EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5987 ISSN 1307-5543 – ejpam.com Published by New York Business Global Abundant Explicit Non-Traveling Wave Solutions to the (3 + 1)-Dimensional Nonlinear Evolution Equation Using a Generalized Variable Separation Method Shami A. M. Alsallami1,∗, Mohammed Alkinidri2 1 Mathematics Department, College of Sciences,Umm Al-Qura University, Makkah 24381, SaudiArabia 2 King Abdulaziz University, College of Science & Arts, Department of Mathematics, Rabigh, Saudi Arabia Abstract. Non-traveling wave solutions allow us to characterize more complex natural phenom- ena across various scientific fields, which may not be easily analyzed using soliton solutions. This study introduces a novel approach for deriving a wide variety of explicit non-traveling wave so- lutions to the (3 + 1)-dimensional nonlinear evolution equation, utilizing a modified generalized variable separation technique. The newly obtained solutions encompass different non-traveling forms, including periodic solitary waves and soliton-like structures. These results underscore the efficacy of the method in tackling complex nonlinear partial differential equations. The findings presented in this article constitute innovative contributions to the equation modeling the velocity of water waves on the surface of shallow water. Furthermore, our employed technique can also pro- vide a foundation for investigating the stability, dynamics, and practical applications of solutions to other similar problems. 2020 Mathematics Subject Classifications: 35Q51, 35C05, 37K40 Key Words and Phrases: Non-traveling wave solutions, Generalized variable separation tech- nique, evolution equation, Wave solutions, Symbolic computation 1. Introduction Nonlinear partial differential equations are essential in modern mathematics, greatly enhancing our comprehension of intricate natural phenomena [1–11]. Recently, there has been a growing focus on developing efficient methods to obtain analytical and approximate solutions for this specific class of differential equations[12–22]. In recent years, numerous researchers have shown keen interest in advancing methods for obtaining exact wave so- lutions of the nonlinear models [23–28]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5987 Email addresses: sasallami@uqu.edu.sa (S. A. M. Alsallami) mohammed.alkinidri@hotmail.com (M. Alkinidri) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 2 of 22 In this paper, we study a nonlinear (3+1)-dimensional evolution equation [29] 3uxxz + (2uxt + uxxxx − 2uxuxx)y + 2 (uxxuy)x = 0, (1) where u = u(x, y, z, t). This equation models the velocity of water waves on the surface of shallow water. The (2+1)-dimensional version of the model was proposed by Geng to describe the (2+1)-dimensional interaction between a Riemann wave traveling along the y-axis and a long wave along the x-axis. In his work, the algebraic-geometrical solutions of the model were explicitly expressed in terms of the Riemann theta functions [29]. Wazwaz also suc- cessfully derived the dispersive relations and phase shifts for the model, as well as identified multiple soliton solutions for each equation [30]. Additionally, some kink breather soliton and multi-soliton solutions to the model have been discovered in [31]. Moreover, the au- thors of [32] explored the application of the linear superposition principle to construct res- onant multiple wave solutions for a (n+1)(n+1)-dimensional nonlinear evolution equation. They derived two types