https://doi.org/10.1007/s10910-022-01412-0 ORIGINAL PAPER
On the modified generalized multidimensional KP equation in plasma physics and fluid dynamics in (3 + 1) dimensions
S. Sáez1
Received: 9 September 2021 / Accepted: 6 October 2022 / Published online: 25 October 2022
© The Author(s) 2022
Abstract
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.
Keywords Similarity reduction· Lie symmetries · Partial differential equations · Conservation laws
1 Introduction
Many important phenomena and dynamic processes can be described by nonlinear partial differential equations [2,3,5]. PDEs arising in many physical and chemical fields like fluid dynamics, condensed matter, biophysics, plasma physics, biogenetics, optical fibers, biology and other areas of engineering. A wealth of methods have been developed to find these exact physically significant solutions of a PDE though it is rather difficult [5,6]. These methods include the sine-cosine method [19], the extended tanh method [13], the simplest equation method [9], non-classical method, the Lie symmetry method [14], the homogeneous balance method [22], sub-equation method [4], multiple exp-function method [12] and others methods [7,8,10].
One of the most efficient methods of studying differential equations is the Lie group method or symmetry analysis [6,11,14–17]. A symmetry group of a system of differential equations transforms solutions of the system to other solutions. Once one
B
S. Sáez1 Department of Mathematics, University of Cádiz, Avda. Universidad de Cádiz, No 10, 11519 Puerto Real, Cádiz, Spain
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 multi- dimensional Kadomtsev–Petviashvili equation given by
ut + λ1unux+ λ2ux x x
x+ λ3uyy+ λ4uzz= 0, (1)
where n andλifor i= 1, . . . , 4 are constants. This equation characterizes the dynam- ics 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 microelec- tronics, 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:
(ut+ 6uux+ ux x x)x+ 3σ2uyy= 0, (2) 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 caseσ = 1 is known as the KPII equation, and models, for instance, water waves with small surface tension. The caseσ = 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
4ut− 6u2ux+ ux x x+ 6ux∂−1uy+ 3∂−1uyy= 0, (3) 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 gener- alized KP equation given as
ux x x y+ 3 uxuy
x+ ut x+ ut y+ ut z− uzz= 0, (4) 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 reduc- tions 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.
2 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 sym- metries of the modified generalized multidimensional KP equation (1), we consider a one-parameter Lie group of infinitesimal transformations acting on independent and dependent variables
ˆt = t + ετ(t, x, y, z, u) + O(ε2), ˆx = x + εξ1(t, x, y, z, u) + O(ε2), ˆy = y + εξ2(t, x, y, z, u) + O(ε2), ˆz = z + εξ3(t, x, y, z, u) + O(ε2),
ˆu = u + εη(t, x, y, z, u) + O(ε2). (5) where ε is the group parameter and τ, ξ1,ξ2,ξ3andη 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
V = τ(t, x, y, z, u)∂
∂t + ξ1(t, x, y, z, u) ∂
∂x + ξ2(t, x, y, z, u) ∂
∂ y + ξ3(t, x, y, z, u) ∂
∂z + η(t, x, y, z, u) ∂
∂u, (6)
where dˆt
dε = τ(t, x, y, z, u), dˆx
dε = ξ1(t, x, y, z, u), dˆy
dε = ξ2(t, x, y, z, u), dˆz
dε = ξ3(t, x, y, z, u), dˆu
dε = η(t, x, y, z, u).
with the initial conditions(ˆt, ˆx, ˆy, ˆz, ˆu) |ε=0= (t, x, y, z, u).
Applying the fourth prolongation pr(4)V to Eq.(1), we obtain the invariance con- dition,
pr(4)V( ) |ε=0= 0. (7)
as the solutions space of (1) is invariant under the point transformation group (5), where
=
ut + λ1unux+ λ2ux x x
x+ λ3uyy+ λ4uzz= 0.
