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An Application of (3+1)-Dimensional Time-Space Fractional ZK Model to Analyze the Complex Dust Acoustic Waves
Dust plasma is a new field of physics which has developed rapidly in recent decades. The study of dust plasma has received much attention due to its importance in the environment of space and the Earth. Dust acoustic waves are generated because of the inertia of dust mass while the restoring force is provided by the thermal pressure of electrons and ions. Since dust acoustic waves were first reported theoretically in unmagnetized dust plasma by Rao et al., they have become a research hot spot. In this paper, the excitation of dust acoustic waves by a gravity field in a dust plasma is analyzed. According to the control equations of dust plasma motion and employing multiscale analysis and perturbation method, we have obtained a (3+1)-dimensional ZK model. Because of the space property of dust plasma, (3+1)-dimensional ZK equation is more suitable than KdV equation and (2+1)-dimensional ZK equation to describe the real dust acoustic waves. Then, the (3+1)-dimensional time-space fractional ZK (TSF-ZK) equation describing the fractal process of nonlinear dust acoustic waves is given for the first time. To further explore how dust acoustic waves change energy as they travel, we discuss the conservation laws of the new model. Moreover, we study the exact solution of (3+1)-dimensional TSF-ZK equation by using extended Kudryashov method. Finally, based on the exact solution, we further investigate the effect of the parameter , the charge properties of dust particle , the fractional order values , , , and , the temperature , the gravity , and the collision frequency and on the properties of dust acoustic waves by a gravity field in dust plasma.
Plasma is a macroscopic system composed of a large number of charged particles, and it is the fourth state in which matter exists. The characteristics of motion of plasma are more complex than those of other substances. In many cases, the production and maintenance of plasma are difficult. Therefore, plasma physics is a branch with research value, which is closely related to the generation of new technology. In recent years, plasma physics has become an important basis for human understanding of the universe and an important guarantee for understanding and controlling the changes of the Earth’s environment. It is hoped that mankind will finally solve the energy problem in the future. As a result, plasmas are getting more attention.
In recent years, the study of nonlinear structures in various kinds of plasmas has been the emphasis of the researchers [1, 2]. Dust plasma is a new field of physics research which has developed rapidly in recent decades. In addition to electrons, ions, and neutral gases, dust plasma also contains dust particles of any shape and size ranging from tens of nm to a few microns. Charged dust grains exist widely in space, plasma equipment in laboratory as well as plasma processing. It is widely believed that dust plasma plays a very important role in the formation of galaxies, such as planetary rings, cometary surroundings, interstellar clouds, and the Earth’s ionosphere. Dust plasma is also important in the laboratory. And the main reason is that dust plasma mode is formed due to the existence of mixtures, which greatly affects the state or behavior of plasma. The dust acoustic waves are generated due to the inertia of dust mass while the restoring force is provided by the thermal pressure of electrons and ions. Low frequency dust acoustic waves were first predicted by Rao  in a dust plasma in 1990. Later, in 1992, Shukla-Silin  obtained the high frequency dust ion-acoustic waves. Merlino  has experimentally confirmed the existence of dust acoustic waves and dust ion-acoustic waves in 1998. Since then, more and more attention has been paid to the study of dust acoustic waves.
At the beginning of the investigations, the (1+1)-dimensional model was used to study the dust acoustic waves. Using the reductive perturbation method, a KdV equation has been derived by Bharuthram  to study the large amplitude ion-acoustic solitons in a dust plasma, and Duan  got a KdV equation to describe the effect of dust size distribution for two ion temperature dusty plasma. Later, Kadomstev and Petviashvili  made an attempt to describe the solitons in (2+1)-dimensional systems by applying Kadomstev-Petviashvili (KP) equation. Singh and Honzawa  studied the effects of ion temperature and relativistic factor on the width and amplitude of ion-acoustic solitons in an unmagnetized two-dimensional weakly relativistic collisionless plasma with finite ion temperature by using the KP equation. Lin  derived the KP equation which is considered as two-dimensional extension of KdV equation to study dust acoustic waves in hot dusty plasma. Employing multiscale analysis and perturbation method, a KP equation was obtained for the stability of dust acoustic waves for host dust plasma by Duan . Gill  proved the existence of compressive and rarefactive dust acoustic solitons under the solution of KP equation in two-dimensional dusty plasma with two-temperature ions. However, in the real environment, dust plasma moves in space, so we must study the higher dimensions of dust acoustic waves. Thus, in this paper, we will be working on the (3+1)-dimensional model.
