EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6012 ISSN 1307-5543 – ejpam.com Published by New York Business Global Unraveling Symmetry Properties in a Three-Dimensional Nonlinear Evolution Model via the Lie Group Method Akhtar Hussain1,∗, M. Usman2, Ahmed M. Zidan3, Jorge Herrera4 1 Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan 2 College of Electrical and Mechanical Engineering (CEME), National University of Sciences and Technology (NUST), H-12 Islamabad 44000, Pakistan 3 Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia 4 Facultad de Ciencias Naturales e Ingenieria, Universidad de Bogota Jorge Tadeo Lozano, Bogota 110311, Colombia Abstract. This research focuses on a (3+1)-dimensional nonlinear evolution model originating from the Jaulent-Miodek hierarchy. Several tools can be used to examine the symmetry of the model. However, our primary emphasis lies in harnessing one of the most significant and powerful analytical tools available for this purpose: the Lie group method. The Lie group method is an effective approach for uncovering the symmetry properties inherent in a model and exploring group- invariant solutions using symmetry algebra. In addition, we applied Ibragimov’s method to study conservation laws relevant to the considered model. Our study is significant as it contributes to the investigation of this model and addresses a particular gap in the group-theoretic approach in this context. Our findings represent novel contributions to the study of the model under consideration. 2020 Mathematics Subject Classifications: 70G65, 35-XX, 70S10, 22E70 Key Words and Phrases: Lie group method, Nonlinear evolution model, Symmetry algebra, Jaulent-Miodek hierarchy, Conservation laws 1. Introduction Many physical phenomena encountered in mechanics, physics, engineering, biology, and chemistry have been effectively described through nonlinear partial differential equa- tions (PDEs). Nonlinear wave equations frequently encompass a multitude of factors, including dispersion, diffusion, dissipation, reaction, and convection. The pursuit of ex- act solutions to these equations plays a significant role in our understanding of the wave ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6012 Email addresses: akhtarhussain21@sms.edu.pk (A. Hussain) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 2 of 20 propagation. Solitons [1, 2], for instance, emerge from a delicate balance between nonlin- earity and linear dispersion, whereas a distinct class of solitons known as “compactons,” characterized by the absence of exponential tails, arises from the interplay between nonlin- earity and genuinely nonlinear dispersion. The solitary wave theory encompasses a wide array of phenomena that researchers are eager to investigate, with the aim of uncovering the underlying structures within the resulting wave solutions. In references [3–5], four (2+1)-dimensional nonlinear models, stemming from the Jaulent-Miodek hierarchy, were formulated in the following manner ψt = −(ψxx − 2ψ3)x − 3 2 (ψx∂ −1 x ψy + ψψy), ψt = 1 2 (ψxx − 2ψ3)x + 3 2 ( − 1 4 ∂−1 x ψyy + ψψy ) , ψt = −1 4 (ψxx − 2ψ3)x − 3 4 ( 1 4 ∂−1 x ψyy + ψx∂ −1 x ψy ) , ψt = 2(ψxx − 2ψ3)x − 3 4 ( ∂−1 x ψyy − 2ψx∂ −1 x ψy − 6ψψy ) , (1) where ∂−1 x is the inverse of ∂x with ∂−1 x ∂x = ∂x∂ −1 x = 1, and (∂−1 x g)(x) = ∫ x −∞ g(t)dt, (2) under decay conditions of infinity. The four models were examined in [3–5] utilizing robust methodologies such as the Wronskian form, simplified Hirota’s method, and the pertur- bation technique introduced by Herman and Nuseir [6]. These investigations resulted in derivation of N soliton solutions for each model. Wazwaz [7] extended the