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Study of Ion-Acoustic Solitary Waves in a Magnetized Plasma Using the Three-Dimensional Time-Space Fractional Schamel-KdV Equation
The study of ion-acoustic solitary waves in a magnetized plasma has long been considered to be an important research subject and plays an increasingly important role in scientific research. Previous studies have focused on the integer-order models of ion-acoustic solitary waves. With the development of theory and advancement of scientific research, fractional calculus has begun to be considered as a method for the study of physical systems. The study of fractional calculus has opened a new window for understanding the features of ion-acoustic solitary waves and can be a potentially valuable approach for investigations of magnetized plasma. In this paper, based on the basic system of equations for ion-acoustic solitary waves and using multi-scale analysis and the perturbation method, we have obtained a new model called the three-dimensional(3D) Schamel-KdV equation. Then, the integer-order 3D Schamel-KdV equation is transformed into the time-space fractional Schamel-KdV (TSF-Schamel-KdV) equation by using the semi-inverse method and the fractional variational principle. To study the properties of ion-acoustic solitary waves, we discuss the conservation laws of the new time-space fractional equation by applying Lie symmetry analysis and the Riemann-Liouville fractional derivative. Furthermore, the multi-soliton solutions of the 3D TSF-Schamel-KdV equation are derived using the Hirota bilinear method. Finally, with the help of the multi-soliton solutions, we explore the characteristics of motion of ion-acoustic solitary waves.
Ion-acoustic solitary waves are well-known to be an important example of nonlinear phenomena in modern plasma research [1–3]. Many researchers have studied ion-acoustic solitary waves in different plasma systems such as thermal, magnetized, and unmagnetized plasmas. Among the different plasma systems, magnetized plasma systems have attracted intense interest. Many authors have studied ion-acoustic solitary waves in magnetized plasma based on the quantum hydrodynamic (QHD) model [4, 5]. The QHD model is derived from the basic system of equations of ion-acoustic solitary waves and is one of the macroscopic mathematical models used to describe the hydrodynamic behavior of quantum plasmas.
For simplicity, 1D and 2D nonlinear partial differential equations have been used to describe the evolution of nonlinear ion-acoustic solitary waves. For the simplest 1D geometry where the ion-acoustic solitary waves become solitons, Washimi and Taniuti  derived the KdV equation by using the reductive perturbation method. Kako and Rowlands  derived the 2D KP equation based on the results of Washimi and Taniuti. However, in the real magnetized plasma environment, 1D and 2D models cannot solve some of the problems encountered in the motion of ion-acoustic solitary waves. Thus, it is necessary to introduce higher-dimensional theories for the nonlinear ion-acoustic solitary waves. Therefore, in this paper, we discuss a new 3D model for nonlinear ion-acoustic solitary waves.
Most of the QHD models, such as the KdV model, mKdV model, and KP model, are integer-order models. Fractional order models have rarely been considered. Fractional calculus is a generalization of integer calculus. Many of the physical processes that have been explored to date are nonconservative. It is important to be able to apply the power of fractional differentiation [8–10]. However, because of its nonlocal character, fractional calculus has not been used in physics and engineering. With the development of nonlinear science, fractional calculus theory has been continuously developed to date. Researchers have discovered that the derivatives and integrals of fractional order models are suitable for describing various physical phenomena. In recent years, the application of fractional differential equations has attracted increasing attention in plasma physics . Thus, research on fractional order models is necessary.
The solution of the integer equation is a research hot spot in the field of research and development of various models [12–14], and similarly, the solution of fractional models has been a focus of our research [15, 16]. Thus, many solution methods have been found and used to solve the fractional order equation. For instance, the iterative method [17–19], Hirota bilinear method [20, 21], trial function method , Homotopy perturbation , and other methods have all been developed in the recent decades. In the past, researchers solved integer-order models by using the Hirota bilinear method. Recently, the Hirota bilinear method has been used to solve fractional models. In this paper, using the Hirota bilinear method, we obtain soliton solutions for the new model. Various phenomena can be explained via the application of the solutions given by the above methods [24–26]. Additionally, the use of these methods enables a better understanding of various magnetized plasma phenomena. Therefore, based on the solutions derived by the abovementioned methods, we seek to determine the properties of ion-acoustic solitary waves. The properties of the model include conservation laws [27, 28], boundary value problems [29, 30], and integrable systems [31, 32].
