Alimirzaluo et al. Advances in Difference Equations https://doi.org/10.1186/s13662-021-03220-3 ( 2021) 2021:60 RESEARCH Open Access Some new exact solutions of (3 + 1)-dimensional Burgers system via Lie symmetry analysis Elnaz Alimirzaluo1 , Mehdi Nadjafikhah1* and Jalil Manafian2 * Correspondence: m_nadjafikhah@iust.ac.ir 1 Department of Pure Mathematics, School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran Full list of author information is available at the end of the article Abstract In this paper, by using the Lie symmetry analysis, all of the geometric vector fields of the (3 + 1)-Burgers system are obtained. We find the 1, 2, and 3-dimensional optimal system of the Burger system and then by applying the 3-dimensional optimal system reduce the order of the system. Also the nonclassical symmetries of the (3 + 1)-Burgers system will be found by employing nonclassical methods. Finally, the ansatz solutions of BS equations with the aid of the tanh method has been presented. The achieved solutions are investigated through two- and three-dimensional plots for different values of parameters. The analytical simulations are presented to ensure the efficiency of the considered technique. The behavior of the obtained results for multiple cases of symmetries is captured in the present framework. The outcomes of the present investigation show that the considered scheme is efficient and powerful to solve nonlinear differential equations that arise in the sciences and technology. Keywords: Analysis burgers equation; Symmetry group; Optimal system; Nonclassical symmetries; Tanh method 1 Introduction The Burgers system describes the propagation processes for nonlinear waves in fluid mechanics such as diverse non-equilibrium, nonlinear phenomena in turbulence, and interface dynamics [12]. Also, this system is used in solitary wave theory to expand integrable models with the extending of famous physical equations. The physicists and mathematicians are looking to study the (1 + 1)-dimensional and the (2 + 1)-dimensional integrable models. Because of the real physical space-time being (3 + 1)-dimensional, researchers have been attempting to detect higher-dimensional models in some ways [29, 31, 32, 51]. Researchers have used many effective techniques to discover the different solution of the Burgers system [10, 48]. One of the Sophus Lie’s significant discoveries in differential equation is to indicate that transforming nonlinear conditions is possible by through infinitesimal invariants which can correspond to the generators of the symmetry group of the system [28]. Having the symmetry group of a system of differential equations has many advantages including the ability to classify the solutions of the differential equations system. This classification is such that we consider both answers in a category that can be con© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Page 2 of 17 verted by some transformational equation. Researchers interested in further reading may refer to recently published articles, including [4, 16, 24, 49]. Another use of these groups is that the differential equations can be classified according to the desired parameter or function. Ovsiannikov [43] gave the strategy for halfway invariant solutions. His technique depends on the idea of an equivalence group, which is a Lie symmetry group working in the extended space of free variables, functions and their subordinates, and saving the class of incomplete differential conditions. The investigation of the correct arrangements has a noteworthy influence in the perusing of nonlinear physical frameworks [37]. Probably the main strategies are the inverse scattering method [18], the Hirota bilinear method [19, 27], Lie symmetry analysis [11, 40], Darboux and Bäcklund transformations [27], the tanh-function method (Duffy and Parkes [15]). In [13], Chao-Qieg and Yue have obtained a new analytical solution of the (3 + 1)-dimensional Burgers system relied on the Riccati equation. And In [12] another mapping equation is utilized to find variable separation solutions of the (3 + 1)-dimensional Burgers system and three class of