research presented in [3–5] by delving into the analysis of four (3+1)-dimensional nonlinear models derived from the Jaulent-Miodek hierarchy. He for- mulated these (3+1)-dimensional models in the following ψt = −(ψxx − 2ψ3)x − 3 2 (ψx∂ −1 x ψy + ψψy) + a∂−1 x ψzz, ψt = 1 2 (ψxx − 2ψ3)x + 3 2 ( − 1 4 ∂−1 x ψyy + ψψy ) + a∂−1 x ψzz, ψt = −1 4 (ψxx − 2ψ3)x − 3 4 ( 1 4 ∂−1 x ψyy + ψx∂ −1 x ψy ) + a∂−1 x ψzz, ψt = 2(ψxx − 2ψ3)x − 3 4 ( ∂−1 x ψyy − 2ψx∂ −1 x ψy − 6ψψy ) − 3 4 ∂−1 x ψzz − 1 4 ψz − 1 2 ψy, (3) where a is a constant. It is evident that these (3+1)-dimensional nonlinear models (3) are constructed by augmenting the first three models in (1) with the term a∂−1 x ψzz and by introducing the terms −3 4∂ −1 x ψzz − 1 4ψz − 1 2ψy to the last model in (1). To eliminate the A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 3 of 20 integral term in the first model of Eq (3), we introduce the potential function as follows ψ(x, y, z, t) = ux(x, y, z, t), which allows us to transform the model into the equation uxt + uxxxx − 6u2xuxx + 3 2 uxxuy + 3 2 uxuxy − auzz = 0, (4) referred to as (4): The main objective of this study is to explore the integrability of nonlinear model (3), which is equivalent to (4). To achieve this, our approach begins with an exploration of the symmetry algebra of the model, which is then used to analyze the associated sym- metry groups. We proceed by performing potential symmetry reductions using symmetry algebra to derive the invariant solutions. Furthermore, we aim to leverage the symmetry algebra to delve into the local conservation laws of the given model Eq. (4), following the principles outlined in Ibragimov’s new conservation theorem. It is worth noting that our main focus is on the first model. However, it is crucial to emphasize that Wazwaz, as mentioned in [7], highlighted the significance of the solutions for the second and third models, perfectly aligned with those of the first model. The Lie symmetry method [8–10], which is based on the principles of symmetry and invariance, offers a systematic approach to analytically solve differential equations. Its origins are traced back to the work of Sophus Lie (1842-1899), and it has since become a fundamental mathematical tool for researchers investigating mathematical models in fields ranging from physics and engineering to natural sciences. Comprehensive discussions of Lie group analysis methods [11] can be found in various books, including those authored by Ovsyannikov [12], Bluman and Kumei [13], Olver [14], and Ibragimov [15], among oth- ers. It is well established that conservation laws play a pivotal role in solving the differen- tial equations. The existence of a substantial number of conservation laws within a set of PDEs signifies their potential for integrability, as elucidated by Bluman and Kumei (1989). An insightful comparison of different approaches for deriving conservation laws for specific PDEs in the field of fluid mechanics can be found in the study conducted by Naz [16] and his team published in 2008. The remainder of this paper is organized as follows. In Section 2, we explore the symmetry algebra of nonlinear model (4). Subsequently, in Section 3, we use it to perform several symmetry reductions on the same model. In section 4, we provide a recap of key defini- tions and theorems related to conservation laws. We then constructed conservation laws for model (4) using Ibragimov’s new conservation theorem. Finally, Section 5 presents the concluding remarks. A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 4 of 20 2. Lie Group Method for Eq (4) In this section, an analysis of the Lie symmetries and the optimal system pertaining to