The research on conservation laws plays an important role in the study of the physical phenomena in nonlinear magnetized plasma. Conservation laws are a mathematical formulation, and they indicate that the total amount of a certain physical quantity remains the same during the evolution of a physical system [33, 34]. In 1918, Noether  proved that each conservation law is associated with an appropriate symmetry and can be derived from the Lagrangian function and the invariance principle. In 1996, Riewe  introduced the Lagrangian function for the fractional derivative. In the past two decades, many different types of fractional Euler-Lagrangian equations have been generalized. Based on these conclusion, some fractional generalizations of Noether’s theorem were proved , and many fractional conservation laws were obtained . To study the conservation laws of the fractional differential equations, we use Lie symmetry analysis to construct the conserved vectors [39, 40]
In this paper, applying the basic system of equations of ion-acoustic solitary waves , we develop a new 3D model. Using the new model, we study the conservation laws and the solution of ion-acoustic solitary waves. The rest of the paper is structured as follows: In Section 2, based on the basic system of equations of ion-acoustic solitary waves, we obtain a new 3D Schamel-KdV equation by using multi-scale analysis and the perturbation method . A new 3D TSF-Schamel-KdV equation is obtained in Section 3 according to the new integer-order model and by using the semi-inverse method and the fractional variational principle [43, 44]. In Section 4, applying the Riemann-Liouville fractional derivative [39, 40], we discuss the conservation laws of the new fractional model. In Section 5, according to the Hirota bilinear method, we obtain the soliton solutions of the 3D TSF-Schamel-KdV equation. The propagation of solitary waves is important because it describes the characteristic nature of the interaction of the waves and the plasmas. Therefore, using soliton solutions [17, 18], we study the characteristics of motion of ion-acoustic solitary waves.
2. Derivation of the 3D Schamel-KdV Equation
We use the basic system of equations of ion-acoustic solitary waves given bywhere is the ion number density, and , , are the ion fluid velocities in the -, -, and -directions, respectively. is the electric field potential, is the electron number density, and is the uniform external magnetic field. Ion-acoustic solitary waves are assumed to propagate in the -direction, and the direction is specified by the unit vector .
We consider the propagation of ion-acoustic solitary waves in 3D space and introduce the following independent stretched variables:where is a small parameter characterizing the strength of the nonlinearity. Thus, we can obtainThe dependent variables are expanded in the following form:and the boundary conditions are given by
Remark 1. Because of the nonlinear term , when and , (12) can be reduced to the 1D Schamel-KdV equation. When , (12) is a 3D equation. Therefore, (12) is called the 3D Schamel-KdV equation. Compared to the KdV and mKdV models , the nonlinearity of the 3D Schamel-KdV equation is relatively weak. Therefore, the 3D Schamel-KdV equation presents a new research direction for the study of ion-acoustic solitary waves.
3. Derivation of the 3D TSF-Schamel-KdV Equation
In Section 2, we have obtained a new 3D integer-order Schamel-KdV equation. To learn more about ion-acoustic solitary waves, we seek to obtain the 3D TSF-Schamel-KdV equation by using the semi-inverse method and the fractional variational principle. First, we introduce some definitions as follows.
Definition 2 (see ). The left Riemann-Liouville fractional derivation of a function is defined as
Definition 3 (see ). The Riemann-Liouville fractional derivation of a function is defined as
According to the integer-order 3D Schamel-KdV equation, assuming , where is a potential function, and therefore, the potential equation of the 3D Schamel-KdV equation can be written in the following form:Then, the function of the potential equation (16) can be described aswhere , , are Lagrangian multipliers which can be obtained later.
