Lie symmetry analysis, optimal system, and new exact solutions of a (3 + 1) dimensional nonlinear evolution equation

1 Introduction

Recently, non-linear governing equations suitable to analyze quartic autocatalysis were presented by Makinde and Animasaun in [1] and [2]. There has been an increasing interest in the study of NLEEs in the past few years. The (3+1) - dimensional nonlinear evolution equations was first introduced by Zhaqilao [3] in the study of algebraic-geometrical solutions. An evolution equation refers to a partial differential equation having partial derivatives of the dependent variable u with respect to the time t and space variables x = (x₁, ..., x_n), which are the independent variables. The (3+1)-dimensional equation possesses the KdV equation u_t − 6uu_x + u_xxx = 0 as its main term under the transformations v(x, t) → u(x′, t′), x′→13x and t′→163t . Based on this, the (3+1)-dimensional nonlinear evolution equation may be used to study the shallow-water waves and short waves in nonlinear dispersive models [4]. A physicist should be well aware of all new aspects of the nonlinear wave theory. It is always a good practice to study a new equation of the theory of nonlinear evolution equations. The proper understanding of qualitative significances of many incidents and procedures can be achieved by exact solitary wave solutions of NLEEs in different fields of applied mathematics, engineering, physics, biology, chemistry, and many more. So, to gain a clear understanding of the qualitative and quantitative properties of these equations, it is necessary to find some exact solutions to these equations. For illustration, the soliton pulse implies an ideal balance between nonlinearity and dispersion effects. The soliton is a crucial character of nonlinearity [5,6,7,8,9,10,11,12,13,14,15,16]. Soliton solutions are of special type PDEs solutions that model phenomena from the balance between nonlinear and dispersive effects in systems like light pulses propagation in optical fibers and water waves. For the nonlinear PDEs, the exact solutions graphically demonstrate and determine the structure of many nonlinear complex phenomena such as absence of multiplicity steady states under different conditions, spatial localization of transfer processes, presence of peaking regimes, and many others. First of all, Geng [13] introduced equation (1) in the algebraic geometrical solutions [17]. In [6] N-soliton solutions of the (3+1)-d NEE was studied by Geng and Wazwaz [5, 18, 19] found some multiple soliton solutions and a collection of traveling wave solutions of the (3 + 1)-d NEE (1). Soliton and rogue wave solutions can be found in [3, 20,21,22,23,24,25]. There are many powerful methods to understand the nonlinear evolution equations that have been used, for instance, the Hietarinta approach [15], Hirota bilinear method [5,6,7,8,9,10,11,12,13,14], the Bäcklund transformation method, Pfaffian technique, Darboux transformation, the inverse scattering method, the generalized symmetry method, the Painlevé analysis, and other methods. To investigate nonlinear dynamical phenomena using a generalised model in shallow water, plasma and nonlinear optics, a generalized (2 + 1)-dimensional Hirota bilinear equation was proposed by Hua et al. [26]. Xin Zhao et al. [27] have investigated the generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics and plasma physics. They have used the Hirota Bilinear method, and obtained bilinear Bäcklund transformation, to construct the Lax pair and obtained Mixed Rogue–Solitary Wave Solutions, Rogue–Periodic Wave Solutions and Lump-Periodic Wave Solutions. They have also explained the interactions between the rogue wave, periodic wave and the solitary wave. Here, we shall study such a (3+1)-dimensional nonlinear evolution equation [5] – [8] and [18].

where

Obviously, ∂x∂x−1f=∂x−1∂xf=1 under the decaying condition at ∞. As per the coefficients of x, y and z, the multiple soliton solutions exist for equation (1) ([5] and [6]). Several soliton solutions, as well as singular soliton solutions, were obtained by the simpler form of the Hirota's method in [1]. An N-soliton solution of a (3+1)-dimensional nonlinear evolution equation is obtained by using the Hirota bilinear method with the perturbation technique in [6]. A new Wronskian condition was set for equation (1), with the aid of the Hirota bilinear transformation, a novel Wronskian determinant solution is presented for the equation (1). The Wronskin determinant is different for both [5] and [6]. We aim to extend the work in [8], where the classical Lie symmetry of the (3+1)-dimensional nonlinear evolution equation (1) was found. Here, we obtained an optimal system for further results and then some new solutions which can explain new nonlinearity features with the approach applied in [28], [29] and [30].