of resonant solutions through parameterization of wave numbers and frequencies and illustrated the resonance phenomena of multiple waves with figures depicting several sample solutions. This research contributes to our understanding of wave interactions in nonlinear systems. The authors of [33] decomposed a (3+1)-dimensional nonlinear evolution equation (1) into three integrable (1+1)-dimensional models, deriving general Nth-order rational solutions through the Darboux transformation method. They then explored doubly-localized lumps and first-order rogue waves, providing insights into their patterns in various planes. In [34], a comprehensive investigation was carried out to study Eq.(1), focusing on non-singular multi-complexiton waves. They ”initially ob- tained multi-shock waves using linear superposition and then constructed non-singular multi-complexiton waves through symbolic computations. In the work of [35], the Hirota bilinear approach was employed to studying the model. They derived NN-soliton solutions and soliton molecules in various planes, examining diverse hybrid interactions and peri- odic wave solutions. This research enhances the understanding of the nonlinear dynamics involved, providing a comprehensive graphical analysis of the dynamic attributes of the solutions. In their research, the authors of [36] introduced new solutions to the model, focusing on resonant multiple soliton solutions (RMSSs) obtained through the linear su- perposition principle and a weight algorithm. They also developed both nonsingular and singular complexiton solutions by introducing conjugate parameter pairs and introduced the bilinear approach to explore complex multiple-soliton solutions. Nowadays, non-traveling wave solutions are applicable in diverse domains such as physics, engineering, and fluid dynamics, offering enhanced insights into the behavior of complex systems influenced by multiple factors. One of the notable techniques that has contributed to the discovery of non-traveling wave solutions for equations is the extended homoclinic test method, which has been employed to address various equations to date [37–40]. The objective of this research is to introduce a novel generalized variable sepa- ration method to obtain a wide range of explicit non-traveling wave solutions for Eq.(1). This article presents a version that is more comprehensive and inclusive than the form con- S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 3 of 22 sidered in [37–39]. To the best of our knowledge, this methodology has yet to be applied to identify non-traveling wave solutions to Eq.(1). This article is organized as follows. The second section introduces the formal structure of the reduction procedure for Eq.(1). Section 3 outlines an alternative scenario to achieve several non-traveling wave solutions to Eq.(1). In Section 4, we present the dynamical analysis of the proposed solutions. Finally, the article concludes with a summary in the last section. 2. The reduction procedure for Eq.(1) In this paper, we investigate a generalized method for variable separation that enables the derivation of non-traveling wave exact solutions to Eq.(1) within the following specified framework: u(x, y, z, t) = p(y, z, t) · φ(ξ, t) + q(y, z, t), (2) where ξ = αx + θ(y, z, t), and p, q, θ and φ are three unknown functions. This frame- work represents a broader version of the structures previously found in the literature for p(y, z, t) = 1. This new assumption could potentially yield solutions that have not been explored or documented in earlier studies of the equation. The subsequent theorem outlines the key findings of the paper. Theorem 1. Through the application of variable transformation defined as u(x, y, z, t) = p(y, z, t) · φ(ξ, t) + q(y, z, t), (3) in Eq.