the equation denoted by (4) is conducted. Consider the one-parameter Lie group of transformation x̃→ x+ εϕ1(x, y, z, t, u) +O(ε2), ỹ → y + εϕ2(x, y, z, t, u) +O(ε2), z̃ → z + εϕ3(x, y, z, t, u) +O(ε2), t̃→ t+ εϕ4(x, y, z, t, u) +O(ε2), ũ→ u+ ες(x, y, z, t, u) +O(ε2), (5) where ε is the group parameter. For Eq (4), the vector field is defined as V = ϕ1 ∂ ∂x + ϕ2 ∂ ∂y + ϕ3 ∂ ∂z + ϕ4 ∂ ∂t + ς ∂ ∂u · (6) The task at hand involves determining the coefficient functions ϕ1, ϕ2, ϕ3, ϕ4, and ς, such that the operator V satisfies the Lie symmetry condition V [4](uxt + uxxxx − 6u2xuxx + 3 2 uxxuy + 3 2 uxuxy − auzz)|(4) = 0, (7) where V [4] denotes the fourth prolongation of V . Solving Eq (7) yields the set of determining equations, expressed as ςzz − 2 3a ϕ2tt = 0, ϕ3zt = 0, ϕ3zz = 0, ςu = 0, ςx − 2 3 ϕ2t = 0, ςy − 2 3 ϕ1t = 0, ϕ4y = 0, ϕ4t − 3 2 ϕ3z = 0, ϕ4u = 0, ϕ4x = 0, ϕ4z = 0, ϕ1u = 0, ϕ1x − ϕ3z 2 = 0, ϕ1y + 8 3 ϕ2t = 0, ϕ1z − ϕ3t 2a = 0, ϕ2u = 0, ϕ2x = 0, ϕ2y − ϕ3z = 0, ϕ2z = 0, ϕ3u = 0, ϕ3x = 0, ϕ3y = 0. (8) Solution of system (8) gives infinitesimals given as ϕ1 = c1 3 x− 8 3 f1ty + f2tz 2α + 3 2 f5 + c3, ϕ2 = 2 3 c1y + f1, ϕ3 = 2 3 c1z + f2, ϕ4 = c1t+ c2, ς = 1 9α ( (−8αy2 + 3z2)f1tt + 3yzf2tt + 9α( 2 3 f1tx+ f5ty + f6z + f7) ) . (9) A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 5 of 20 By assuming f2 = c4, f6 = c5, f7 = c6 and f1 = f5 = 0, we obtain the following symmetry generators V1 = ∂ ∂x , V2 = ∂ ∂z , V3 = ∂ ∂t , V4 = ∂ ∂u , V5 = z ∂ ∂u , V6 = x 3 ∂ ∂x + 2y 3 ∂ ∂y + 2z 3 ∂ ∂z + t ∂ ∂t · (10) The commutator relation for the symmetry algebra (10) is defined as [Vi, Vi] = ViVj − VjVi, (11) where [Vi, Vj ] denotes the Lie product. The commutator relation for symmetry algebra (10) is given in Table 1. Table 1: Commutator Table. [Vi, Vj ] V1 V2 V3 V4 V5 V6 V1 0 0 0 0 0 1 3V1 V2 0 0 0 0 V4 2 3V2 V3 0 0 0 0 0 V3 V4 0 0 0 0 0 0 V5 0 −V4 0 0 0 −2 3V5 V6 −1 3V1 −2 3V2 −V3 0 2 3V5 0 Theorem 1. The vector fields Vi (i = 1, 2, · · · , 6) for (4) constitute six-dimensional Lie algebra. Proof. As indicated in Table 1, an antisymmetrical pattern is noticeable, and zero diagonal elements are evident. The determination of the structure constants is easily ac- complished by examining the commutator table, and Jacobi identity verification is straight- forward. Lie group transformations associated with the symmetry generators (10) are written as S1 : (x̃, ỹ, z̃, t̃, ũ) → (x+ ε, y, z, t, u), S2 : (x̃, ỹ, z̃, t̃, ũ) → (x, y, z + ε, t, u), S3 : (x̃, ỹ, z̃, t̃, ũ) → (x, y, z, t+ ε, u), S4 : (x̃, ỹ, z̃, t̃, ũ) → (x, y, z, t, ε+ u), S5 : (x̃, ỹ, z̃, t̃, ũ) → (x, y, z, t, εz + u), S6 : (x̃, ỹ, z̃, t̃, ũ) → (e ε 3x, e 2ε 3 y, e 2ε 3 z, eϵt, u). 3. Group-Invariant Solutions Case 1: V3 + V4 = ∂ ∂t + ∂ ∂u . In the case of the symmetry generator V3 + V4, the Lagrange equation can be expressed A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 6 of 20 as shown below, du 1 = dx 0 = dy 0 = dz 0 = dt 1 , and produces a similarity transformation, u(x, y, z, t) = t + ϑ(α, β, γ), α = x, β = y, γ = z. Through the application of this transformation, we continue with equation (4) in its reduced form as shown below, 2ϑαααα + 3 ( −4ϑ2α + ϑβ ) ϑαα − 2aϑγγ + 3ϑαϑαβ = 0, (12) The application of the Lie symmetry method to this (2+1) dimensional PDE unveils additional infinitesimals ϕα = c3α 2 + c6, ϕβ = c3β + c5, ϕγ = c3γ + c4, ςϑ = c1γ + c2. (13) Case 1.1: For c2 = c5 = 1, the symmetry generator ∂ ∂β + ∂ ∂ϑ leads to characteristic form dϑ 1 = dα 0 = dβ 1 = dγ 0 , which gives rise to similarity variables ϑ(α, β, γ) = β + p(k, τ), where k = α, τ = γ. Applying these invariants, Eq (12) transforms into the following (1+1) PDE 2pkkkk − 12pkkp 2 k + 3pkk − 2apττ = 