To remove the integral term in equation (1) by introducing the potential

we get

Generally, it is not easy to get every possible combination of group generators to obtain the invariant solutions, as there may be infinitely many solutions. Researchers have always discussed relatively independent solutions, this inspires many other researchers to obtain a new system called an optimal system. Thus, in this paper, we constructed a one-dimensional optimal system of subalgebra for equation (4). The Norwegian mathematician Sophus Lie introduced the term invariant solutions and developed the Lie point symmetry analysis (1842–1899). The research conducted so far motivates us to obtain some new exact solutions using an optimal system, of equation (4), which has not been found in research yet.

One may find in this article some acceptable answers, as a result, shown in the graphs and solutions presented in the closed-form. Do the soliton solutions of the given equation exist? If so, how do they behave? Can one speculate the “soliton” nature of the solution even if solutions are not well known in some real systems? How can one find some precise solutions that can be useful “if the complexity of the methods affects the solution results”? Are there solutions to test stability and estimate errors for the newly proposed numerical algorithms”? The authors have tried to find the answers of the above mentioned questions in the present article.

This work has two main objectives. The first is to obtain an optimal system, and the second is to obtain several types of new exact solutions. In section (2), we have applied the Lie group approach to obtain the symmetries of equation (4). An optimal system of vector fields is established in section (3). In section (4), we investigated the reduced equations to find exact solutions, and in the end, some remarks are presented in the conclusion.

2 Lie point symmetries

Lie group of transformations with parameter (ɛ) acting on variables (dependent and independent) for equation (4) are as follows

where ɛ is a small Lie group parameter andψ^x, ψ^y, ψ^z, τ and η are the infinitesimals of the transformation which are to be found for independent and dependent variables, respectively. Thus, the associated Lie algebra will be of the form Olver [31]

The above vector field generates a symmetry of equation (4). Also, for the invariance, pr⁽⁵⁾ℙ(Δ) = 0, when Δ = 0 for equation (4), where pr⁽⁵⁾ℙ is the fifth prolongation of ℙ. To obtain an overdetermined system of the coupled PDEs, we applied pr⁽⁵⁾ℙ to equation (4)

and get

After this, we use a computer algebra software (Maple) to obtain the following system of PDEs:

and, thus, we obtained the required infinitesimal generator as follows:

where c_i′s, (i = 1, 2, 3, 4) and f_j′s, (j = 1, 2, 3) are arbitrary. Following the Lie symmetry method explained in [31], we get the Lie algebra of symmetries for equation (4) as follows:

Now, for convenience, we obtain the Table 1 of commutator with entries as [ℙ_i, ℙ_j] = ℙ_i · ℙ_j − ℙ_j ·ℙ_i (see [31]).

Clearly, the infinite-dimensional Lie algebra spanned by vector fields (11) generates an infinite continuous group of transformations of equation (4). These generators are linearly independent. Thus, it is very much appropriate to represent any infinitesimal of equation (4) as a linear combination of ℙ_i, given as

The group of transformation Gi:(x,y,z,t,u)→(x˜,y˜,z˜,t˜,u˜) which is generated by the infinitesimal generator ℙ_i for i = 1, 2, 3, 4, 5, 6, 7 are as follows [31]:

The right hand side gives the transformed point exp(ϵℙi)(x,y,z,t,u)=(x˜,y˜,z˜,t˜,u˜) . As each group G_i is a symmetry group (by [31]), if u = f (x, y, z, t) is a solution of equation (4) so are the functions

where ɛ is any real no. For a detailed description, the reader can see [31].

