Download| PPT slide Figure 5.Evolution plots of equation (91) by choosing k1=1.5, p=1.5, q=0.5, λ=1.9, h1=1, h3=1, ω=1, η=0, ζ=0.
Figure 6.
New window|Download| PPT slide Figure 6.Evolution plots of equation (91) by choosing k1=1.5, p=1.5, q=0.5, λ=1.9, h1=1, h3=1, η=0, ζ=0.
Figure 7.
New window|Download| PPT slide Figure 7.Evolution plots of equation (91) by choosing k1=1.5, p=1.5, q=0.5, λ=1.9, h1=1, h3=1, η=0, ζ=0.
Figure 8.
New window|Download| PPT slide Figure 8.Evolution plots of equation (91) by choosing k1=1.5, p=1.5, q=0.5, λ=1.9, h1=1, h3=1.
According to ${n}_{{e}}={\left(1-\tfrac{{T}_{\mathrm{eff}}}{{T}_{{e}}}\tfrac{\phi }{{\kappa }_{{e}}-\tfrac{3}{2}}\right)}^{-{\kappa }_{{e}}+\tfrac{1}{2}}$, ${n}_{{i}}\,={\left(1-\tfrac{{T}_{\mathrm{eff}}}{{T}_{{i}}}\tfrac{\phi }{{\kappa }_{{i}}-\tfrac{3}{2}}\right)}^{-{\kappa }_{{i}}+\tfrac{1}{2}}$, we obtain C1, C2. According to equations (13) and (14), C1 and C2 are both related to λ. By choosing appropriate parameters, we discuss the effect of λ on dust acoustic rogue waves. As shown in figures 9(a) and (b), when the values of the time-fractional order ω are 1 and 0.9, respectively, the change trend of waves with λ is almost the same, which also shows that when the values of ω is closer to 1, dust acoustic rogue waves will appear in the model of fractional dusty plasma.
Figure 9.
New window|Download| PPT slide Figure 9.Evolution plots of equation (91) by choosing k1=1.5, p=1.5, q=0.5, h1=1, h3=1, ω=1, η=0, ζ=0.
6. Conclusions
In this paper, the (3+1)-dimensional MKP equation is derived by using the multi-scale analysis and reduced perturbation method. Compared with one-dimensional or two-dimensional models, the three-dimensional model is more in line with the actual situation. According to the obtained integer-order equation, the (3+1)-dimensional TF-MKP equation is derived by using the semi-inverse method and the fractional variational principle. The fractional differential equation can better describe dust acoustic rogue waves in dusty plasma. The symmetry and conservation laws of (3+1)-dimensional TF-MKP equation are analyzed, and conservation vectors are constructed. The exact solution of (3+1)-dimensional TF-MKP equation is obtained by using the definition and properties of Bell polynomials. Based on the obtained solution, we analyze and study dust acoustic rogue waves in dusty plasma, and find the effect of the time-fractional order ω on dust acoustic rogue waves.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grants 11975143).
,1,∗1College of Mathematics and System Science, 
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
