EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6030 ISSN 1307-5543 – ejpam.com Published by New York Business Global Symmetries and Novel Exact Solutions for (2+1)-D QZK equation via Lie-Symmetry and Kudryashov-Auxaliry Method Ahmed A. Gaber1,∗, Tawfik M. Younis2, Mona F.Alharbi3 1Department of Mathematics, College of Science El-Zulfi, Majmaah University, Majmaah 11952, Saudi Arabia 2 Department of Business Administration, College of Business Administration, Majmaah University, Majmaah 11952, Saudi Arabia 3 College of Administrative Sciences and Humanities, Mustaqbal University, Buraydah, Saudi Arabia 2 Department of Operations Research Faculty of Graduate Studies of Statistical Research, Cairo University, Egypt Abstract. In this paper, the (2+1)-D quantum Zakharov-Kuznetsov (QZK) equation that de- scribes how nonlinear ion-acoustic waves diffuse in magnetized plasma is studied. Firstly, The governing equation was transformed into a number of ordinary differential equations using symme- try analysis. After that, We used Kudryashov-Auxaliry Method (KAM) to develop a new kind of accurate answers for the QZK equation. The discovered solutions included a number of arbitrary constants that improved their dynamic characteristics. The resulting solutions represent solitary wave, single wave, and multisolitons solutions and include hyperbolic and trigonometric functions. 2020 Mathematics Subject Classifications: 35C05, 35Q60, 35A30, 35B06 Key Words and Phrases: Lie-Symmetry method, Quantum Zakharov-Kuznetsov, Kudryashov- Auxaliry Method, Wave solutions 1. Introduction Over the past centuries, scientists have sought to understand and explain many natural and physical phenomena. The discovery of partial differential equations (PDEs) has led to the explanation of many of these events. Many scientists have turned to discovering the types of PDEs and their approximate and accurate solutions. This has helped in the ease of understanding and developing many models and how to use them in scientific development. Through that, mathematicians have made remarkable progress in inventing ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6030 Email addresses: a.gaber@mu.edu.sa; aagaber6@gmail.com (A. A. Gaber), tawfik younis@hotmail.com (T. M. Younis), monaf.alharbi@gmail.com (M. F.Alharbi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. A. Gaber, T. M. Younis, M. F. Alharbi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6030 2 of 12 many methods to solve nonlinear PDEs for explaining many physical phenomena. Many nonlinear PDEs have been used in the fields of engineering, physics, and other scientific and social sciences. Modern science has produced many equations, such as Korteweg-de-Vries [1, 2] , Burgers [2] , Boltzmann [4] and other famous equations. On the other hand, scientists have discovered and developed many methods to solve nonlinear partial differential equations and find accurate or approximate solutions. These methods include Lie-Symmetry method [5− 9], Kudryashov method [7, 10], Exp-function method [11], tan(φ/2)-expansion method [12], Jacobi elliptic function [13] , (G′/G)- expansion method [14, 15] and others [16− 19]. The main challenge of this work are studying symmetries and obtaining a novel exact solutions for the (2+1)-dimensional QZK equation [15, 20 − 22], which can be written in the form vt + pv vx + q(vx,x,x +vy,y,y) + r(vx,y,y + vx,x,y) = 0, (1) where v = v(x, y, t). Many researchers have been interested in this equation [20 − 22]. Nuruddeen et al. [21] reduced the governing equation to a single form and then used tanh method to obtain solutions. On the other hand, Vinita and Ray [20] reduced the governing equation and obtained some solutions 2. Symmetries Proposition 1: The QZK equation (1) has the following five Lie point symmetries: W1 = t ∂ ∂t + 1 3 x ∂ ∂x + 1 3 y ∂ ∂y − 2 3 v ∂ ∂v , W2 = ∂ ∂t , W3 = t ∂ ∂x + 1 a ∂ ∂v , W4 = ∂ ∂x , W5 = ∂ ∂y , (2) Proof : A Lie group with infinitesimal on the space of independent and dependent variables with one parameter ε is examined as follows: t∗ = t+ εA(x, y, t, v) + ϑ(ε2), x∗ = x+ εB(x, y, t, v) + ϑ(ε2), y∗ = y + εC(x, y, t, v) + ϑ(ε2), v∗ = v + εΦ(x, y, t, v) + ϑ(ε2). (3) The third vector field which can generated the Lie algebra of the (2+1)-dimensional QZK equation can be expressed as follows: Γ(3) = χ+Φ[x] ∂ ∂vx +Φ[xxx] ∂ ∂vx,x.x +Φ[yyy] ∂ ∂vy,y,y +Φ[xxy] ∂ ∂vx,x,y +Φ[xyy] ∂ ∂vx,y,y . (4) A description of the infinitesimal vector χ associated with the aforementioned trans- formations is given. χ = T ∂ ∂t +A ∂ ∂x +B ∂ ∂y + C ∂ ∂v . (5) A. A. Gaber, T. M. Younis, M. F. Alharbi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6030 3 of 12 where the components Φ[t],Φ[x], Φ[xxx], Φ[xxy], Φ[yxx].... establish as the expressions: Φ[x] = DxΦ− vtDxA− vxDxB − vyDxC, Φ[xy] = DyΦ− vtxDyA− vxxDyB − vxxDyC. (6) The invariance condition is satisfied [5, 6] Γ(3)(∆) = 0, (7) where ∆ = vt + pv vx + q(vx,x,x +vy,y,y) + r(vx,y,y + vx,x,y) = 0, (8) This invariance condition leads to a specific system of PDEs. When we solve this condition, we get A = c1t+ c2, B = 1 3 x+ c3t+ c4, C = 1 3 c1y + c5, Φ = −2 3 c1v − 1 a c3. (9) where Ω is sum of W1, ...W5. The commutator relations are given by Table 1. Table 1: The commutator table of Ω W1 W2 W3 W4 W5 W1 0 −1 3 W2 0 −1 3 W4 −1 3 W5 W2 1 3W2 0 −W3 W4 0 W3 0 W3 0 0 0 W4 1 3W4 −W4 0 0 0 W5 1 3W5 0 0 0 0 Table 2: The adjoint table of Ω W1 W2 W3 W4 W5 W1 W1 e ϵ 3 2W2 W3 e ϵ 3 4W4 e ϵ 3 5W5 W2 W1 − 1 3W2 W2 eϵ3W3 e−ϵ 4 W4 W5 W3 W1 W2 −W3 W3 W4 W5 W4 W1 − 1 3W4 W2 −W4 W3 W4 W5 W5 W1 − 1 3W5 W2 W3 W4 W5 where Adj(exp(εWi))Wj = Wj−ϵ[Wi,Wj ]+ ϵ2 2 [Wi, [Wi,Wj ]], i = 1, 2, 3 and [Wi,Wj ] = WiWj −WjWi. The asymmetric probability are derived from the preceding table under the following scenarios: A. A. Gaber, T. M. Younis, M. F. Alharbi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6030 4 of 12 (I) W1 +m W2 (II) W1 +W3 (III) W2+ m1W4 +m2W5 (IV) W4 +mW5 3. The reductions and exact solutions The constant transformation can be obtained by applying the characteristic equation that follows: dt c1t+ c2 = dx 1 3x+ c3t+ c4 = dy 1 3c1y + c5 = du −2 3c1v − 1 ac3 . (10) Case I: Substituting W1,and W2 into Eq.(10) correspondingly, the invariant variables are ζ1 = x (t+m) 1 3 , ζ2 = y (t+m) 1 3 , v = F (ζ1, ζ2) (t+m) 2 3 , (11) where m = c3. Utilizing Eq.(11) into Eq.(1), then Eq.(1) is reduced to the posterior equation −2F − ζ1Fζ1 − ζ2Fζ2 + 3pFFζ1 + 3q(Fζ1ζ1ζ1 + Fζ2ζ2ζ2) + 3r(Fζ1ζ2ζ2 + Fζ2ζ1ζ1) = 0. (12) In this case, by putting θ = hζ1 + kζ2, Eq.(12) can be written in the following form: −2F − θF ′ + 3pFF ′ + 3[q(k3 + h3) + r(k2h+ kh2)]F ′′′ = 0. (13) Taking the solution to the previous equation in the following form: F = a0 + m∑ i=1 (aiθ i + biθ −i) (14) Substituting Eq.(14) into Eq.