Generally, there is an infinite number of subalgebras for this Lie algebra formed from linear combinations of generators ℙ₁, ℙ₂, ℙ₃, ℙ₄, ℙ₅, ℙ₆ and ℙ₇. If two subalgebras are equivalent, i.e., each has conjugate in the symmetry group, then their corresponding invariant solutions are connected by the same transformation. Thus, it is sufficient to place all similar subalgebras in one class and select a representative for every class. The set of all these representatives is called an optimal system (for details, see [31] and [32]). A detailed discussion is given in the next section.

3 Optimal system of subalgebra

Now, we find an optimal system of one dimensional Lie subalgebra. As an application of Lie group analysis, the primary use of an optimal system is to classify the group invariant solutions of partial differential equations to shorten the problem of categorizing subgroups of the complete symmetry group. A set of subalgebras forms an optimal system if each subalgebra of the Lie algebra is equivalent to a unique member of the set of subalgebras under some element of adjoint representation. Ovsiannnikov and Olver [31, 32] suggested the construction of an optimal system for the Lie subalgebra. The method made useful progress under the work of Petera, Winternitz, and Zassenhaus [33, 34], where various illustrations of an optimal system of subgroups can be seen for the Lie groups of mathematical physics. Based on the systematic algorithm [35], we find an optimal system of one-dimensional subalgebras of the equation (4). The symmetry Lie algebra having a basis {ℙ₁, ℙ₂, ℙ₃,ℙ₄, ℙ₅, ℙ₆, ℙ₇} of section (2) and identify this with ℝ⁷ as a vector space using the map ℙ_i → 𝔢_i where {𝔢₁, 𝔢₂, 𝔢₃, 𝔢₄, 𝔢₅, 𝔢₆, 𝔢₇} is the standard basis of ℝ⁷. Then, from the Table 1, we obtain the following matrix description of Ad(ℙ_i):

where [ℙ_i, ℙ_j] is the commutator of the two operators. A real function ϕ on the Lie algebra 𝔤 is called an invariant if it satisfies the following condition:

For the Lie algebra 𝔤, we consider any subgroup g = exp(ɛ𝒮), where S=∑j=17bjℙj to act on ℳ=∑i=17aiℙi , we get

where θ_i = θ_i(a₁, a₂, a₃, a₄, a₅, a₆, a₇, b₁, b₂, b₃, b₄, b₅, b₆, b₇), i = 1, 2, 3, 4, 5, 6, 7 can be obtained from the commutator table (1), and for invariance

Expanding the right-hand side of eq. (15), we obtain

where

Substitution of equations (17) into equation (16) and collection of the coefficients of all bi′s gives the following linear over determined system of PDEs in ϕ:

Looking the solutions of the above system, we get the invariant form given as, ϕ(a₁, a₂, a₃, a₄, a₅, a₆, a₇) = F(a₁, a₃), where F can be chosen as an arbitrary function. Thus, the following two basic invariants of the Lie algebra 𝔤 exist:

also the function η(ℙ)=53a12+2a32+43a1a3 , is invariant of the full adjoint action known as the Killing's form for 𝔤 ([31] and [36]). It can be seen that the Killing form is a combination of the basic invariants of the Lie Algebra 𝔤. Thus, the basic invariants of the Lie algebra 𝔤 are used to find the one-dimensional optimal system of the equation (4).

Now, we need to prepare the general adjoint transformation matrix A, which is obtained by the product of the individual matrices of the adjoint actions A₁, A₂, A₃, A₄, A₅, A₆, A₇, which are the adjoint action of ℙ₁, ℙ₂, ℙ₃, ℙ₄, ℙ₅, ℙ₆, ℙ₇ to A.

Let ɛ_i, i = 1, 2, 3, 4, 5, 6, 7 be real constants and g = e^ɛ_iℙ_i, then we get

The adjoint action of ℙ_j on ℙ_i can be obtained from the adjoint representation, (see Table 2) for more detail, one may refer to Hu et al. [35].