0. (14) The application of the Lie symmetry method to this (1+1) dimensional PDE unveils additional infinitesimals ϕk = c3, ϕτ = c4, ςp = c1τ + c2. (15) Case 1.1.1: For c3 = 1, the symmetry generator ∂ ∂k leads to characteristic form dp 0 = dk 1 = dτ 0 , which gives rise to invariant variables p(k, τ) = h(s), where s = τ . Applying these invariants, Eq (14) transforms into the following ODE −2ah′′ = 0 with solution h(s) = c1s+ c2, (16) this gives, p(k, τ) = c1τ + c2, (17) A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 7 of 20 which yields, ϑ(α, β, γ) = c1γ + c2 + β. (18) As a outcome, the invariant solution for the (3+1)-dimensional model (4) is formulated as follows u(x, y, z, t) = c1z + c2 + y + t, (19) where c1, c2 are constants. Case 2: V2 + V3 + V4 = ∂ ∂z + ∂ ∂t + ∂ ∂u · In the case of the symmetry generator V2+V3+V4, the Lagrange equation can be expressed as shown below, du 1 = dx 0 = dy 0 = dz 1 = dt 1 , and produces a similarity transformation, u(x, y, z, t) = z + ϑ(α, β, γ), α = x, β = y, γ = −z + t. Through the application of this transformation, we continue with equation (4) in its reduced form as shown below, 2ϑαααα + 3 ( −4ϑ2α + ϑβ ) ϑαα − 2aϑγγ + 3ϑαϑαβ + 2ϑαγ = 0. (20) The application of the Lie symmetry method to this (2+1) dimensional PDE unveils additional infinitesimals ϕα = c1α 2 − c1γ 4a + c6, ϕβ = c1β + c2, ϕγ = c1γ + c5, ςϑ = −c1β 6a + c3γ + c4. (21) Case 2.1: For c2 = c3 = 1, the symmetry generator ∂ ∂β + γ ∂ ∂ϑ leads to characteristic form dϑ γ = dα 0 = dβ 1 = dγ 0 , This gives rise to variables ϑ(α, β, γ) = βγ + p(k, τ), where k = α, τ = γ. Applying these invariants, Eq (20) transforms into the following (1+1) PDE pkkkk + 3 ( −4p2k + τ ) pkk − 2apττ + 2pkτ = 0. (22) The application of the Lie symmetry method to this (1+1) dimensional PDE unveils additional infinitesimals ϕk = c3, ϕτ = 0, ςp = c1τ + c2. (23) A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 8 of 20 Case 2.1.1: For c1 = c2 = c3 = 1, the symmetry generator (1 + τ) ∂ ∂p + ∂ ∂k leads to characteristic form dp τ + 1 = dk 1 = dτ 0 , which gives rise to invariant variables p(k, τ) = kτ + k + h(s), where s = τ . Applying these invariants, Eq (22) transforms into the following ODE −2ah′′ +2 = 0, with solution h(s) = s2 2a + c1s+ c2, (24) this gives, p(k, τ) = τ2 2a + c1τ + c2 + kτ + k, (25) which yields, ϑ(α, β, γ) = γ2 2a + c1γ + c2 + αγ + α+ βγ. (26) As a outcome, the invariant solution for the (3+1)-dimensional model (4) is formulated as follows u(x, y, z, t) = (t− z)2 2a + c1(t− z) + c2 + x(t− z) + x+ y(t− z) + z, (27) where c1, c2 are any constants. Case 3: V2 + V3 + V5 = ∂ ∂z + z ∂ ∂u + ∂ ∂t . In the case of the symmetry generator V2+V3+V5, the Lagrange equation can be expressed as shown below, du z = dx 0 = dy 0 = dz 1 = dt 1 , and produces a similarity transformation u(x, y, z, t) = z2 2 + ϑ(α, β, γ), α = x, β = y, γ = −z + t. Through the application of this transformation, we continue with equation (4) in its reduced form as shown below, 2ϑαααα + (−12ϑ2α + 3ϑβ)ϑαα − 2aϑγγ + 3ϑαϑαβ − 2a+ 2ϑαγ = 0, (28) The application of the Lie symmetry method to this (2+1) dimensional PDE unveils additional infinitesimals ϕα = c3, ϕβ = c4, ϕγ = c5, ςϑ = c1 + c2γ. (29) Case 3.1: For c1 = c3 = c4 = 1 and the symmetry generator ∂ ∂β + ∂ ∂ϑ + ∂ ∂α leads to characteristic form dϑ 1 = dα 1 = dβ 1 = dγ 0 , A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 9 of 20 which yields variables ϑ(α, β, γ) = α+ p(k, τ), where k = γ, τ = −α+ β. Applying these invariants, Eq (28) transforms into the following (1+1) PDE 2pττττ + (−12p2τ + 30pτ − 15)pττ − 2apkk − 2a− 2pkτ = 0. (30) The application of the Lie symmetry method to this (1+1) dimensional PDE unveils