For a long time, people have been committed to the study of integral calculus. Therefore, researchers conduct scientific research based on integer order models [13–18]. However, with the continuous development of calculus theory, the appearance and development of fractional order have become a major trend of integral science [19–21]. As we know, Leibniz was the first to study the theory of fractional derivatives. So far, the development of fractional derivatives has been very mature, and fractional calculus and the corresponding fractional partial differential equations have attracted wide attention in many subjects [22–28]. Compared with integer order models, fractional models can better explain nonlinear physical processes and propagation characteristics in real environment [29–33]. However, fractional models were rarely used to study the dust plasma in the past. Therefore, it is of great research value to construct the (3+1)-dimensional TSF-ZK equation to discuss the influence of fractional order for the dust acoustic waves in gravity field for dust plasma.
The solution of the nonlinear partial differential equation is an important research topic in nonlinear physical phenomena. Similarly, the solutions of fractional differential equations have attracted the attention of researchers in many fields. Thus, many methods have been proposed one after another, such as the first integral method [34, 35], (G’/G)-expansion method [36, 37], the Hirota method [38, 39], the trial function method , the subequation method , and others [42–46]. In this paper, using extended Kudryashov method , the exact solution of (3+1)-dimensional TSF-ZK equation is obtained.
The rest of the paper is organized as follows. In Section 2, based on the control equations of motion, a (3+1)-dimensional integer order ZK equation is derived by employing multiscale analysis and perturbation method [48, 49]. Applying the semi-inverse method and the fractional variational principle , the integer order ZK equation is transformed into a time-space fractional ZK (TSF-ZK) equation. Next, we study the exact solution of (3+1)-dimensional TSF-ZK equation by using extended Kudryashov method. Then, the conservation law of (3+1)-dimensional TSF-ZK equation is got by applying Lie symmetry analysis method. In the end, on the basis of the exact solution of (3+1)-dimensional TSF-ZK equation, we further investigate the property of dust acoustic waves. The influence of the parameter , the charge properties of dust particle , fractional order values , , , , temperature , gravity , and collision frequency , on the properties of dust acoustic waves by a gravity field in dust plasma are studied.
2. Derivation of the (3+1)-Dimensional ZK Equation
For low frequency dust acoustic wave, the control equations are composed of mass conservation equation, momentum conservation equation, and Poisson equation [51, 52] followingwhere is three-dimensional Laplace operator, is the velocity of dust particles, is the electrostatic potential, , , and are the number densities of the dust particles, electrons, and ions in the dusty plasma, respectively, is the temperature of dust particles, is the basic unit electric quantity of dust particles, and its positive and negative represent the electrical properties of dust particles. is expressed by , where , , and are the unit vectors in the , , and directions, respectively. and , where , , is the effective temperature, where and are the temperatures of electrons and ions, respectively, , and . In a state of complete constant temperature, the ions and electrons will obey the Boltzmann condition. The equilibrium state satisfies the neutral condition , where , , , and are the unperturbed number densities of dust particles, ions, electrons, and the numbers of electrons residing on the dust particles, respectively. is normalized by , and is normalized by . The space coordinates , time , velocity , electrostatic potential , collision frequency , and gravity are normalized by Debye length , the characteristic dust period , the dust acoustic speed , and , respectively, and the collision frequency refers to the number of collisions between one plasma and another over a period of time. The detailed description is given in .
We assume that the charge of dust particles is constant; that is, , is constant. Then the dimensionless forms of the control equations are as follows:We introduce the following slow space-time variables:Then we haveThe dependent variables , , , , and are expanded as follows:Substituting (4) and (5) into (3), the following equations can be got:In (6), lettingand using the Taylor’s series, can be approximately rewritten asA series of approximate equations for in the following form can be obtained
When , (18) is the KdV equation. When , (18) is the ZK equation. On the basis of and , (18) is a (3+1)-dimensional ZK equation. It is well known that the KdV equation reflects the propagation of dust acoustic waves along a line, and the ZK equation can describe the propagation of dust acoustic waves on one surface. Dust acoustic waves in dust plasma move around in space. Therefore, the (3+1)-dimensional ZK equation is more suitable to describe the dust acoustic waves in the real dust plasma.