(13) and obtainig the arbitrary constant a0, ai and bi .The closed form solution of Eq.(13), takes the following expression: F = −1 pk (ζ1 + ζ2). (15) The general solution of Eq. (1) writtes in the form: v(x, y, t) = −1 pk(t+m) 2 3 ( k x (t+m) 1 3 + h y (t+m) 1 3 ) (16) Case II: Similarly, in the earlier case, substituting W1,and W3 in Eq.(2). The form of the invariant variables as follows ζ1 = x− 2 3 t t 1 3 , ζ2 = y t 1 3 , v = 1 t 2 3 F (ζ1, ζ2). (17) A. A. Gaber, T. M. Younis, M. F. Alharbi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6030 5 of 12 Figure 1: The solitary wave solution of (16). Figure 2: The Double wave solution of (21) We obtained the following equation by substituting Eq. (17) into Eq. (1). −2F − ζ1Fζ1 − ζ2Fζ2 + 3pFFζ1 + 3q(Fζ1ζ1ζ1 + Fζ2ζ2ζ2) + 3r(Fζ1ζ2ζ2 + Fζ2ζ1ζ1) = 0. (18) We take θ = kζ1 + hζ2, then Eq.(18) takes the form −2F − θF ′ + 3pFF ′ + 3[q(k3 + h3) + r(k2h+ kh2)]F ′′′ = 0. (19) The following formula represents the closed form solution of Eq. (13): F = −1 pk (ζ1 + ζ2). (20) The exact solution of Eq. (1) writtes in the form: v(x, y, t) = −1 pkt 2 3 ( k (x− 3 2 t) t 1 3 + h y t 1 3 ) (21) Case III: The vectors ofW2,W4 andW5 are substituted in the characteristic equation, then the following is the form of the invariant variables. ζ1 = x−m1t, ζ2 = y −m2t, v = F (ζ1, ζ2), (22) A. A. Gaber, T. M. Younis, M. F. Alharbi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6030 6 of 12 Utilizing the relations of Eq.(22) to reduce Eq.(1) in the form −m1Fζ1 −m2Fζ2 + pFFζ1 + q(Fζ1ζ1ζ1 + Fζ2ζ2ζ2) + r(Fζ1ζ2ζ2 + Fζ2ζ1ζ1) = 0. (23) We take θ = kζ1 + hζ2, then Eq.(23) takes the form (−m1k −m2h)F ′ + pkFF ′ + [q(k3 + h3) + r(k2h+ kh2)]F ′′′ = 0. (24) We employ Kudryashov-Auxaliry Method [23], which is expressed as follows: F (θ) = A0 + m∑ i=1 Ai [1 + Ψ(θ)]i , (25) where Ψ(θ) satisfies the following auxaliry equation [23] Ψ′(θ) = R+Q Ψ2(θ) + P Ψ4(θ), (26) Balncing between linear term F ′′′ and non linear term FF ′, we get F (θ) = A0 + A1 1 + Ψ(θ) + A2 [1 + Ψ(θ)]2 . (27) Substituting (27) into (24) and equating the coefficients of all powers of Ψ(θ) to zero. Utilizing Maple for solving the algebraic equations of Ai, we get: A1 = 6 pm1k (2qk3P + 2rk2hP + 2rkh2P + rkh2Q+ qh3Q+ 2qh3P + qk3Q+ rk2hQ), R = −(P +Q) m2 = 1 h (6qk3P + 6rk2Ph(1 + h) + rkhQ(k + h) + qh3Q+ 6qh3P + pm1kA0 −m1k + qk3Q). (28) The appropriate solitary wave solutions of Eq.(2) yield: Case 1: P = 1, Q = 1, R = 0, v(x, y, t) = A0 + 6h2(qm3 + 2qm3 − q − 2q −m2r − 2m2r + 2rm+ rm) pm[1 + csc (k(x−m1) + h(y −m2))] . (29) where A0= −1 pmh(−6h3(rm2 + h3q) + h3qm3 + 6h3qm(m2 + 1) + hQrm(1−m)− h3q − k). Case 2: P = 1, Q = −1, R = 1, v(x, y, t) = A0 + h2(qm3 − 2qmP − q + 2q −m2r + 2m2r − 2rm+ rm) pm[1 + tan (k(x−m1) + h(y −m2))] . (30) where A0= −1 pmh(6h 3(rm2 + h3q) + h3qm3 − 6h3qm(m2 + 1) + h3rm(1−m)− h3q − k). Case 3: P = 0, Q = −1, R = 1, v(x, y, t) = A0 + 6h2(qm3 − q −m2r + rm)6h2(qm3 − q −m2r + rm) pm[1 + sin (k(x−m1) + h(y −m2))] . (31) A. A. Gaber, T. M. Younis, M. F. Alharbi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6030 7 of 12 Figure 3: The single wave solution of (29). Figure 4: The multi-soliton solution of (30). where A0= −1 pmh(h 3qm3 + h3rm(1−m)− h3q − k). Case IV: Utilizing vectors W4 and W5 for obtainig the invariant variables as follow ζ1 = t, ζ2 = y −mx, v = F (ζ1, ζ2), (32) Using the invariant variables in Eq.(32). Eq.