additional infinitesimals ϕk = c3, ϕτ = c4, ςp = c1 + c2τ. (31) Case 3.1.1: For c1 = c4 = 1 and the symmetry generator ∂ ∂τ + ∂ ∂p leads to characteristic form dp 1 = dk 0 = dτ 1 , with invariant variables p(k, τ) = τ + h(s), where k = s, applying these invariants, (30) transforms into ODE −2ah′′ − 2a = 0. The solution for this ODE is h(s) = −s 2 2 + c1s+ c2, (32) this gives, p(k, τ) = −k 2 2 + c1k + c2 + τ, (33) which yields, ϑ(α, β, γ) = −γ 2 2 + c1γ + c2 + (−α+ β) + α. (34) As a outcome, the invariant solution for the (3+1)-dimensional model (4) is formulated as follows u(x, y, z, t) = −(−z + t)2 2 + c1(−z + t) + c2 + y + z2 2 , (35) where c1, c2 are any constants. Case 4: V2 + V5 = ∂ ∂z + z ∂ ∂u · In the case of the symmetry generator V2 + V5, the Lagrange equation can be expressed as shown below, du z = dt 0 = dz 1 = dy 0 = dx 0 , and produces a similarity transformation, u(x, y, z, t) = z2 2 + ϑ(α, β, γ), α = t, β = x, γ = y. Through the application of this transformation, we continue with equation (4) in its reduced form as shown below, 2ϑββββ + 3 ( −4ϑ2β + ϑγ ) ϑββ + 3ϑβϑβγ − 2a+ 2ϑαβ = 0. (36) A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 10 of 20 The application of the Lie symmetry method to this (2+1) dimensional PDE unveils additional infinitesimals ϕα = c1α+ c2, ϕβ = 3f1(α) 2 + (−16aαγ + β) c1 3 − 8c3γ 3 + c5, ϕγ = ( 3aα2 + 2γ ) c1 3 + c3α+ c4, ςϑ = f1αγ + f2(α) + 2 (2aαc1 + c3)β 3 − 16c1a γ 2 9 . (37) Case 4.1: For c4 = 1, the symmetry generator ∂ ∂γ leads to characteristic form dϑ 0 = dα 0 = dβ 0 = dγ 1 , which yields variables ϑ(α, β, γ) = p(k, τ), where k = α, τ = β. Applying these invariants, Eq (36) transforms into the following (1+1) PDE 2pττττ − 12pττp 2 τ − 2a+ 2pkτ = 0. (38) The application of the Lie symmetry method to this (1+1) dimensional PDE unveils additional infinitesimals ϕk = c1, ϕτ = c2, ςp = f1(k) . (39) Case 4.1.1: For c1 = 1 and the symmetry generator ∂ ∂k leads to characteristic form dp 0 = dk 1 = dτ 0 , which gives rise to the invariant variables p(k, τ) = h(s) where, s = τ . Applying these invariants, Eq (38) transforms into the ODE h(iv) − 6h′ 2 h′′ − a = 0. (40) Eq (40) admits a Lagrangian L = h′′2 2 + h′4 2 − ah, which in turn exhibits variational symmetries given by ∂ ∂s , ∂ ∂h · (41) For ∂ ∂h , the corresponding Noether first integral is I = 2h′ 3 + as. (42) A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 11 of 20 Now, DsI = 0 implies I = c, that is 2h′ 3 + as = c, (43) where c denotes a constant. The solution of equation (43) is h(s) = 3 ι (as− c) ( ι± √ 3 ) (−4as+ 4c) 1 3 16a , (44) this gives, p(k, τ) = 3 ι (aτ − c) ( ι± √ 3 ) (−4aτ + 4c) 1 3 16a , (45) which yields, ϑ(α, β, γ) = 3 ι (aβ − c) ( ι± √ 3 ) (−4aβ + 4c) 1 3 16a . (46) As a outcome, the invariant solution for the (3+1)-dimensional model (4) is formulated as follows u(x, y, z, t) = z2 2 + 3 ι (ax− c) ( ι± √ 3 ) (−4ax+ 4c) 1 3 16a , (47) where c denotes a constant. Case 5: V3 = ∂ ∂t . In the case of the symmetry generator V3, the Lagrange equation can be expressed as shown below, dt 1 = dz 0 = dy 0 = dx 0 = du 0 , and produces a similarity transformation, u(x, y, z, t) = ϑ(α, β, γ), α = x, β = y, γ = z. Through the application of this transformation, we continue with equation (4) in its reduced form as shown below, 2ϑαααα + (−12ϑ2α + 3ϑβ)ϑαα − 2aϑγγ + 3ϑαϑαβ = 0, (48) The application of the Lie symmetry method to this (2+1) dimensional PDE unveils additional infinitesimals ϕα = c4α+ c3, ϕβ = 2c4β + c5, ϕγ = 2c4γ + c6, ςϑ = c1 + c2γ. (49) Case 5.1: For c6 = 1 and the symmetry generator ∂ ∂γ leads to characteristic form dϑ 0 = dα 0 = dβ 0 = dγ 1 , A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 12 of 20 