The fourth prolongation is given by
pr(4)V = V +
J
φJ(t, x, y, z, u(4)) ∂
∂uJ, (8)
where
φJ(t, x, y, z, u(4)) = ∂J(η − τut− ξ1ux+ ξ2uy+ ξ3uz)
+τuJ,t+ ξ1uJ,x + ξ2uJ,y+ ξ3uJ,z, (9)
with J = ( j1, . . . , jk), 1 ≤ jk ≤ 4, 1 ≤ k ≤ 4, uJ,xi = ∂xi∂x∂kj 1+1...∂xu jk and∂J =
∂k
∂xj1∂xj2...∂xjk.
Specifically the fourth prolongation can be given as
pr(4)V = V + φx ∂
∂ux + φx x ∂
∂ux x + φt x ∂
∂ut x
+φyy ∂
∂uyy + φzz ∂
∂uzz + φx x x x ∂
∂ux x x x. (10)
whereφx, φx x, φt x, φyy, φzz, φx x x xare given explicitly in terms ofτ, ξ1,ξ2,ξ3,η and the derivatives ofη. By using (10) we find the coefficient functionsτ(t, x, y, z, u), ξ1(t, x, y, z, u), ξ2(t, x, y, z, u), ξ3(t, x, y, z, u) and η(t, x, y, z, u). From (7) and (10), the invariance condition reads as
φt x+ λ1n(un−1ux x+ (n − 1)un−2u2x)η + 2λ1nun−1uxφx
+λ1unφx x + λ3φyy+ λ4φzz+ λ2φx x x x = 0. (11) where
φx = Dxη − utDxτ − uxDxξ1− uyDxξ2− uzDxξ3
φt x = Dxφt− ut tDxτ − ut xDxξ1− ut yDxξ2− ut zDxξ3
φx x = Dxφx− ut xDxτ − ux xDxξ1− ux yDxξ2− ux zDxξ3
φyy = Dyφy− ut yDyτ − ux yDyξ1− uyyDyξ2− uzyDyξ3
φzz= Dzφz− ut zDzτ − ux zDzξ1− uyzDzξ2− uzzDzξ3
φx x x x = Dxφx− ut x x xDxτ − ux x x xDxξ1− ux x x yDxξ2− ux x x zDxξ3 (12) and Dt, Dx, Dyand Dzt denote the total differential operators with respect to t, x, y and z
Dxi = ∂
∂xi + uxi
∂
∂u + uxixj
∂
∂uxj
+ . . .
where i = 1, 2, 3, 4 and (x1, x2, x3, x4) = (t, x, y, z).
2.1 Classification of Lie point symmetries
In this section we calculate the Lie point symmetries admitted by the modified general- ized 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 infinitesimalsτ(t, x, y, z, u), ξ1(t, x, y, z, u), ξ2(t, x, y, z, u), ξ3(t, x, y, z, u) and η(t, x, y, z, u) and the parameters a, b. Solving this system we obtain the next theorem:
Theorem 1 The point symmetries admitted by Eq. (1) are generated by:
1. Forλ3= 0 and λ4= 0:
V1= ∂t, V2= ∂x, V3= ∂y, V4= ∂z, V5= t∂t +1
3x∂x+2
3y∂y+2
3z∂z− 2 3nu∂u,
V6= z∂x− 2λ4t∂z, V7= λ3z∂y− λ4y∂z, V8= y∂x− 2λ3t∂y (13) 2. Forλ3= 0 and λ4= 0:
V1, V3, V5, Vf1 = f1(y)∂x,
Vg1 = zg1(y)∂x− 2λ4tg1(y)∂z, Vh1 = h1(y)∂z (14) 3. Forλ3= 0 and λ4= 0:
V1, V4, V5, Vf2 = f2(z)∂x,
Vg2 = yg2(z)∂x− 2λ4tg2(z)∂y, Vh2 = h2(z)∂y (15) 4. Forλ3= 0 and λ4= 0:
V1, V5, Vf = F(y, z)∂x, VG1 = G1(z)∂y, VG2 = G2(y)∂z (16)
Table 1 Commutator table of Lie algebra of (13)
[V i, Vj] V1 V2 V3 V4 V5 V6 V7 V8
V1 0 0 0 0 V1 −2λ4V4 0 −2λ3V3
V2 0 0 0 0 13V2 0 0 0
V3 0 0 0 0 23V3 0 −λ4V4 V2
V4 0 0 0 0 23V4 V2 λ3V3 0
V5 −V1 −13V2 −23V3 −23V4 0 13V6 0 13V8
V6 2λ4V4 0 0 −V2 −13V6 0 λ4V8 0
V7 0 0 λ4V4 −λ3V3 0 −λ4V8 0 λ3V6
V8 2λ2V3 0 −V2 0 −13V8 0 −λ3V6 0
Proof By using (11), letting the coefficientsτ(t, x, y, z, u), ξ1(t, x, y, z, u), ξ2(t, x, y, z, u), ξ3(t, 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