Using integration by parts for (17) and taking , we obtainUsing the variation of the above function, integrating each term by parts and applying the variation optimum condition, we obtain
Similarly, the Lagrangian form of the 3D TSF-Schamel-KdV equation is given bywhere . Thus, the function of the 3D TSF-Schamel-KdV equation can be obtained as
Using the fractional integration by parts,we can obtainOptimizing the variation Eq. (24), , we can obtain the Euler-Lagrange equation of the 3D TSF-Schamel-KdV equation as
4. Conservation Laws of the 3D TSF-Schamel-KdV Equation
4.1. Lie Symmetry Analysis
In the previous section, we have obtained the 3D TSF-Schamel-KdV equation. To learn about the properties of the new model, we study the conservation laws [48, 49]. First, we convert (30) to the following fractional partial differential equation form:
We assume that (31) is invariant under a one parameter Lie group of point transformations in the following form:where , , , , and are infinitesimal functions, and , , , , and are the prolongations of infinitesimal functions defined aswhere and are the total derivative operators given by
Applying the generalized Leibnitz rule as given bywhereand the chain rule for a compound function defined aswe can obtain the following equation:
Similarly, using the generalized Leibnitz rule and the chain rule for a compound function, we also obtain the following equation:where
The infinitesimal generator can be defined as follows:Under the infinitesimal transformations, the invariance of the system (31) leads to the following invariance condition:According to (42) and (43), we can obtainThen, we can obtain the following invariance criterion:
Substituting (33), (34), (41), and (42) into (47) and equating the coefficients of alike partial derivatives, fractional derivatives and powers of , the set of determining equations can be obtained asBy solving the above equations, we can obtain a series of Lie algebra of point symmetries asHence, a series of Lie algebra of point symmetries can be written as
4.2. Conservation Laws
We have obtained the Lie symmetry generator in Section 4.2. In this section, we will discuss conservation laws of the 3D TSF-Schamel-KdV equation based on the obtained Lie symmetry generator. We know that the conservation laws of (30) satisfy the following equation:where , , and are the conserved vectors.
A formal Lagrangian for the 3D TSF-Schamel-KdV equation can be presented aswhere is a new dependent variable. According to the formal Lagrangian, an action integral is defined asTherefore, we can obtain the adjoint equation of (30) as the Euler-Lagrange equationwhere is the Euler-Lagrange operator defined aswhere , , , , and are the adjoint operators of the Riemann-Liouville fractional differential operators , , , , and , respectively. These are given bywhere and are the right-sided fractional integral operators of orders and , respectively. and are the right-sided Caputo fractional differential operators of orders and , respectively. Therefore, the adjoint equation (54) can be rewritten as
Using the Riemann-Liouville fractional derivative, the components of the conserved vectors of (30) are defined aswhere , and is the integral given bywith the property thatThe conservation laws of the 3D TSF-Schamel-KdV equation are explained in detail below (see the appendix).
5. Multi-Soliton Solutions for the 3D TSF-Schamel-KdV Equation
The solution of the model is a relatively broad research area in science [50, 51]. In this section, using the simplified Hirota bilinear method [24, 52], we seek multiple soliton solutions of the 3D TSF-Schamel-KdV equation.
First, we introduce the following fractional transforms:where , , and are constants. Using the above transformations and omitting the apostrophe, we can convert the derivatives into classical derivatives,Then, (30) can be described as
5.1. Single-Soliton Solution
We assume that the single-soliton solution of (65) has the following form:where is the auxiliary function defined as
5.2. Two-Soliton Solution
We assume that the two-soliton solution has the following form:where and are defined in (69). We know thatand substituting this expression into (65), the coefficient can be obtained asTherefore, the two-soliton solution for (65) has the following form:where
5.3. Three-Soliton Solution
According to the pattern obtained in Section 5.3, the -soliton solutions for the TSF-Schamel-KdV equation can be obtained, where . Based on the single-soliton solution and the two-soliton solution, we can study the characteristics of the motion of the ion-acoustic solitary waves.
In this section, we describe the interaction of two small ion-acoustic solitary waves with finite amplitude in a weakly relativistic 3D magnetic plasma. Then, we can study the characteristics of motion of the solitary waves by changing the coefficients. Based on the single-soliton solution of ion-acoustic solitary waves, we obtain the evolution plots of the ion-acoustic solitary waves (see Figure 1). Figure 1 shows that the solitonic amplitude increases with an increase in the ratio, and the initial superimposed solitons travel different distances over a period of time for the different choices of and . Therefore, we conclude that the soliton moves along the positive -axis with constant amplitude and velocity.
Examination of Figure 2(a) shows that the propagation trajectory of the soliton exhibits a periodic oscillation. Figure 2(a) shows the curve propagation trajectory with constant amplitude and constantly changing velocity, where the velocity changes with time. Furthermore, Figure 2(b) shows the two-soliton interaction with constantly changing velocity. When , the trajectory is sinusoidal with periodic oscillation. Otherwise, when is far from the origin, the trajectory is parabolic-like. It can be seen from Figure 2(c) that the soliton generates a peak at the time of the interaction. Based on this, we conclude that, in addition to the periodic oscillation of the solitons in the local region, the large-scale propagation trajectories for such a structure show parabolic-type curves. Thus, if the variable coefficients are taken to have other forms, the corresponding characteristic curves will present different behaviors.