which yields variables ϑ(α, β, γ) = p(k, τ), where k = α, τ = β. Applying these invariants, Eq (48) transforms into the following (1+1) PDE 2pkkkk + (−12p2k + 3pτ )pkk + 3pkpkτ = 0. (50) The application of the Lie symmetry method to this (1+1) dimensional PDE unveils additional infinitesimals ϕk = c2 + c3k, ϕτ = 2c3τ + c4, ςp = c1. (51) Case 5.1.1: For c4 = 1 and the symmetry generator ∂ ∂τ leads to characteristic form dp 0 = dk 0 = dτ 1 , which gives rise to invariant variables p(k, τ) = h(s), where k = s. Applying these invariants, Eq (50) transforms into the following ODE h(iv) − 6h′ 2 h′′ = 0. (52) Eq (52) admits a Lagrangian L = h′′2 2 + h′4 2 , which in turn exhibits variational symmetries given by ∂ ∂s , ∂ ∂h · (53) For ∂ ∂h , the corresponding Noether first integral is I = h′′′ − 2h′ 3 . (54) Now, DsI = 0 implies I = c, that is h′′′ − 2h′ 3 = c, (55) where c is any constant. The polynomial solution of equation (55) is h(s) = ( − (−4c) 1 3 4 ± i √ 3 4 (−4c) 1 3 ) s+ c1. (56) By back substitution of variables, we get p(k, τ) = ( − (−4c) 1 3 4 ± i √ 3 4 (−4c) 1 3 ) k + c1, (57) this implies, ϑ(α, β, γ) = ( − (−4c) 1 3 4 ± i √ 3 4 (−4c) 1 3 ) α+ c1. (58) A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 13 of 20 As a outcome, the invariant solution for the (3+1)-dimensional model (4) is formulated as follows u(x, y, z, t) = ( − (−4c) 1 3 4 ± i √ 3 4 (−4c) 1 3 ) x+ c1, (59) where c1 denotes a constant. Case 6: V2 + V4 = ∂ ∂u + ∂ ∂z . In the case of the symmetry generator V2 + V4, the Lagrange equation can be expressed as shown below, du 1 = dx 0 = dy 0 = dz 1 = dt 0 , and produces a similarity transformation, u(x, y, z, t) = z + ϑ(α, β, γ), α = t, β = x, γ = y. Through the application of this transformation, we continue with equation (4) in its reduced form as shown below, 2ϑββββ + (−12ϑ2β + 3ϑγ)ϑββ + 3ϑβϑβγ + 2ϑαβ = 0, (60) The application of the Lie symmetry method to this (2+1) dimensional PDE unveils additional infinitesimals ϕα = 3c7α+ c5, ϕβ = 3 2 αc2 + c7β − 4γc3 + c6, ϕγ = 3 2 αc3 + 2γc7 + c8, ςϑ = c2γ + c3β + c4α+ c1. (61) Case 6.1: For c8 = 1 and the symmetry generator ∂ ∂γ leads to characteristic form dϑ 0 = dα 0 = dβ 0 = dγ 1 , with the invariant variables ϑ(α, β, γ) = p(k, τ), where k = α, τ = β. Applying these invariants, Eq (60) transforms into the following (1+1) PDE 2pττττ − 12p2τpττ + 2pkτ = 0, (62) The application of the Lie symmetry method to this (1+1) dimensional PDE unveils additional infinitesimals ϕk = 3c5k + c3, ϕτ = c5τ + c4, ςp = c2k + c1. (63) Case 6.1.1: For c3 = c4 = 1 and the symmetry generator ∂ ∂k + ∂ ∂τ leads to characteristic form dp 0 = dk 1 = dτ 1 , A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 14 of 20 which gives rise to invariant variables p(k, τ) = h(s), where −k + τ = s. Applying these invariants, Eq (62) transforms into the following ODE h(iv) − 6h′ 2 h′′ − h′′ = 0. (64) Eq (64) admits a Lagrangian L = h′′2 2 + h′4 2 + h′2 2 , which in turn exhibits variational symmetries given by ∂ ∂s , ∂ ∂h · (65) For ∂ ∂h , the corresponding Noether first integral is I = h′′′ − 2h′ 3 − h′. (66) Now, DsI = 0 implies I = c, that is h′′′ − 2h′ 3 − h′ = c, (67) where c is any constant. The polynomial solution of equation (67) is h(s) = ( − (−54c+ 6 √ 81c2 + 6) 1 3 12 + 1 2(−54c+ 6 √ 81c2 + 6) 1 3 − i √ 3 2 ((−54c+ 6 √ 81c2 + 6) 1 3 6 + 1 (−54c+ 6 √ 81c2 + 6) 1 3 )) s+ c1. (68) By back substitution of variables, we get p(k, τ) = ( − (−54c+ 6 √ 81c2 + 6) 1 3 12 + 1 2(−54c+ 6 √ 81c2 + 6) 1 3 − i √ 3 2 ((−54c+ 6 √ 81c2 + 6) 1 3 6 + 1 (−54c+ 6 √ 81c2 + 6) 1 3 )) (−k + τ) + c1, (69) this implies, ϑ(α, β, γ) = ( − (−54c+ 6 √ 81c2 + 6) 1 3 12 + 1 2(−54c+ 6 √ 81c2 + 6) 1 3 − i √ 3 2 ((−54c+ 6 √ 81c2 + 6) 1 3 6 + 1 (−54c+ 6 √ 81c2 + 6) 1 3 )) (−α+ β) + c1. (70) A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 15 of 20 As a outcome, the invariant solution for the (3+1)-dimensional model (4) is formulated as follows u(x, y, z, t) = ( − (−54c+ 6 √ 81c2 + 6) 1 3 12 + 1 2(−54c+ 6 √ 81c2 + 6) 1 3 − i √ 3 2 ((−54c+ 6 √ 81c2 + 6) 1 3 6 + 1 (−54c+ 6 √ 81c2 + 6) 1 3 )) (−t+ x) + c1 + z, (71) where c1 is any constant. Comment 1. 