variable separation solutions are driven. Also in [22, 23], Ibragimovs new technique for finding conservation laws and the idea of nonlinear self-adjointness were explained that widely implemented to find conservation laws of equations (for example see [1, 5]). Especially In [3], the nonlinear self-adjointness and conservation laws of the (3 + 1)-dimensional Burgers equation has been obtained by the aid of Ibragimovs method, BS : ut = 2uuy + 2vux + 2wuz + uxx + uyy + uzz , ux = vy , uz = wy . This equation describes the propagation processes for nonlinear waves in fluid mechanics such as diverse non-equilibrium, nonlinear phenomena in turbulence and inter-face dynamics. Lie symmetries of BS will present some additional results which are also obtained from the Lie algebra structure of Lie symmetry group. We, therefore, plan to make an optimal system of 1-subalgebras of the BS which is useful for classifying of group invariant solutions and to construct an optimal system of 2-subalgebras for the BS, also for 3-subalgebra of the BS which are effectively helping to classify the group invariant solutions. After that, the reduced equation for each element of the optimal system is obtained. Then, using the method of Lie symmetry group a solution will be presented for reduced equations. Afterward, the group invariant solutions of BS is achieved. In [30], the Lie point symmetries of (3 + 1)-dimensional Burgers system is obtained and the system is reduced by them. Finally, we found the nonclassical symmetries that were first discussed in Bluman and Cole [8] in their treatment of generalized self-similar solutions of the linear heat equation [42]. This method and its correlation to direct reduction methods of Clarkson [14] and Galaktionov [17] have become the focus of much research and many applications to physically significant partial differential equations. Obviously, other related points, such as partially invariant solutions differentially partially invariant solutions of group foliation, will give rise to effective and delicate methods of treating differential equations. In [33, 50], they have been able to find a powerful solution method, By using the tanh method for the computation of traveling wave solutions. First an ansatz of power series in tanh was used to obtain solutions of traveling wave of certain nonlinear evolution equations [34]. Recently, Fan and Hon [16] introduced a generalized tanh method to investigate special types of nonlinear equations. Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Page 3 of 17 Hereafter, the symmetry reductions, explicit solutions, convergence analysis and conservation laws to the Chen–Lee–Liu model in nonlinear optics were been studied by the authors in [4]. Inc and co-authors surveyed the time fractional generalized Burgers–Huxley equation with Riemann–Liouville derivative via Lie symmetry analysis and the power series expansion method [24]. Tchier and et al. presented an investigation and analysis for the space-time Carleman equation (STCE) in nonlinear dynamical system [49]. Also, some researchers utilized the Lie symmetry analysis for various nonlinear PD equations. See references [2–5]. Some authors studied the stochastic influenza model with constant vaccination strategy [6], the stochastic meme epidemic model with investigate effect of threshold number [47], a reliable numerical analysis of a stochastic HIV/AIDS model in a two-sex population considering counseling and anti-retroviral therapy [46]. Some research as regards nonlinear study of equations has been made covering nonlinear vibrations of Euler– Bernoulli beams [44], parametrically excited nonlinear oscillators [45], and nonlinear free vibration analysis of tapered beams [7]. This article is organized as follows. In Sect. 2, the Lie symmetries of BS equations have been obtained. By the aid of obtained symmetries, the invariant groups of BS equations are given in Sect. 3. In Sects. 