(1) is converted to the format Fζ1 − pmFFζ1 + q(−m3Fζ1ζ1ζ1 + Fζ2ζ2ζ2) + r(m2Fζ1ζ2ζ2 + Fζ2ζ1ζ1) = 0. (33) Taking θ = kζ1 + hζ2, then Eq. (33) can be written as following kF ′ − pmhFF ′ + [qh3(−m3 + 1) + rh3(m2 − 1)]F ′′′ = 0, (34) Substituting Eq.(27) into Eq.(34), equating to zero the coefficients of all powers of Ψ(θ) yields a set of algebraic equations for Ai. By applying Maple we solve this algebraic equation, to yield A0 = −1 pmh (−6h3P (rm2 + h3q) + h3qm3Q+ 6h3qPm(m2 + 1) + h3Qrm(1−m)− h3qQ− k), A. A. Gaber, T. M. Younis, M. F. Alharbi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6030 8 of 12 Figure 5: The oscillating-solitons solution of (31). Figure 6: The wave solution solution of (36). A1 = 6h2 pm (qm3Q+ 2qm3P − qQ− 2qP −m2rQ− 2m2rP + 2rmP + rmQ) (35) The general exact solutions of Eq.(2) taking form Case 1: P = −1, Q = 1, R = 0, v(x, y, t) = A0 + 6h2(qm3 − 2qmP − q + 2q −m2r + 2m2r − 2rm+ rm) pm[1 + sech (kt+ h(y −m2x))] . (36) where A0= −1 pmh(6h 3(rm2 + h3q) + h3qm3 − 6h3qm(m2 + 1) + h3rm(1−m)− h3q − k). Case 2: P = 1, Q = 1, R = 0, v(x, y, t) = A0 + 6h2(qm3 + 2qm3 − q − 2q −m2r − 2m2r + 2rm+ rm) pm[1 + csch (kt+ h(y −m2x))] . (37) where A0= −1 pmh(−6h3(rm2 + h3q) + h3qm3 + 6h3qm(m2 + 1) + hQrm(1−m)− h3q − k). Case 3: P = 0, Q = 1, R = −1, v(x, y, t) = A0 + 6h2(qm3 − q −m2r + rm)6h2(qm3 − q −m2r + rm) pm[1 + cosh (kt+ h(y −m2x))] . (38) A. A. Gaber, T. M. Younis, M. F. Alharbi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6030 9 of 12 where A0= −1 pmh(h 3qm3 + h3rm(1−m)− h3q − k). Figure 7: The wave solution of (37). Figure 8: The single wave solution of (38). 4. Discussion and Results We contrast our exact answers and similarity reduction findings with those of earlier research: 1) Nuruddeen et al. [20] reduced the govern equation the one case of ordinary differ- ential equations. They used the tanh technique to get precise answers for these cases. In our study, we employed a novel traveling wave method and achieved numerous accurate answers for four common examples. 2) Vinita and Ray[21] investigated the similarity reductions for the gonvering equation. Their reductions, however, are subcases of ours. We obtained new accurate solutions for four general examples in our study. A. A. Gaber, T. M. Younis, M. F. Alharbi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6030 10 of 12 5. Conclusion In this work, we reduced the governing equation to four distinct ordinary differential equations using symmetry analysis. In order to generate new types of solutions, we finally used traveling wave approach called KAM. We graphed the solutions to display their attributes. The solutions that were discovered are entirely different from those that were obtained in earlier research. Acknowledgements The author Ahmed A. Gaber would like to thank the Deanship of Scientific Research, Majmaah University, Saudi Arabia, for supporting this work under project No. R-2025- 1786 References [1] Zang, Y. Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY, (2012). https://doi.org/10.1007/978- 1-4614-1806-1 52 [2] Yusuf A. , Sulaiman T. A. , Dynamics of Lump-periodic, breather and two- wave solutions with the long wave in shallow water under gravity and 2D nonlinear lattice, Comm. Non. Sci. and Num. Simul., 99 (2021), 105846, https://doi.org/10.1016/j.cnsns.2021.105846 [3] Gaber A. A., Wazwaz A. and Mousa M. 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