ξ3u = 0, ξ3x = 0, ξ2u = 0, ξ2x = 0, ξ1u = 0, τu = 0, τx = 0, ηuu= 0, τz = 0, τy= 0, ηux = 0, ξ1x x = 0,
λ3(ξ3y) + λ4(ξ2z) = 0, ξ3t + 2λ4(ξ1z) = 0, ξ3z − 2(ξ1x) = 0, ξ2t + 2λ3(ξ1y) = 0, ξ2y− 2(ξ1x) = 0, 3(ξ1x) − τt = 0, 2λ4(ηuz) − λ4(ξ3zz) − λ3(ξ3yy) = 0,
2λ3(ηuy) − λ4(ξ2zz) − λ3(ξ2yy) = 0, 2λ1(ξ1x)un+1+ λ1nηun− (ξ1t)u = 0, (ηu)u + 2(ξ1x)u + nη − η = 0,
λ1(ηx x)un+ λ4(ηzz) + λ3(ηyy) + λ2(ηx x x x) + ηt x = 0 2λ1n(ηx)un+ (ηt u)u − λ4(ξ1zz)u − λ3(ξ1yy)u − (ξ1t x)u = 0,
(17)
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 Table1, where[Vi, Vj] is the commutator for the Lie algebra defined by
[Vi, Vj] = ViVj− VjVi.
Then we build the adjoint table for each pair of elements Vi and Vj, with i, j = 1, . . . , 7, where
Ad(exp(εV ))W0=∞
n=0εn
n!(comV )n(W0)
= W0− ε[V , W0] +ε22[V , [V , W0]] − · · ·
The adjoint relationship of the Lie algebra is shown in Table2. We will use the adjoint representation to decompose all the subalgebras of the Lie algebra in equiv- alence 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 subal- gebras. 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 equa- tion (1). We consider a linear combination of Vi
V = α1V1+ α2V2+ α3V3+ α4V4+ α5V5+ α6V6+ α7V7+ α8V8 (18) and we define
fi : G → G
V → Ad(exp( iVi))V (19)
The function fi is a linear maps for i = 1, 2, . . . , 8, also we define the matrix A i with respect to basis{V1, V2, V3, V4, V5, V6, V7, V8}, for i = 1, 2, . . . , 8 as follows:
Ad(exp( iVi))V
= (α1, α2, α3, α4, α5, α6, α7, α8)Ai(V1, V2, V3, V4, V5, V6, V7, V8)t (20) with
A1=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
− 1 0 0 0 1 0 0 0 0 0 0 2λ4 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 2λ3 1 0 0 0 0 1
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
, A2=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 −13 2 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
A3=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −23 3 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 λ4 3 0 0 1 0 0 − 3 0 0 0 0 0 1
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
, A4=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 −23 4 0 0 1 0 0 0 0 0 0 − 4 0 1 0 0 0 0 −λ3 4 0 0 0 1 0
0 0 0 0 0 0 0 1
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
Table2AdjointtableoftheLiealgebra AdV1V2V3V4V5V6V7V8 V1V1V2V3V4V5−εV1V6+2λ4εV4V7V8+2λ3εV3 V2V1V2V3V4V5−1 3εV2V6V7V8 V3V1V2V3V4V5−2 3εV3V6V7+λ4εV4V8−εV2 V4V1V2V3V4V5−2 3εV2V6−εV4V7−λ3εV3V8 V5eεV1e1 3εV2e2 3εV3e2 3εV4V5e−1 3εV6V7e−1 3εV8 V6V1−2λ4εV4V2V3V4+εV2V5+1 3εV6V6V7−λ4εV8V8 V7V1V2V3−λ4εV4V4+λ3εV3V5V6+λ4εV8V7V8−λ3εV6 V8V1−2λ3εV3V2V3+εV2V4V5+1 3εV8V6V7+λ3εV6V8
A5=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