4, 5, and 6 the optimal systems of order one, two, and three of BS equations are presented, respectively. By considering the new coordinates the BS equations are reduced in Sect. 7. In Sect. 8, the nonclassical symmetries of the BS equation are driven. In Sect. 9, by implementing of tanh-function method ansatz solution of BS equations has been found and plotted. Also, in Sect. 10 the results and discussion of graphs and their behavior are given. 2 Lie symmetry of BS The method which is conventionally applied to determine the classical symmetries of a partial differential equation is standard and is explained in [9, 40, 41] we consider an infinitesimal generator of Lie symmetry from the following form in order to get the Lie symmetry: X = ξt ∂t + ξx ∂x + ξy ∂y + ξz ∂z + η1 ∂u + η2 ∂v + η3 ∂w , where ξ t , ξ x , ξ y , ξ z are functions of t, x, y, z, and ηu , ηv , ηw are functions of t, x, y, z, u, v, w. By using the invariant condition, such as applying pr(2) X the second prolongation of X to BS, we have the following system of equations: 3 3 2 ηzt = ηz3 = ηz2 = ηwz = ηwz = ηw3 = ηw2 = ηx1 = ηy2 = ηy3 = 0, ηw1 = ξx1 = ξx3 = ξy1 = ξy2 = ξy4 = ξz1 = ξz3 = ξu1 = ξu2 = ηv1 = ηu3 = 0, ξu3 = ξu4 = ξv1 = ξv2 = ξv3 = ξv4 = ξw1 = ξw2 = ξw3 = ξw4 = ηz1 = ηu2 = 0, 3 ηwt = 2ηz3 , 2 ηwt = –2ηz2 , ξt3 – 2ηw3 u = 2η1 , ξx2 = –ηw3 , ηu1 = ηw3 , ηt1 = 2ηz3 u = 0, ξy3 = –ηw3 , ηv2 = ηw3 , ξt1 = –2ηw3 , ξz4 = –ηw3 , ηv3 = –ηw2 , ξt2 – 2ηw2 w – 2ηw3 v + 2η2 = 0, ξt4 + 2ηw3 v = 2ηw3 w – 2η3 , ξz2 = ηw2 , ηy1 = –ηz3 , ηx2 = ηz2 , ηx3 = –ηz2 . ηx3 = –ηz2 , Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Therefore, the Lie symmetry group of BS includes a Lie algebra generator in the form of the vector field υ having these functional coefficients: c1 2 c1 t + c2 c1 c4 t + c2 t + c3 , η1 = – u– y– , 2 2 4 2   c1 c1 t + c2 1 c1 t + c2 ξ2 = –F1 z + F3 + x, η2 = – F1′ + F3′ – F1′ w – v – x, 2 2 2 4  c 1 ′ c1 y + 2c4 t + c c2 c 1 2 1 ξ3 = η3 = – F1 + F2′ + F1′ v + t + y + c5 , w – z, 2 2 2 2 4 c1 t + c2 z, ξ4 = F1 + F2 + 2 ξ1 = where ci , i = 1, . . . , 5 are arbitrary constants and Fi (t), i = 1, 2, 3 are arbitrary smooth functions. In the case the above system is solved, the following theorem is introduced. Theorem 2.1 The Lie algebra L(G) of projectable Lie symmetries of BS is spanned by 1 1 1 t X1 = (t∂t + x∂x + y∂y + z∂z ) – (2tu + y)∂u – (2tv + x)∂v – (2tw + z)∂w , 2 4 4 4 1 X2 = ∂t , X3 = t∂t + (x∂x + y∂y + z∂z – u∂u – v∂v – w∂w ), 2 1 1 1 X4 = t∂y – ∂u , X5 = ∂y , Af = f ∂x – f ′ ∂v , Bg = g∂z – g ′ ∂v , 2 2 2   1 1 Ch = h(–z∂x + x∂z ) + zh′ – wh ∂w – xh′ – vh ∂w , 2 2 where f (t), g(t), h(t) are arbitrary smooth functions. Having functional coefficients, these vector fields produce a Lie pseudo-group L(G). This Lie pseudo-algebra L(G) has a 5-subalgebra h generated by v1 , . . . , v5 , and an infinite dimensional ideal i generated by Af , Bg , Ch . Therefore L(G) ≃ h × i. To having a reduction in BS, a Lie subgroup of the above pseudo-group is chosen. For executing this chose, simpler forms for every one of the coefficients in the previously mentioned vector fields are chosen. 3 Group invariant solutions of BS In order to have the group of transformations which are generated by vector fields vi , i = 1, . . . , 5, we need to at first solve first order system including the first order equation which is in agreement whit each of the same generators. If k (s) is the parametric group represented by vk , k = 1, . . . , 5, then Pk = k (s)(t, x, y, z, u, v, w) are, respectively,   2 u sy v sx w sz P1 = at, ax, ay, az, + , + , + , a= ; a 4 a 4 a 4 st + 2   P2 = [t + s, x, y, z, u, v, w]; P3 = es/2 t, x, y, z, e–s u, e–s v, e–s w ;   s P5 = [t, x, y + s, z, u, v, w]; P4 = t, x, y + st, z, u – , v, w ; 2 Page 4 of 17 Alimirzaluo et al. Advances in Difference Equations and if h , g , f ( 2021) 2021:60 Page 5 of 17 are the 1-parameter group generated by Ah , Bg , Cf , respectively, then     s s g = t, x, y, z + sg, u, v, w – g ′ ; h = t, x + sh, y, z, u, v – h′ , w ; 2 2  2 s 2 zf , z, u, g = t, x – szf , y + sxf – 2  g  cos(sf )v – sin(sf )w + z(z + 1) sin(sf ) – x cos(sf ) + x – szf , 2f   g  sin(sf )v + cos(sf )w + –x sin(sf ) + z(1 – z) cos(sf ) + z2 + z . 