4. Conservation Laws Let Λ denote the multiplier or characteristic function [16] depending on x, y, z, t, u and the first- and second-order derivatives of u. Then for (4), we have the following relation DtT t+DxT x+DyT y+DzT z = Λ(uxt+uxxxx−6u2xuxx+ 3 2 uxxuy+ 3 2 uxuxy−auzz). (72) By applying the Euler operator δ δu on (72), we acquire δ δu (uxt + uxxxx − 6u2xuxx + 3 2 uxxuy + 3 2 uxuxy − auzz) = 0. (73) The determining equations obtained using (73) are described as Λzz = 0, Λx = 0, Λu = 0, Λuxt = 0, Λuxx = 0, Λuxy = 0, Λuzz = 0. (74) The solution of (3) yields the multipliers for (4), which are stated as follows Λ = f1(y, t)z + f2(y, t). (75) To obtain the local conservation laws for (4), the results of (75) indicate further classifica- tion of the functions f1(y, t) and f2(y, t). In addition, the multiplier can be used to obtain the potential symmetries. 5. Non-Local Conservation Laws for the Eq (4) In the referenced work [17], Ibragimov’s pioneering theorem centers on the conserved flow of differential equations. This theorem demonstrates significant applicability to sys- tems of differential equations, where the number of equations aligns with the number of dependent variables. We consider a kth order PDE given as G = G(x, u, u1, u2, ..., up). (76) A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 16 of 20 In this context, u = u(x) and x = x(x1, x2, ..., xm). Given the formal Lagrangian for Eq (76), an adjoint equation can be followed by G⋆ ≡ δ δu (vG), (77) where the Euler-Lagrange operator δ δu in symmetric form is given by δ δu = ∂ ∂u + ∞∑ i=1 (−1)sDi1 ...Dis ∂ ∂ui1...is , (78) and the differential operator Di is defined by Di = ∂ ∂xi + ui ∂ ∂u + uij ∂ ∂uj + . . . . (79) Theorem 2. For any symmetry Lie point, Lie-Bäcklund, or nonlocal of Eq (76) given by V = ϕi ∂ ∂xi + ς ∂ ∂u , (80) where L serves as a formal Lagrangian, the conserved vectors for Eq (77) can be formally defined as W i =ϕiL+N [ ∂L ∂ui −Dj ( ∂L ∂uij ) +DjDk ( ∂L ∂uijk ) + ... ] +Dj(N) [ ∂L ∂uij −Dk ( ∂L ∂uijk ) + ... ] +DjDk(N) [ ∂L ∂uijk + ... ] · · · (81) In this context, N is defined as N = ς − ϕiui, (82) where Di (W i) = 0. Theorem 3. [17] The adjoint equation for the nonlinear model described in Eq (4) is represented as follows G⋆ = 12uxuxxvx + 3uxyvx + 3 2 uyvxx − 6vxxu 2 x + 3 2 uxvxy + vxt − αvzz + vxxxx = 0, (83) where L = v(x, y, z, t) ( uxt + uxxxx − 6u2xuxx + 3 2 uxxuy + 3 2 uxuxy − auzz ) . (84) According to the Ibragimov theorem, every symmetry generator corresponds to a con- served vector. Consequently, we calculated the conserved vectors using Theorem 2. A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 17 of 20 (I) When considering the vector field V1 = ∂ ∂x and N = −ux, the resulting conserved vectors are as follows Wt 1 =vxux − uxxv, Wx 1 =L − ux (3 2 uxyv + 6u2xvx + 3 2 uyvx − 3 2 uxvy − vt − vxxxx ) − uxx ( vxx − 6u2xv + 3 2 uyv ) − 3 2 uxuxyv − vuxt + vxuxxx − vuxxxx, Wy 1 = 3 2 ux ( uxvx − uxxv ) , Wz 1 =− avzux + avuxz. (85) (II) When considering the vector field V2 = ∂ ∂z and N = −uz, the resulting conserved vectors are as follows Wt 2 =uzvx − vuxz, Wx 2 =− uz (3 2 uxyv + 6u2xvx + 3 2 uyvx − 3 2 uxvy − vt − vxxxx ) − uxz ( vxx − 6u2xv + 3 2 uyv ) − 3 2 uxuyzv − vuzt + vxuxxz − vuxxxz, Wy 2 = 3 2 ux(uzvx − uxzv), Wz 2 =v ( uxt + uxxxx − 6u2xuxx + 3 2 uxxuy + 3 2 uxuxy − auzz ) (86) (III) When considering the vector field V3 = ∂ ∂t and N = −ut, the resulting conserved vectors are as follows Wt 3 =v ( uxxxx − 6u2xuxx + 3 2 uxxuy + 3 2 uxuxy − auzz ) + utvx, Wx 3 =− ut (3 2 uxyv + 6u2xvx + 3 2 uyvx − 3 2 uxvy − vt − vxxxx ) − uxt ( vxx − 6u2xv + 3 2 uyv ) − 3 2 uxuytv − vutt + vxuxxt − vuxxxt, Wy 3 = 3 2 ux ( utvx − uxtv ) , Wz 3 =− avzut + avuzt. (87) (IV) When considering the vector field V4 = ∂ ∂u and N = 1, the resulting conserved A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 18 of 20 vectors are as follows Wt 4 =− vx, Wx 4 = 3 2 uxy + 6vxu 2 x + 3 2 uyvx + vxt + 3 2 uxvy − vt − vxxxx, Wy 4 =− 3 2 uxvx, Wz 4 =avz. (88) (V) When considering the vector field V5 = z ∂ ∂u and N = z, the resulting conserved vectors are as follows Wt 5 =− zvx, Wx 5 =z (3 2 uxyv + 6u2xvx + 3 2 uyvx − 3 2 uxvy − vt − vxxxx ) , Wy 5 =− 3 2 zuxvx, Wz 5 =− av. (89) (VI) When considering the vector field V6 = x ∂ ∂x +2y ∂ ∂y +2z ∂ ∂z +3t ∂∂t and N = −(xux+ 2yuy + 2zuz + 3tut), the resulting conserved vectors are as follows Wt 6 =(3t)L+ ux(xux + 2yuy + 2zuz + 3tut)− vDx(xux + 2yuy + 2zuz + 3tut), Wx 6 =xL+ (xux + 2yuy + 2zuz + 3tut) (3 2 uxyv + 6u2xvx + 3 2 uyvx − 3 2 uxvy − vt − vxxxx ) + Dx(xux + 2yuy + 2zuz + 3tut) ( vxx − 6u2xv + 3 2 uyv ) + 3 2 uxvDy(xux + 2yuy + 2zuz + 3tut) + vDt(xux + 2yuy + 2zuz + 3tut)− vxD2 x(xux + 2yuy + 2zuz + 3tut) + vD3 x(xux + 2yuy + 2zuz + 3tut), Wy 6 =(2y)L+ 3 2 uxvx(xux + 2yuy + 2zuz + 3tut)− 3 2 uxvDx(xux + 2yuy + 2zuz + 3tut), Wz 6 =(2z)L − avz(xux + 2yuy + 2zuz + 3tut) + avDz(xux + 2yuy + 2zuz + 3tut). (90) Note that in all the cases discussed above, L is the formal Lagrangian deduced from Eq (84), and the divergence relation DtWt +DxWx +DyWy +DzWz = 0, (91) holds for all the conserved vectors. 6. Conclusions In this research, our primary focus was on examining a (3+1)-dimensional nonlinear model (4) derived from the Jaulent-Miodek hierarchy. We utilized the Lie group method A. Hussain et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6012 19 of 20 to analyze the integrability features of this model. This method led to the identification of six-dimensional symmetry algebra, which allowed us to determine the symmetry groups associated with the nonlinear model (4). Using this symmetry algebra, we successfully derived polynomial group-invariant solutions. In the literature, various approaches have been used to obtain solitary wave, soliton, and other types of solutions for the desired model [3–6]. Notably, in a prior study conducted by Wazwaz [7], rational function solu- tions were obtained including single, two, and three soliton solutions. However, our results differ significantly from those of previous studies. Our research underscored the effective- ness of the Lie group method in unveiling not only the inherent symmetry properties of the model, but also in facilitating the exploration of group-invariant solutions through symmetry algebra. Furthermore, we apply Anco’s method to investigate the conservation laws pertinent to model (4). Our study assumed considerable significance as it contributed to the understanding of this model and addressed a specific gap in the group theoretic ap- proach within this context. Consequently, our findings represent pioneering contributions to the study of the examined model. Motivated by these outcomes, we hope to apply the same technique to other nonlinear models in the future. Acknowledgements The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through large group Research Project under grant number RGP.2/51/46. 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