On the modified generalized multidimensional KP equation in plasma physics and fluid dynamics in (3 + 1) dimensions

This paper considers a modified generalized multidimensional Kadomtsev–Petviashvili equation depending on several real constants, which characterizes the dynamics of solitons and the nonlinear waves in plasma physics and fluid dynamics. We derive the Lie point symmetry generators and Lie symmetry groups and we will apply this symmetry analysis to the equation in order to obtain exact solutions. Finally by means ot the multiplier method, we classify all low-order conservation laws of the equation that have been obtained by applying the corresponding symmetries of the family.

has determined the symmetry group of a system of differential equations, a number of applications become available.
In this paper, based on the Lie group method, we will investigate a modification and generalization of an important equation to find an optimal system of one-dimensional subalgebras and we used them to perform symmetry reductions and determine new group-invariant solutions of this equation. We consider a modified generalized multidimensional Kadomtsev-Petviashvili equation given by u t + λ 1 u n u x + λ 2 u x x x x + λ 3 u yy + λ 4 u zz = 0, where n and λ i for i = 1, . . . , 4 are constants. This equation characterizes the dynamics of solitons and the nonlinear waves in plasma physics and fluid dynamics. The physics of dusty plasmas has been an important topic of rapidly interest from academic point of view and the view of its new applications in space and modern astrophysics because of its huge existence in different areas such as magnetosphere, plasma crystals, cometary tails and atmosphere of lower part of the earth, with potential applications in modern technology, such as metallic and semiconductor nanostructures or microelectronics, quantum dots and carbon nanotubes. During the last decade, the researchers found a great interest to study the propagation of linear and nonlinear characteristics of electrostatic and electromagnetic modes based on the quantum hydrodynamic models. Several authors have studied some special cases of KP equation, which can be written in normalized form as follows: where u(t, x, y) is a scalar function, t is the time coordinate and x and y are respectively the longitudinal and transverse spatial coordinates. The case σ = 1 is known as the KPII equation, and models, for instance, water waves with small surface tension. The case σ = i is known as the KPI equation, and may be used to model waves in thin films with high surface tension. Several authors [1,18,23] have studied Eq. (2), using bifurcation analysis, an extended homogeneous balance method and the multiple rogue wave solutions technique respectively. In [20] Wazwaz proposed the modification of KP equation as the below form in which the propagation of the ion-acoustic waves in a plasma with non-isothermal electrons has been utilized, and in [21] Wazwaz and El-Tantawy proposed the generalized KP equation given as Equation (1) characterizes the dynamics of solitons and the nonlinear waves in plasma physics and fluid dynamics and in [4] the authors employ the sub-equation method to obtain exact solutions to the proposed strongly nonlinear time-fractional differential equations of conformable type.
The outline of this paper is as follows: In the following section we perform a study of Lie symmetries of the modified generalized multidimensional Kadomtsev-Petviashvili equation (1) and we establish our main results about it. We deal the point symmetries classification, commutators table of Lie algebra, Lie symmetry groups and new solutions that we obtain using these groups. Also we study the symmetry reductions that we can obtain by using the generators calculated previously. In Sect. 2, we employ the similarity variable and similarity solution to obtain symmetry reductions to ODE's for the modified generalized multidimensional KP equation (1). In Sect. 3 we derive low-order local conservation laws admitted by (1) on the whole solution space, by employing the multiplier method. Finally, concluding remarks are presented in Sect. 4.

Lie symmetries
In this section, we will perform Lie symmetry analysis for (1) firstly. The main task of the classical Lie method is to seek some symmetries and look for exact solutions of a given partial differential equation. According to the Lie theory, to obtain Lie symmetries of the modified generalized multidimensional KP equation (1), we consider a one-parameter Lie group of infinitesimal transformations acting on independent and dependent variablest where ε is the group parameter and τ, ξ 1 , ξ 2 , ξ 3 and η are the infinitesimal of the transformations for the independent and dependent variables respectively. The vector field associated with the above group of transformations can be written as where with the initial conditions (t,x,ŷ,ẑ,û) | ε=0 = (t, x, y, z, u).
Applying the fourth prolongation pr (4) V to Eq.(1), we obtain the invariance condition, as the solutions space of (1) is invariant under the point transformation group (5), where The fourth prolongation is given by where φ J (t, x, y, z, u (4) Specifically the fourth prolongation can be given as pr (4) where φ x , φ x x , φ t x , φ yy , φ zz , φ x x x x are given explicitly in terms of τ, ξ 1 , ξ 2 , ξ 3 , η and the derivatives of η. By using (10) we find the coefficient functions τ (t, x, y, z, u), x, y, z, u) and η(t, x, y, z, u). From (7) and (10), the invariance condition reads as where (12) and D t , D x , D y and D z t denote the total differential operators with respect to t, x, y and z x, y, z).