2f It should be mentioned that, generally, for each, a parameter of the subgroup of complete symmetry group’s system offers a set of invariant solutions [35, 40, 41]. Theorem 3.1 If s be a small real number, (u(t, x, y, z), v(t, x, y, z), w(t, x, y, z)) be a solution of the BS and i = 1, . . . , 5, there are the functions ϕi := (U, V , W ) = k (s)(u, v, w): ϕ1 = a.(u, v, w)(at, ax, ay, az) – ϕ2 = (u, v, w)(t – s, x, y, z), as .(x, y, z), 2 a= 2 , st + 2   ϕ3 = e–s/2 .(u, v, w) te3s/2 , x, y, z , s ϕ4 = (u, v, w)(t, x, y – st, z) + .(1, 0, 0), 2 ϕ5 = (u, v, w)(t, x, y – s, z). Furthermore, if h , g , f are the 1-parameter group generated by Af , Bg , ch , respectively, and u = u(t, x, y, z), v = v(t, x, y, z) and w = w(t, x, y, z) are a solution of the BS, so are the functions φh := (U, V , W ) = h (u, v, w), ξg := (U, V , W ) = g (u, v, w) and θf := (U, V , W ) = θf (u, v, w):   sh′ (t) .(0, 1, 0), ψh = (u, v, w) t, x + sh(t), y, z + 2   sg ′ (t) ξg = (u, v, w) t, x, y, z + sg(t) + .(0, 0, 1), 2           θf = u, cos sf (t) v + sin sf (t) w, – sin sf (t) v + cos sf (t) w  s2 t, x – szf (t), y + sf (t)x – zf (t)2 , z 2     g(t)  . 0, –z(z + 1) sin sf (t) + x cos sf (t) 2f (t)      + x + sf (t)z, x sin sf (t) + z(1 – z) cos sf (t) + z2 + z . + This theorem has several useful consequences. For example, by using the command PDETools[TWSolutions] of MAPLE 2016© for the system BS, we find a seven parametrized set of tanh-solutions for BS as u = a3 tanh(a2 x + a3 y + a4 z + 2a5 t + a1 ) + a6 , v = a2 tanh(a2 x + a3 y + a4 z + 2a5 t + a1 ) + a7 , (1) Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 w = a4 tanh(a2 x + a3 y + a4 z + 2a5 t + a1 ) – Page 6 of 17 a2 a7 + a3 a6 – a5 , a4 where c1 , . . . , c7 ∈ IR are arbitrary numbers with c4 = 0. Now, we find a large set of solutions of BS by using Theorem 3.1. For example, by the ψh of Theorem 3.1 for (1), we find   u = a3 tanh a2 h(t) + 2a5 t + a2 x + a3 y + a4 z + a1 + a6 ,   h′ (t) , v = a2 tanh a2 h(t) + 2a5 t + a2 x + a3 y + a4 z + a1 + a7 – 2  a2 a7 + a3 a6 – a5  , w = a4 tanh a2 h(t) + 2a5 t + a2 x + a3 y + a4 z + a1 – a4 where c1 , . . . , c7 ∈ R are arbitrary numbers with c4 = 0, and h is an arbitrary smooth function of t. 4 Optimal system of 1-subalgebras In this part, we take the advantage of symmetry group in order to obtain the OS (optimal system) of 1-subalgebras of BS. Regarding the fact that every linear combination of symmetries is a symmetry, there will be an endless number of 1-subgroups for G. Therefore, determining the subgroups which give different types of solutions is emphasized. So, we need to look symmetry transformations for invariant solutions which are unable to be to convert to each other in the full symmetry groups. This, in turn, leads to the notional of an OS of subalgebra. The problem of classifying this 1-subalgebra is identical to the problem which exists in the classification of the orbits of the adjoint representation [36, 40]. The optimal set of subalgebras is achieved by selecting just one representative from any class of equivalent subalgebras. By using a general member in the Lie algebra and simplifying it via diverse adjoint transformations, it is possible to solve the problem of orbits classifications. The following Lie series are considered to include the adjoint representation [38]:   s2 Ad exp(s.Xi ).Xj = Xj – s.[Xi , Xj ] + .[Xi , [Xi , Xj ]) + · · · , 2 where s is the group parameter and adXi Xj = [Xi , Xj ] is the Lie algebra communicator and i, j = 1, . . . , 9 An adjoint action is considered for Lie algebra L(G) so we consider the following theorem, Theorem 4.1 The OS of 1-subalgebras for BS is (1) X4 , (2) X1 + X2 , (3) X2 – X4 , (4) X1 + X5 , (5) X5 , (6) X1 – X2 , (7) X2 + X4 , (8) X1 , (9) X3 , (10) X2 . [20]. Proof Let Fis : L(G) → L(G) be the linear map, by X −→ Ad(exp(sXi )X), i = 1, . . . , 5. The matrix Mis of Fis , i = 1, . . . , 5, with respect to the basis {X1 , . . . , X5 } is M1s = I + s1 s21 E12 – s1 E13 – s1 E32 + E45 , 2 2 M4s = I – s4 E43 – s4 E52 , 2 Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 M2s = I + 2s22 E22 + s2 E23 + s2 E31 + s2 E54 , Page 7 of 17 M5s = I + s5 (2E41 – s5 E53 ), 4 M3s = e–s3 (E11 – E66 ) + es3 /2 (E44 – E55 ), respectively. Let X = 5 i=1 ai Xi , s then (F55 ◦ F4s4 ◦ · · · ◦ F1s1 )(X) is   1 1 2 1 s s2 – s1 1 + s21 s22 – s1 s2 es3 a1 + s21 e–s3 a6 + 4 2 2 1   1 1 –s3 /2 s6 /2 + (( s4 s2 e a5 a1 – e s4 a2 + – s2 es3 /2 s4 + 2 2  + s2 es3 /2 a4 + e–s3 /2 a5 X5 .  