e 5 0 0 0 0 0 0 0
0 e 53 0 0 0 0 0 0 0 0 e2 53 0 0 0 0 0 0 0 0 e2 53 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 e− 53 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 e− 53
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠ , A6=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
1 0 0 −2λ4 6 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 6 0 1 0 0 0 0 0 0 0 0 1 13 6 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 −λ4 6
0 0 0 0 0 0 0 1
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
A7=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 −λ4 7 0 0 0 0 0 0 λ3 7 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 λ4 7
0 0 0 0 0 0 1 0
0 0 0 0 0 λ3 7 0 1
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠ , A8=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
1 0 −2λ3 8 0 0 0 0 0 0 1 0 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 13 8
0 0 0 0 0 1 0 0 0 0 0 0 0 λ3 8 1 0 0 0 0 0 0 0 0 1
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
Finally the general adjoint matrix A, calculated using the previous matrix, is given by
A=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎝
e 5 2 6 7 8λ3λ4e31 5 −2 8λ3e23 5 A14 0 0 0 0
0 e13 5 0 0 0 0 0 0
0 A32 e23 5 − 7λ4e23 5 0 0 0 0 0 6e13 5 7λ3e23 5 e23 5 0 0 0 0
− 1 A52 A53 A54 1 A56 0 13 8e−13 5 0 − 3 7λ4e−13 5 2 1 7λ3λ4e−13 5 A64 0 e−13 5 0 7λ4e−13 5 0 A72 A73 A74 0 8λ3e−13 5 1 A78 0 − 3e−13 5 2 1λ3e−13 5 A84 0 7λ3e−13 5 0 e−13 5
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎠
(21) where
A14= 2( 7 8λ3λ4− 6λ4)e23 5, A32= −( 6 7λ4− 8)e13 5 A52= −13 3 8e−13 5 −13 2−23 4
A53=23 1 7λ3λ4e−13 5−23 3
A54=23( 7 8λ3+ 6) 1λ4e−13 5 −13( 7 8λ3+ 6) 4e−13 5 A56=13( 7 8λ3+ 6)e−13 5
A64= 2 1λ4e−13 5− 4e−13 5 A72= −( 7 8λ3λ4− 6λ4) 3e−13 5
A73= 2( 7 6λ3λ4− 6λ4) 1λ3e−13 5− 4λ3
A74= 2 1 8λ3λ4e−13 5− 4 8λ3e−13 5+ 3λ4
A78= ( 7 8λ3λ4− 6λ4)e−13 5 A84= 2 1 7λ3λ4e−13 5− 4 7λ3e−13 5
By using (21) the adjoint transformation equation to (1) is
(β1, β2, β3, β4, β5, β6, β7, β8) = (α1, α2, α3, α4, α5, α6, α7, α8)A.
Then we have the equation system β1= α1e 5− α5 1
β2= α12 6 7 8λ3λ4e13 5 + α3e13 5+ α3A32+ α4 6e13 5 +α5A52+ α6− 3 7λ4e−13 5 + α7A72+ α8− 3e−13 5 β3= α12 8λ3e23 5+ α3e23 5+ α4 7λ3e23 5+ α5A54
+α62 1 7λ3λ4e−13 5+ α7A73+ α82 1λ3e−13 5
β4= α1A14− α3 7λ4e23 5+ α4e23 5+ α5A54+ α6A64+ α7A74+ α8A84
β5= α5
β6= α5A56+ α6e−13 5+ α7 8λ3e−13 5 + α8 7λ3e−13 5 β7= α7
β8= α5
1
3 8e−13 5+ α6 7λ4e−13 5 + α7A78+ α8e−13 5 (22) and it must have solutions for ifor i= 1, 2, . . . , 8, for certain values of αiandβi, i = 1, 2, . . . , 8. Then we can obtain the generators of the optimal one-dimensional system:
α1V1+α2V2+α3V3+α4V4, V5, V6+αV1+βV3, V7+αV1+βV2, V8+αV1+βV4.