(1) where ξ = αx + θ(y, z, t), with unknown functions p, q, θ, and φ, we obtain the following results: (i) For any arbitrary continuous two-dimensional functions f1 and Λ that depend on z and t, if p(y, z, t) = 1 holds along with θ(y, z, t) = f1(z, t), q(y, z, t) = −3f1z(z, t) 2α y + Λ(z, t), (4) then the solution of Eq.(1) can be expressed as u(x, y, z, t) = φ(αx+ f1(z, t), t)− 3 ( ∂ ∂zf1(z, t) ) y 2α + Λ(z, t), (5) where φ is an arbitrary continuous two-variable function. (ii) For any arbitrary continuous two-dimensional functions f1, f2, and Λ, if p(y, z, t) = 1 holds with θ(y, z, t) = f1(z, t) + f2(y, z), q(y, z, t) = −3f1z(z, t)αy − 3 (∫ f2z(y, z)dy ) α+ 2f2(y, z)f1t(z, t) 2α2 + Λ(z, t), (6) then Eq.(1) simplifies to α3φξξξ − 2α2φ2 ξ + 2φt = 0. (7) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 4 of 22 (iii) For any arbitrary continuous two-dimensional functions f1, f2, and Λ, if it holds that p(y, z, t) = f1(z, t), θ(y, z, t) = f2(z, t), q(y, z, t) = ( 3f1z(z, t) 2αf1(z, t) − 3f2z(z, t) 2α ) y+Λ(z, t), (8) then the solution of Eq.(1) is given by u(x, y, z, t) = f1(z, t)g(t)e −αx−f2(z,t) ( 3f1z(z, t) 2αf1(z, t) − 3f2z(z, t) 2α ) y + Λ(z, t). (9) Proof. First, by inserting the symbolic structure Eq.(3) into Eq.(1), it is simplified as δ1φξξξξξ + δ2φξξξξ + δ3φξξξ + δ4φξξξφξ + δ5φξξξφ+ δ6φξξ + δ7φξξφξ + δ8φξ + δ9φ 2 ξξ + δ10φξ t + δ11φξξ t = 0, (10) where δ1 = −pθyα 4, δ2 = −pyα 4, δ3 = pα ( 2qyα 2 − 2θyθt + 3αθz ) , δ4 = 4p2θyα 3, δ5 = 2ppyα 3, δ6 = −2pαθty − 2θtαpy + 3pzα 2 − 2ptαθy, δ7 = 6α3ppy, δ8 = −2αpty, δ9 = 4p2θyα 3, δ10 = −2αpy, δ11 = −2αpθy. (11) At this point, we are seeking the criteria that would transform Eq.(10) into a more straight- forward and solvable form. To accomplish this, we need to analyze the scenario in which the term φξξξ in Eq.(10) is removed. This leads us to set δ3 = 0, resulting in q(y, z, t) = 1 2α2 ∫ (2θyθt − 3αθz)dy + Λ(z, t) . (12) As the first assumption, we consider the case of py = pz = pt = 0. Without loss of generality, it can be assumed that p(y, z, t) = 1. A direct consequence of this assumption is that δ2 = δ5 = δ7 = δ8 = δ10 = 0. Further, Eq.(10) reduces to −θyα 3φξξξξξ + 4α2θyφξξξφξ − 2θtyφξξ + 4α2θyφ 2 ξξ − 2θyφξξt = 0. (13) By executing a double integration of Eq.(13) with respect to ξ, and then eliminating the resulting integral along with applying some simple algebraic rearrangements, we obtain the following equation θy ( α3φξξξ − 2α2φ2 ξ + 2φt ) + 2θtyφ = 0. (14) (i) By setting θy = 0, the entire equation (14) simplifies to zero, resulting in the following conclusion θ (y, z, t) = f1(z, t) . (15) Furthermore, by substituting Eq.(15) into Eq.(12), the solution is simplified to q(y, z, t) = −3f1z(z, t) 2α y + Λ(z, t) . (16) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 5 of 22 Thus, a general non-soliton wave solution for Eq.(1) is obtained as follows u(x, y, z, t) = φ(αx+ f1(z, t) , t)− 3 ( ∂ ∂zf1(z, t) ) y 2α + Λ(z, t) , (17) where φ, f1, and Λ are arbitrary continuous two-variable functions. (ii) To further simplify Eq.(14), let us consider θty = 0, which implies θ (y, z, t) = f1(z, t) + f2(y, z) . (18) Inserting Eq.(18) into Eq.(12), the solution becomes q(y, z, t) = −3f1z(z, t)αy − 3 (∫ f2z(y, z) dy ) α+ 2f2(y, z) f1t(z, t) 2α2 + Λ(z, t) . (19) Furthermore, Eq.