a 3 X1 + · · · 1 –s3 /2 e s5 a3 2 We can simplify X as follows: • If a5 = a3 = a2 = a1 = 0, then X is decreased to the case (1). • If a5 = a2 = a1 = 0 and a3 = 0, so we can get the coefficient of X4 vanish by using 2a /a F4 4 3 . In this case is reduced to the case (9). • If a5 = a3 = a2 = 0 and a1 = 0, so we can get the coefficient of X4 vanish by using F52a4 /a1 . In this case is reduced to the case (8). s • If a5 = a4 = a3 = 0 and a2 = 1, then coefficient of X1 can be vanished or be ±1 by F33 for s3 = – ln |1/a1 |. So X is reduced to the case (2, 6). s • If a5 = a3 = a1 = 0, then we can make the coefficient of X4 vanish or be ±1 by using F33 for s3 = (2/3) ln |a2 /a4 |. In this case is reduced to the case (3, 7). • If a1 = a3 = a2 = 0 and a5 = 0, then we can assume that a5 = 1 so we can get the coefficient of X4 vanish by using F12a4 . In this case it is reduced to the case (5). • If a5 = a2 = a1 = 0, then we can suppose that a1 = a5 = 1, so we can get the coefficient –a of X4 and X3 vanish by using F12a4 and F2 3 . In this case is to the case (4). • If a1 = a3 = 0 and a5 = a2 = 0, then we can make the coefficient of X4 vanish by using F12a4 , and X5 vanish by using F41/a2 . In this case is decreased to the case (10).  5 OS of 2-subalgebras Here, we get OS of 2-subalgebras for BS, choose X 1 or X 2 , as an element of the OS of 2subalgebras in Theorem 3.1 and regard X = c1 v1 +· · ·+c5 v5 as an optimal vector field where c5i s are smooth functions of (t, x, y, z, u, v, w). By such selection, we have [vi , X] = λX 1 + MX. The following system (2) is a computation of both sides of these equations: Cjki aj ak = λai + μai , i = 1, . . . , 5. (2) The elements of the OS of 2-subalgebras are reached by solving the system of linear equations for each choice of the OS of 1-subalgebras from the Theorem 3.1. When these elements are in the combination form, they could be simplified as a 1-dim case by acting the adjoint matrices to each of them and the following theorem is suggested. Theorem 5.1 An OS of 2-subalgebras from the BS is (1) X1 , X2 + X3 , (2) X1 , X3 – X4 , (3) X1 , X3 + X4 , (4) X3 , X5 – X2 , (5) X3 , X1 – X5 , (6) X3 , X1 + X5 , Alimirzaluo et al. Advances in Difference Equations (7) X3 , X4 + X2 , ( 2021) 2021:60 (8) X1 , X3 + X4 , Page 8 of 17 (9) X2 , X3 + X5 , (10) X3 , X4 – X2 , (11) X1 , X3 – X4 , (12) X3 , X5 , (13) X4 , X2 + X5 , (14) X3 , X1 + X4 , (15) X1 , X4 , (16) X4 , X2 – X5 , (17) X3 , X1 – X4 , (18) (19) X3 , X5 + X2 , (20) X4 , X3 , (22) X3 , X2 , X2 , X5 , (21) X4 , X5 , (23) X3 , X1 . Proof Each 2-subalgebra needs two generators. By choosing one of the generators of the OS of 1-subalgebras that have explained in the previous theorem and the second one arbitrary, 2-subalgebras can be classified. let h = Span{X, Y } is a 2-subalgebra where X is a 1subalgebra that is selected from the 1-subalgebra list and Y is an arbitrary vector described by Y = b1 v1 + · · · b5 v5 . Now, we need to simplify h as much as possible by implementing various adjoint transformations on it and proceed algebraically [38]. Every adjoint transformation is a linear map Flsi : L(G) → L(G) defined by X −→ Ad(exp(svi )X) for i = 1, . . . , 5, we only illustrate one of the cases in the following: If X = v1 + av2 then 5 h = X, Y  = X1 + aX2 , b i Xi i=1 = X1 + aX2 , b1 X1 + b2 X2 + b3 X3 + b4 X4 + b5 X5  So, we have: • If b3 = b4 = b5 = 0 then we have h = X1 , b2 X2  = X1 , X2  and [X1 , b2 X2 ] = b3 X3 = rX1 + s(b2 X2 ) for any r, s ∈ RR. Since h, is not closed under the Lie bracket, So there are no two-dimensional subalgebras in this case. • If b5 = 0 and b4 = 0 so we can get the coefficient of X2 vanish by F2s2 ; By setting s3 = (–2/3) ln |b4 /b2 |. Then we have h = X1 , b3 X3 + b4 X4  and [X1 , b3 X3 + b4 X4 ] = –b3 X1 . Therefore h is closed under the Lie bracket and we have the case X1 , X3 ± X4 . • If b5 , b4 = 0 and a3 = 0. Then we have h = X1 , b2 X2 + b3 X3  and [X1 , b2 X2 + b3 X3 ] = b2 X3 + b1 X1 So h is closed under the Lie bracket and we have the case X1 , X2 + X3 . s • If b3 = 0 and b5 = 0 and b5 = 1 so we can get the coefficient of X4 vanish by F55 ; By setting s5 = –2a4 also, we have scaling if necessary, we can assume that b5 = 1. Then we have h = X1 , b2 X2 + X5  and [X1 , b2 X2 + X5 ] = b2 X3 + b5 X4 . So, h is not closed under the Lie bracket and we have no two-dimensional subalgebras in this case. It should be noted that there is no other way to check, and in each case the h is simplified as much as possible. In the same way, two-dimensional algebras can be obtained for other states.  