2.2 Lie symmetry groups and new solutions
In this part, by solving the following initial problems, we can get the Lie symmetry group from the related symmetries to get some new exact solutions from the known ones. To calculate the one-parameter Lie symmetry group g(t, x, y, z, u) generated through the general vector field (6), we consider
g(t, x, y, z, u) = (ˆt, ˆx, ˆy, ˆz, ˆu) (23) and we solve the following initial problems
∂ ˆt
∂ = τ(t, x, y, z, u), ˆt | =0= t
∂ ˆx
∂ = ξ1(t, x, y, z, u), ˆx | =0= x
∂ ˆy
∂ = ξ2(t, x, y, z, u), ˆy | =0= y
∂ ˆz
∂ = ξ3(t, x, y, z, u), ˆz | =0= z
∂ ˆu
∂ = η(t, x, y, z, u), ˆu | =0= u (24)
Therefore, from (24) we can obtain the corresponding Lie symmetry group, that is to say the one-parameter Lie symmetry groups gi, i = 1, . . . , 8, which are generated through Vi, i = 1, . . . , 8, respectively are given by
g1: (t, x, y, z, u) → (ˆt, ˆx, ˆy, ˆz, ˆu) = (t + , x, y, z, u) time translation
g2: (t, x, y, z, u) → (ˆt, ˆx, ˆy, ˆz, ˆu) = (t, x + , y, z, u) space-translations along the x-axis
g3: (t, x, y, z, u) → (ˆt, ˆx, ˆy, ˆz, ˆu) = (t, x, y + , z, u) space-translations along the y-axis
g4: (t, x, y, z, u) → (ˆt, ˆx, ˆy, ˆz, ˆu) = (t, x, y, z + , u) space-translations along the z-axis
g5: (t, x, y, z, u) → (ˆt, ˆx, ˆy, ˆz, ˆu) = (te , xe 3, ye23 , ze23 , ue−23n nonhomogeneous scaling group
g6: (t, x, y, z, u) → (ˆt, ˆx, ˆy, ˆz, ˆu) = (t, −λ4 2t+ z + x, y, −2λ4 t + z.u) time and space dependent shift
g7: (t, x, y, z, u) → (ˆt, ˆx, ˆy, ˆz, ˆu) = (t, x,1+λy+λ3 z
3λ4 2,1z+λ−λ4 y
3λ4 2, u) space-dependent shift
g8: (t, x, y, z, u) → (ˆt, ˆx, ˆy, ˆz, ˆu) = (t, −λ3t 2+ y + x, y − 2λ3 t, z, u) time and space dependent shift
(25)
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 gi, i= 1, . . . 8, we can calculate the new solutions of (1).
Based on (25), the corresponding new solutions of (1) can be given by:
ˆu1= f (t − , x, y, z) (26)
ˆu2= f (t, x − , y, z) (27)
ˆu3= f (t, x, y − , z) (28)
ˆu4= f (t, x, y, z − ) (29)
ˆu5= f (te− , xe− 3 , ye−2 3 , ze−2 3 )e−2 3n (30) ˆu6= f (t, x − z − λ4 2t, y, z + 2λ4 t) (31) ˆu7= f (t, x, y − λ3 z, z + λ4 y) (32) ˆu8= f (t, x − y − λ3 2t, y + 2λ3 t, z) (33)
3 Symmetry reductions
In this section we mainly consider the one-dimensional subalgebras computed in the previous subsection and obtain symmetry reductions of modified generalized mul- tidimensional Kadomtsev–Petviashvili equation. Similarity variables and similarity solutions associated with any vector field V = τ∂t + ξ1∂x+ ξ2∂y+ ξ3∂z+ η∂ucan