The formation of an optimal system of subalgebras of a Lie algebra is not an easy assignment. An optimal system of Lie subalgebras can be obtained by solving the system of algebraic equations, and the equivalent Lie subalgebras can be identified by the use of adjoint action on the set of these Lie subalgebras. Let

where c₁, c₂, c₃, c₄, c₅, c₆, c₇ are the real constants. Here, X can be considered as a column vector with entries c₁, c₂, c₃, c₄, c₅, c₆, c₇. Let A(ɛ₁, ɛ₂, ɛ₃, ɛ₄, ɛ₅, ɛ₆, ɛ₇) = A₇A₆A₅A₄A₃A₂A₁, which gives

Now, to construct an optimal system of equation (4), we consider X=∑j=17cjℙj and Y=∑j=17djℙj as two elements of Lie algebra 𝔤. Adjoint transformation equation for equation (4) is

In addition, A(ɛ₁, ɛ₂, ɛ₃, ɛ₄, ɛ₅, ɛ₆, ɛ₇) transform X as follows

By definition, X and A(ɛ₁, ɛ₂, ɛ₃, ɛ₄, ɛ₅, ɛ₆, ɛ₇)·X generate equivalent one dimensional Lie subalgebras for any ɛ₁, ɛ₂, ɛ₃, ɛ₄, ɛ₅, ɛ₆, ɛ₇. This provides the liberty of choosing various values of ɛ_i to represent the equivalence class of X that might be much simpler than X. In order to distinguish the one dimensional Lie subalgebras of equation (4), we consider the cases as follows:

4 Invariant solutions

After the formation of one-dimensional optimal system of equation (4), we reach the equivalence class of group invariant solutions of equation (4). We will present the details of the calculation for some of the vector fields and directly give the calculation results for the remaining vector fields.

4.1 ℙ˜1=ℙ1+c3ℙ3

Solving the characteristic equation, similarity variables can be obtained as

(25) dxx3=dyc3y=dz(c3+23)z=dtt=du−u3,

then, we get the invariants as X=xt13 , Y=ytc3 , Z=ztc3+23 , u=t−13F(X,Y,Z) with X, Y, Z as new independent variables and F as a new dependent variable. Thus, equation (4) transforms to

(26)  6c3YFXYY+6c3ZFXYZ+2XFXXY+4ZFXYZ+6FXFXXY+12FXYFXX+6c3FXY+6FXXXFY+9FXXZ+4FXY −3FXXXXY=0.

Again, reducing the equation (26) by point symmetries, the following vector fields are found to span the symmetry group of equation (26):

(27) ξX=f1(Z), ξY=n1Y+n2Z(11+23c3), ξZ=n1Z,ηF= (27 z3 c33 c3+2n2 c3 x+18 z(c3+23)((x(c3+23)z+32y)ddzf1(z)−13 xf1(z)+f2(z)))(18 c3+12)z,

where f₁(Z) and f₂(Z) are the arbitrary functions. By the appropriate choice of the arbitrary functions of the above equation, if f₁(Z) = Z, f2(Z)=f1'(Z) and c₃ = 1, it leads to the following characteristic equations:

(28) dXZ=dYn1 YZ+n2 Z35=dZn1Z     =dF27 n2Z35 X+30 Z((53 XZ+32 Y)ddZf1(Z)−13 Xf1(Z)+f2(Z))30 Z,

which yields

(29) F=23 Z2n12+43 rZn1+32 sZn1−19 n2 Z3/54 n12+ln(Z)n1−94 n2 rn1 Z25+G(r,s),

where G(r, s) is a similarity function of variables r and s, which are given by

(30) r=X+Zn1, s=YZ+52n2n1Z25.

Thus, the characteristic equation for the second reduction of equation (4) is

(31) 12GrrrsGr+(4r−18s)Grrs+24GrsGrr+12GrrrGs−6Grrrrs−8sGrss=0.

We can see that, this is a nonlinear PDE with two independent and one dependent variable. After applying the similarity transform again, the following vector fields are found to span the symmetry group of equation (31):

(32) ξr=0, ξs=0, ηG=m1.