6 OS of 3-subalgebras In this section, we get OS of 3-subalgebras for BS. To find 3-subalgebras, we must consider one 3-subalgebra as h = X, Y , Z of symmetry group, so that Z = 5i=1 ci vi . Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Page 9 of 17 Table 1 Some similarity reduced equations OPi P Ui Similarity reduced equations X3 , X2 , X5 z x xu xv xw X1 , X3 , X4 z x x(2tu+y) 2t x(2tv+x) 2t x(2tw+z) 2t ϕ ′′ + ϕ ′′ p2 + 2pϕ ′ η + 4pϕ ′ + 2ϕ = 2pϕ ′ ψ + 2ψϕ ϕ′p + ϕ = 0 ϕ′ = 0 ϕ ′′ + p2 ϕ ′′ + 2pϕ ′ η + 4pϕ ′ + 2ϕ = 2pϕ ′ ψ + 2ψϕ ϕ′p + ϕ = 0 ϕ′ = 0 To classify 3-subalgebras, we need to be chosen two of the generators from the OS of 2-subalgebras, and another generator should be taken arbitrary, then we should check that [X, Z] and [Y , Z] are closed under the Lie bracket. Theorem 6.1 The OS of 3-subalgebra from the BS are (1) X2 , X3 , X5 , (2) X2 , X3 + X5 , X1 , (3) X2 , X3 , X4 , (5) X5 , X2 , X1 , (6) X3 , X1 , X4 + X5 , (4) X2 , X3 + X5 , X4 , (7) X2 , X3 + X5 , X1 + X4 , (8) X3 , X1 + X5 , X4 , (10) X1 , X3 + X4 , X5 , (11) (13) X3 , X1 , X2 , (16) X3 , X1 + X4 , X2 , (17) X3 , X1 , X4 , (19) X1 , X3 + X4 , X2 , (20) X3 , X5 + X2 , X1 , (22) X3 , X5 + X2 , X4 . (14) X3 , X1 , X5 , (9) X3 , X1 + X5 , X2 , (12) X3 , X4 + X2 , X1 , X3 , X4 + X2 , X5 , (15) X3 , X4 , X5 , (18) X3 , X1 + X4 , X5 , (21) X1 , X4 , X2 , 7 Similarity reduction of BS If expressed in the new coordinates, BS is reduced. The BS is presented in the coordinates of (t, x, y, z, u, v, w) and we have to look for this equation’s form in the appropriate coordinates in order to make it reduced. These new coordinates will come in hand by searching independent invariants (p, ϕ, ψ, η) which correspond to the generators of the symmetry group. Hence, If we use the new coordinates and apply chain derivative role into account, we can have the reduced equation. We introduce the present procedure for one of the infinitesimal generators in the OS of Theorem 6.1 and provide the list of results for some other instances. This equation has an independent variable, named p, and three dependent variables labeled as (ϕ, ψ, η). Similarly, it is possible to compute all of the identically reduced equation which correspond to the infinitesimal symmetries that were mentioned in Theorem 6.1. Some of them are listed in Tables 1 and 2. Then we can reduce some equations obtained in the previous section to ODEs, ϕ ′′ + p2 ϕ ′′ + 2pϕ ′ (η – ψ + 2) + 2ϕ(1 – ψ) = 0, pϕ ′ + ϕ = 0, ϕ ′ = 0. Therefore, ϕ = 0, ψ = ψ(z/x), η = η(z/x). We have a non-trivial solution of equation BS: u = 0,  1 z v= ψ , x x  1 z w= η . x x Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Page 10 of 17 Table 2 The commutator table of L(G) [·, ·] X1 X2 X3 X4 X5 Af̃ Bg̃ Ch̃ X1 X2 X3 X4 X5 Af Bg Ch 0 X3 X1 0 1 2 X4 –A(tf ′ –f )/2 –B(tg′ –g)/2 –Ct2 h′ /2 –X3 0 0 –X5 0 –Af ′ –Bg′ –Ch′ –X1 0 0 – 12 X4 1 2 X5 –Atf ′ –f /2 –Btg′ –g/2 –Cth′ 0 X5 1 2 X4 0 0 0 0 0 – 12 X4 0 – 12 X5 0 0 0 0 0 At(tf̃ ′ –f̃ )/2 Af̃ ′ Atf̃ ′ –f̃ /2 0 0 0 0 –Bhf̃ B(tg̃′ –g̃)/2 Bg̃′ Btg̃′ –g̃/2 0 0 0 0 Ahg̃ Ct2 h̃′ /2 Ch̃′ Cth̃′ 0 0 Bh̃f –Ah̃g 0 By applying Theorem 3.1 we are able to obtain a new solution of the equation as follows. Theorem 7.1 Let h, g, ϕ, ψ are arbitrary smooth functions and s be a real number. Then, each of the sets (1) u = 0,  1 z v= ϕ , x x  1 z w= ψ , x x –sx + (4/x)ϕ(z/x) –sz + 4(xψ(z/x))–1 v= , w= ; 2st + 4 2st + 4   1 1 z z , w= ψ ; ϕ x x x x   sh′ (t) sg ′ (t) 1 1 z z + sg(t) + + ϕ , w= ψ , v= x + sh(t) x + sh(t) 2 x x 2 –sy , 2st + 4 s v= (3) u = , 2 (2) u = (4) u = 0, are solutions of the