Classification of Lie point symmetries
In this section we calculate the Lie point symmetries admitted by the modified generalized multidimensional Kadomtsev-Petviashvili equation (1). A point symmetry of (1) is a one-parameter Lie group of transformations on (t, x, y, z, u) generated by a vector field of the form (6), whose prolongation leaves invariant equation (1). The condition for a vector field (6) to generate a point symmetry of Eq. (1) is given by (11), that splits with respect to the x, y, z and t derivatives of u giving an overdetermined linear system of equations for the infinitesimals x, y, z, u) and η(t, x, y, z, u) and the parameters a, b. Solving this system we obtain the next theorem: The point symmetries admitted by Eq. (1) are generated by: 1. For λ 3 = 0 and λ 4 = 0: 2. For λ 3 = 0 and λ 4 = 0: 3. For λ 3 = 0 and λ 4 = 0: 4. For λ 3 = 0 and λ 4 = 0: Proof By using (11), letting the coefficients τ (t, x, y, z, u), x, y, z, u) and η(t, x, y, z, u) of the polynomial be zero yields a set of differential equations of the functions. By simplifying the system, we obtain Solving system (17) on can arrive at the previous generators.
The previous vector fields are closed under the Lie bracket. Thus the symmetry generators form a closed Lie algebra. The commutation relationships of Lie algebras determined by the symmetry generators (13) are shown in Table 1 is the commutator for the Lie algebra defined by Then we build the adjoint table for each pair of elements V i and V j , with i, j = 1, . . . , 7, where The adjoint relationship of the Lie algebra is shown in Table 2. We will use the adjoint representation to decompose all the subalgebras of the Lie algebra in equivalence classes of conjugated subalgebras. From the action attached infinitesimal of a Lie algebra over itself, we can rebuild the adjoint representation to the underlying Lie group adding the Lie series.
To calculate the reductions of Eq. (1) we use elements of the optimal system of subalgebras. An optimal system of subalgebras is a list of subalgebras that are not equivalent or conjugated. Also, any other subalgebra of the Lie algebra is conjugated or equivalent with it. For calculate the optimal system, we first calculate the adjoint transformation matrix of the modified generalized multidimensional Kadomtsev-Petviashvili equation (1). We consider a linear combination of V i and we define The function f i is a linear maps for i = 1, 2, . . . , 8, also we define the matrix A i with respect to basis Finally the general adjoint matrix A, calculated using the previous matrix, is given by where and it must have solutions for i for i = 1, 2, . . . , 8, for certain values of α i and β i , i = 1, 2, . . . , 8. Then we can obtain the generators of the optimal one-dimensional system:

z, u) time and space dependent shift
where is the group parameter. The theory of Lie assures that a group of symmetry transforms solution into solutions, then we can conclude that if u = f (t, x, y, z) represents a known solution of the differential equation (1), by applying the different groups of symmetry g i , i = 1, . . . 8, we can calculate the new solutions of (1). Based on (25), the corresponding new solutions of (1) can be given by: (30)
(ii) We now consider the Lie point symmetry V 5 and Eq. (34) becomes We obtain the similarity variables and similarity solution and the new reduction equation with the independent variables w 1 , w 2 and w 3 Fig. 1 Exact solution of (1) given by (45) and .
(iv) When we consider the Lie point symmetry V 7 +α 1 V 1 +α 2 V 2 we obtain for λ 4 = 0 the similarity variables and similarity solution and the reduction equation with the independent variables w 1 , w 2 and w 3 as follows (v) When we use the generator V 8 + α 1 V 1 + α 4 V 4 we obtain the similarity variables and similarity solution Substituting (35) into (1) and the following reduction equation

Conservation laws
Conservations laws are very important in the study of differential equations. They have also been used in the development of numerical methods and in obtaining exact solutions for some partial differential equations. They describe chemical and physical conserved quantities such as energy, electric charge, mass, etc. A local conservation law for the modified generalized multidimensional Kadomtsev-Petviashvili equation (1) is a divergence expression holding on the whole solution space ε of equation (1) The conserved density T and the spatial fluxes X , Y and Z are functions of t, x, y, z, u and derivatives of u. Here D t , D x , D y and D z denote the total derivative operators with respect t, x, y and z respectively. This method makes use of the concept of multiplier, that is, a function which satisfies that is a divergence expression for solutions of (1) and for any function u(t, x, y, z). There is a one-to-one correspondence between non-trivial multipliers and non-trivial conservation laws in characteristic form. All non-trivial conservation laws arise from multipliers. When we move off of the set of solutions of Eq. (1), every non-trivial local conservation law (59) is equivalent to one that can be expressed in the characteristic form D t T + D x X + D y Y + D z Z = u t + λ 1 u n u x + λ 2 u x x x x + λ 3 u yy + λ 4 u zz Q (60) that vanishes on the set of solutions of Eq. (1) where ( T , X , Y , Z ) differs from (T , X , Y , Z ) by a trivial conserved current. We find all multipliers by solving the determining equation δ δu u t + λ 1 u n u x + λ 2 u x x x x + λ 3 u yy + λ 4 u zz Q = 0 where δ δu is the Euler-Lagrange operatorÊ [u] given bŷ and the general form for a low-order multiplier for the generalized (2+1)-dimensional nonlinear evolution equation (1) is given by Q(t, x, y, u, u t , u x , u y , u z , u x x , u x x x ). The determining equation (61) yields a linear determining system for the multipliers Q(t, x, y, z, u).
We solve this determining system and we get the multiplier Q = 1. For each of the conserved currents obtained, Eq. (59) is satisfied when the modified generalized multidimensional KP equation holds. By solving the determining equation (61), the solution multiplier Q give us these conserved densities and fluxes of Eq. (1). The low-order local conservation laws admitted on the whole solution space ε + by the modified generalized multidimensional KP equation (1) are given by T 1 = u x , X 1 = λ 1 uu x + λ 2 u x x x , Y 1 = λ 3 u y Z 1 = λ 4 u z . (64)

Concluding remarks
In this paper, by using the Lie symmetry analysis method we studied the modified generalized multidimensional Kadomtsev-Petviashvili equation (1). For the first time, the classical Lie point symmetries were used to construct an optimal system of onedimensional subalgebras. This system was then used to obtain symmetry reductions and new group-invariant solutions of (1). Also we have calculated the Lie symmetry group from the related symmetries to get some new exact solutions from the known ones. Again for the first time, we have derived the conservation laws for (1) by employing the multiplier method.