Thus, the second reduction by similarity of equation (4) gives

(33) dr0=ds0=dGm1,

which leads to G = m₂, where m₁ and m₂ are some real constants. Thus, the invariant solution of equation (4) is given by

(34) u(x,y,z,t)=112 n12 t113z25(12 m2 n12 t103 z25+12 n1 t103 z25ln(z)−20 n1 t103 z25ln(t)+16 n1 z75t43 x−8 z125+18 n1 z25t73 y+15 n2 t73 z−27 n1 n2 t113 x).

Figure 1

Solution profiles for equation (34) with m₂ = 20, n₁ = 10, n₂ = 20 and y = 5.

4.2 ℙ˜2=ℙ1+c5ℙ5

For this subalgebra, the similarity variables can be obtained by the following characteristic equation:

(35) dxx3=dyc5=dz23z=dtt=du−u3,

then, we get the invariants as X=xt13 , Y = y − c₅ ln(t), Z=zt23 and u=t−13 F(X,Y,Z) with X, Y, Z as new independent variables and F as new dependent variable. Thus, equation (4) is then transformed to

(36) 3FXXZ+23XFXXY+2c5FXYY+43ZFXYZ+43FXY−FXXXXY+4FXYFXX+2FXFXXY+2FYFXXX=0.

Again, reducing the equation (36) by point symmetries, the following vector fields are found to span the symmetry group of equation (36):

(37) ξX=f1(Z), ξY=n1Y−32 n1 c5 ln(Z)+n2, ξZ=n1Z,ηF= 32 Y f1'(Z)+112 XZ(8 Z2 f1'(Z)−4 Z f1(Z)−27 n1 c5)+f2(Z),

where f₁(Z)and f₂(Z) are the arbitrary functions. It leads to the following characteristic equations:

(38) dXf1(Z)=dYn1Y−32 n1 c5 ln(Z)+n2=dZn1Z=dF32 Y f1'(Z)+X12 Z(8 Z2 f1'(Z)−4 Z f1(Z)−27 n1 c5)+f2(Z).

Figure 2

Solution profiles for equation (44) with n₁ = 1, n₂ = 2, m₃ = 3, z = 2 and R = 20sin(w).

By the appropriate choice of the arbitrary functions of the above equation, if f₁(Z) = Z, f2(Z)=f1'(Z) and c₅ = 1, we obtain

(39) F=9 (ln(Z))28 n1+9r4Z+rZ3n1+3sZ2n1+Z26n12+ln(Z)n1−3n2 ln(Z)2n12+G(r,s),

where G(r, s) is a similarity function of variables r and s, which are given by

(40) r=X−Zn1, s=(Y−32 ln(Z)−32+n2n1)⋅1Z.

Thus, the second reduction by similarity of equation (4) gives

(41) 3sGrrs+23rGrrs−43sGrss−Grrrrs+4GrsGrr+2GrGrrs+2GrrrGs=0.

We can see that, this is a nonlinear PDE with two independent and one dependent variable. After applying the similarity transform again, the following vector fields are found to span the symmetry group of equation (41),

(42) ξr=m1, ξs=0, ηG=−13 m1r+m2.

Thus, the characteristic equation for the second reduction of equation (4) is

(43) drm1=ds0=dG−13 m1r+m2,

where m₁ and m₂ are some real constants.

This leads to G=−r26+m3 r+R(w) , where R(w) is an arbitrary function of w with w = s and m3=m2m1 . Thus, an invariant solution of equation (4) is given by

(44) u(x,y,z,t)=1t3(98 n1(ln(zt23))2+94 t23z(xt3−zn1t23)+13 zn1t23(xt3−zn1t23)+32 1n1(y−ln(t)−32 ln(zt23)−32+n2n1)+16 z2n12t43+1n1ln(zt23)−32 n2n12ln(zt23)−16 (xt3−zn1t23)2+m3(xt3−zn1t23)+R(w)),

where

w=t23z(y−ln(t)−32 ln(zt23)−32+n2n1).