equation BS. 8 Nonclassical symmetries of BS A kth order system of differential equations is naturally treated as a submanifold E ⊂ J k of the kth order jet space on the space of independent and dependent variables. Consider a kth order system E of differential equations [21]:   ν x, u, u(k) = 0, ν = 1, . . . , l, (3) involving x = (x1 , . . . , xn ) and u = (u1 , . . . , uq ) as independent and dependent variables, respectively, and with u(k) denoting the derivatives of the u, s with respect to the x, s up to order k. Suppose that V is a vector field on the space Rn × Rq of independent and dependent variables: q n ξ i (x, u)∂xi + V= i=1 φj (x, u)∂uj . (4) j=1 A n-dimensional submanifold of RRn × RRq of the space of independent and dependent variables is defined by a map of the solution uα = f α (x1 , . . . , xn ) = 0, α = 1, . . . , q, (5) Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Page 11 of 17 to the system. Then we must add the invariant surface conditions of these equations: n   Qα x, u, u(1) = ϕ α (x, u) – ξ i (x, u) i=1 ∂uα = 0, ∂xi α = 1, . . . , q, (6) where Q = (Q1 , . . . , Qn ) is known as the characteristic of the vector field (5). The kth prolongation of the invariant surface condition (6) will be denoted by EQk . If the kth prolongation of the V (k) vector field V is tangent to the intersection E ∩ EQk then the system (3) and (6) are compatible: V k (△ν ) |E∩Ek = 0, Q ν = 1, . . . , l. (7) The vector field (4) is called a nonclassical infinitesimal symmetry of the system (3) If Eqs. (7) are satisfied [30]. For finding the nonclassical symmetries, according to the system E of BS we do as follows. Case 1: If we suppose that the coefficient of ∂t in the (4) is not equal to zero, we can set coefficient of ∂t equal to 1 without changing the totality, then for the vector field V = ∂t + ξ ∂x + η∂y + ζ ∂z + φ∂u + ψ∂v + θ ∂w , the invariant surface conditions are as follows: (φ, ψ, θ ) = (ut + ξ ux + ηuy + ζ uz , (8) vt + ξ vx + ηvy + ζ vz , wt + ξ wx + ηwy + ζ wz ). According to Eq. (7), we can find the variables ut , . . . , vt , . . . , wt from (BS) and (4): ut = –2ζ uwy + 2ηvvy + 2ηwwy – 2ξ uvy + ηuxx + ηuyy + ηuzz + 2φu , η + 2u ζ wy + ξ vy + 2vvy + 2wwy – φ + uxx + uyy + uzz , η + 2u ux = vy , uy = uz = wy , vt = –ζ vz – ηvy – ξ vx + ψ, wt = –ζ wz – ηwy – ξ wx + θ . By using nonclassical methods for the BS, we have the following determining equations: ηx = ηz = ηu = ηv = ηw = 0, ζy = ζu = ζw = 0, ψy = ψu = ψwz = ψww = ψzz = 0, θy = θu = θ zz = θww = θwz = 0, φx = φz = φv = φw = 0, ξy = ξu = ξv = ξw = 0, ψx = θ z , θv = –ψw , θx = –ψz , φy = θ z , ξz = ψw , θzt = 2θw θz , ηy = –θw , ξx = –θw ,   2 θwt = 2 θw + θz , ζz = –θw , ψv = θ w , φu = θ w , ζx = –ψw , ψwt = 2(θw ψw – ψz ), ξt = 2(wψw + ξ θw + vθw – ψ), ηt = 2(ηθw + uθw – φ), ζt = 2(ζ θw + wθw – vψw – θ ), φt = 2(φθw + uθz ). Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Page 12 of 17 Therefore we have the solutions (ξ , η, ζ , φ, ψ, θ ) =  1 2tx – z + 1, –1 + (2y – 1)t, 2tx – z + 1, 2t 2 + 1 1 – 2tu – y, –2tv – x – w, –2tw – z + v . 2 Case 2: Now we suppose that the coefficient of ∂t in (4) is equal to zero and try to obtain the infinitesimal nonclassical symmetries of the form V = ∂x + η∂y + ζ ∂z + φ∂x + ψ∂v + θ ∂w . So we have the category of solutions η = 1, φ = ψ = –1/2 and θ = ζ = 0. Case 3: Again we suppose that the coefficient of ∂x in (4) is equal to zero and try to obtain the infinitesimal nonclassical symmetries of the form V = ∂y + ζ ∂z + φ∂x + ψ∂v + θ ∂w . So we obtain the solutions φ = –1/2 and ψ = θ = ζ = 0. Case 4: In the end, we assume that the coefficient of ∂y in (4) is equal to zero and try to obtain the infinitesimal nonclassical symmetries of the form V = ∂z + φ∂x + ψ∂v + θ ∂w . Then we have φ = ψ = 0, and θ = F(t, x, z, w), where F is an arbitrary function. 