3. Derivation of the (3+1)-Dimensional Time-Space Fractional ZK (TSF-ZK) Equation
Based on the above section, we get a (3+1)-dimensional ZK equation. In this section, the (3+1)-dimensional TSF-ZK equation is obtained by using the semi-inverse method and the fractional variational principle.
Firstly, let us introduce some basic fractional definitions.
Definition 1 (see ). The Riemann-Liouville fractional derivative operator is defined as
Definition 2 (see ). The modified Riemann-Liouville fractional derivative operator is defined asLetting , where is a potential function, then the (18) can be written asThe functional of the above potential equation can be given aswhere are Lagrangian multipliers, which can be obtained later, and . Integrating (23) by parts and letting , we can getwhere . Using the variation of (24) and integrating each term by parts, we getThus, the following comparison expression is given asAt this point, (26) is equivalent to (22), so the Lagrangian multipliers are as follows:The Lagrangian form of the (3+1)-dimensional ZK equation can be written as follows:Similarly, the following Lagrangian form of the (3+1)-dimensional TSF-ZK equation can be obtainedwhere , , , and are fractional order values. Thereby the functional of (3+1)-dimensional TSF-ZK equation is given as follows:where
Using the fractional integration by partsoptimizing the above functional (30), and letting , we can get the following Euler-Lagrange equation of (3+1)-dimensional TSF-ZK equationAccording to (29), (33) can be rewritten asLetting , we get
Equation (35) is a new model, namely, the (3+1)-dimensional TSF-ZK equation. When , (35) is the integer order (3+1)-dimensional ZK equation. This shows that the integer order model is the special type of fractional model. The fractional derivatives of the (3+1)-dimensional TSF-ZK equation are related to the dust acoustic waves propagation with fractal properties. Equation (35) can describe fractal processes of the dust acoustic waves. The fractional order model could simulate various real plasma environments more adequately than the integer order model and provide an excellent tool for the description of dynamical processes. Therefore, (3+1)-dimensional TSF-ZK equation can better describe the dust acoustic waves in the real dust plasma.
4. Conservation Law of the (3+1)-Dimensional TSF-ZK Equation
In the last section, we extend the integral order equations describing dust acoustic waves to the fractional order of time-space and obtain some new properties of dust acoustic waves propagation. In this section, in order to research the energy changes during the propagation of dust acoustic waves, we use the Lie symmetry method to obtain multiple conservation law of the (3+1)-dimensional TSF-ZK equation.
Considering that, under one-parameter Lie group of point transformations, (35) is invariant with the dependent and independent variables, the transformations are given bywhere are infinitesimal function of the transformations, is the group parameter, and are extended infinitesimal functions in the following explicit expression:where and are the total derivative operators as follows:
Applying the generalized Leibnitz rule and the chain rule, the extended symmetry operator can be introduced in the following form:where
Similarly, we also have the following equation:where
Introducing the Lie algebra associated with (35) is composed of the following infinitesimal generator
Applying the infinitesimal transformations, the (35) is invariable that results in the invariance conditions given aswhere represents the (3+1)-dimensional TSF-ZK equation.
We get following symmetry determining equation by using the third prolongation to (35)
Substituting (37), (38), (39), and (41) into this equation, letting the same coefficients of derivatives to zero, and then solving the series of determining equations, we can obtain the following infinitesimal functionswhere , , are arbitrary constants, since the corresponding infinitesimal generators can be expressed as
The conserved vector of (35) is which satisfies the conservation equation
A formal Lagrangian for the (3+1)-dimensional TSF-ZK equation is given aswhere is the new dependent variable.
In the considered linear case, the adjoint equation to (35) as the Euler-Lagrange equation is represented asHere is the Euler-Lagrange operator which is defined aswhere , and are the adjoint operators.
Hence, the adjoint equation (50) can be rewritten as
We introduce the Lie characteristic function for generator as follows:The fractional Noether operator for the variables , and are given bywhere and is an integral equation.
Consequently, the conservation law of the (3+1)-dimensional TSF-ZK equation is obtained. This indicates that the dust acoustic wave described by this new model is conserved in energy during its propagation, no matter in the fractal process or in the interaction.
5. The Solution of the (3+1)-Dimensional TSF-ZK Equation
In this section, we seek the exact solution of (3+1)-dimensional TSF-ZK equation by using the extended Kudryashov method. Firstly, for a given nonlinear partial differential equation