9 Ansatz solutions of BS Now we consider the most important ansatz method (specially, the tanh-function method, [16, 34, 50]) for constructing exact traveling wave solutions of this nonlinear system of PDEs. For this, we introduce a new variable τ = tanh(c0 + c1 t + c2 x + c3 y + c4 z) and the ansatz u = A1,0 + A1,1 τ , v = A2,0 + A2,1 τ and w = A3,0 + A3,1 τ , where Ai,k and ci are arbitrary constants. Substituting expansions into the BS equations, we obtain the following system of algebraic equations: c3 w′ (τ ) – c4 u′ (τ ) = 0, c3 v′ (τ ) – c2 u′ (τ ) = 0,  2 2 2  2  c2 + c3 + c4 τ + 2τ – 1 u′′ (τ ) = 0,   2c3 u(τ ) – 2c2 v(τ ) + 2c4 w(τ ) – c1 u′ (τ ) = 0. Finally, we obtain three sets of exact solutions, with linear algebra and required simplifications: (1) u = a1 , v = a2 , (2) u = a3 tanh θ + a4 a5 , w = a3 , v = a2 tanh θ + a4 a7 , w = a4 tanh θ + a1 – a2 a7 – a3 a5 , (3) u = a5 a6 + a7 sinh(2θ ) , sinh2 θ + cos2 (αx) w = a2 – a4 a6 + where α =  v = a8 + a7 α sinh(2αx) , sinh2 θ + a4 cos2 (αx) a5 a7 sinh θ , sinh2 θ + a4 cos2 (αx) a24 + a25 , θ = a1 + 2a2 a5 t + a4 y + a5 z and the ai are constants. Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Figure 1 For solution of (2) u, v, ci = 1, t = 1, z = –10, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Figure 2 For solution of (2) u, v, ci = 1, t = 1, z = 0, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Figure 3 For solution of (2) u, v, ci = 1, t = 1, z = 10, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 10 Results and discussion The graphical representation of BS equations produces the substantial information to interpret the phenomena physically. This section deals with a physical interpretation of the solutions given via choosing the appropriate amounts of parameters, the graphic display of traveling wave solution is presented in Figs. 1 to 12 including 3D plot, density plot, and 2D plot when three spaces arise at spaces z = –10, z = 0, and z = 10 for u, v and z = –20, z = 0, and z = 20 for w. The solutions contain various arbitrary constants and functions and their appropriate choices are crucial to describe the significant behavior of the phenomena. The simulation is performed on MATLAB for Figs. 1–12, when x = –3, 0, 3. Page 13 of 17 Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Figure 4 For solution of (2) w, ci = 1, t = 1, z = –20, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Figure 5 For solution of (2) w, ci = 1, t = 1, z = 0, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Figure 6 For solution of (2) w, ci = 1, t = 1, z = 20, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Figure 7 For solution of (3) u, v, w, ci = 1, t = 1, z = –10, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Page 14 of 17 Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Figure 8 For solution of (3) u, v, w, ci = 1, t = 1, z = 0, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Figure 9 For solution of (3) u, v, w, ci = 1, t = 1, z = 10, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Figure 10 For solution of (3) u, v, w, ci = 0, t = 1, z = 10, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Figure 11 For solution of (3) u, v, w, ci = 0, t = 1, z = 10, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Page 15 of 17 Alimirzaluo et al. Advances in Difference Equations ( 2021) 2021:60 Figure 12 For solution of (3) u, v, w, ci = 0, t = 1, z = 10, (f1) 3D plot (f2) density plot and (f3) 2D plot with spaces x = –3, 0, 3 Conflict of interest The authors declare that they have no conflict of interest. Acknowledgements Not applicable. Funding This work was not supported by any specific funding. Availability of data and materials The data sets supporting the conclusions of this article are included within the article and its additional file. Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors equally made contributions to this work. EA and MN made the numerical simulations and wrote the article. JM and EA provided the group invariant solutions of BS and solved the problem with it in Sect. 3. MN and JM provided Sects. 4–6 in the paper. Also, Sects. 7–9 have been provided by EA and MN. The remain sections have been written by all authors. The authors read and approved the final manuscript. Author details 1 Department of Pure Mathematics, School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran. 2 Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran. 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