(14) is expressed in a simpler form as follows α3φξξξ − 2α2φ2 ξ + 2φt = 0. (20) (iii) Here, we consider another way of simplifying Eq.(10), as py = θy = 0, or equivalently p(y, z, t) = f1(z, t), θ(y, z, t) = f2(z, t), (21) f1 and f2 are arbitrary continuous functions. Then Eq.(10) reduces to p (2qyα+ 3θz)φξξξ + 3pzφξξ = 0. (22) Further, we integrate Eq.(22) once with respect to ξ and null the integral constant as p (2qyα+ 3θz)φξ + 3pzφ = 0. (23) Now, if we assume that p (2qyα+ 3θz) = 3pz and thanks to (19) and (21), we have q(y, z, t) = ( 3f1z(z, t) 2αf1(z, t) − 3f2z(z, t) 2α ) y + Λ(z, t) . (24) Taking Eqs.(21), (24) into account in Eq.(23), we derive the following equation φ(ξ, t) + φξ(ξ, t) = 0. (25) The recent equation clearly possesses the following solution φ (ξ, t) = g(t) e−ξ, (26) where g is an arbitrary continuous non-zero function. Thus, a general non-soliton solution for Eq.(1) is obtained as follows u(x, y, z, t) = f1(z, t) g(t) e −αx−f2(z,t) ( 3f1z(z, t) 2αf1(z, t) − 3f2z(z, t) 2α ) y + Λ(z, t) . (27) This solution can also be considered a more generalized form of the solution u1(x, y, z, t) given in the formula (17). S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 6 of 22 3. Further non-traveling wave solutions to Eq.(1) As demonstrated in the second part of Theorem 1, under the specified conditions, the main equation is reduced to the Eq.(20). This latter equation is significantly more straightforward and likely easier to solve than the original problem (1). In this section, we aim to obtain analytical solutions for the beta equation, which will subsequently allow us to derive solutions for the original equation using an efficient technique. To this purpose, we introduce the new variable φ(ξ, t) = φ(℘) along with ℘ = κξ + ωt. (28) Applying this transformation to Eq.(20) gives α2κ3 ( d3 d℘3 φ (℘) ) + 3ακ2 ( d d℘ φ (℘) )2 + ω ( d d℘ φ (℘) ) = 0. (29) We now utilize the modified version of the generalized exponential rational function method (abbreviated as mGERFM), which is an innovative analytical approach intro- duced by Ghanbari in [41]. Based on this method, the solution to Eq.(29) takes the form of the following structure φ(℘) = ε0 + n∑ j=1 εj ( Γ′(℘) Γ(℘) )j + n∑ j=1 γj ( Γ(℘) Γ′(℘) )j , (30) where Γ(℘) = ς1e ϑ1℘ + ς2e ϑ2℘ ς3eϑ3℘ + ς4eϑ4℘ , (31) and n is the balance number of the given equation. By applying the balance rule given in Eq.(29), we may infer that 2(n+ 1) = n+ 3, which gives n = 1. Thus, from Eq.(30), we obtain φ(℘) = ε0 + ε1 ( Γ′(℘) Γ(℘) ) + γ1 ( Γ(℘) Γ′(℘) ) . (32) Upon inserting the expression from Eq.(32) along with Eq.(31) in Eq.(29) and solving the resultant for the unknown parameters, the following solutions are obtained. Set 1: For [ς1, ς2, ς3, ς4] = [2, 0, 1,−1] and [ϑ1, ϑ2, ϑ3, ϑ4] = [2, 0, 2, 0], Eq.(31) reduces to Γ(℘) = 2e2℘ e2℘ − 1 . (33) Moreover, the rest of the parameters can be attained as ω = −2κ3α3, ε1 = 0, γ1 = 3κα, (34) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 7 of 22 where κ, ε0 are free-chosen parameters. Taking the obtained results into account in Eqs.(32) and (33), we have φ(℘) = (−3κα+ ε0) e 2℘ + 3κα+ ε0 e2℘ + 1 . (35) In light of the recent result along with Eq.(28), the solution for Eq.(20) is obtained as φ(ξ, t) = (−3κα+ ε0) e −4α3κ3t+2κξ + 3κα+ ε0 e−4α3κ3t+2κξ + 1 . (36) As a result, by substituting Eqs.(36), (18), and (19) into (3), we can derive a non-soliton solution for Eq.(1) as u(x, y, z, t) = (−3κα+ ε0) e −4α3κ3t+2κ(αx+f2(y,z)+f1(z,t)) + 3κα+ ε0 e−4α3κ3t+2κ(αx+f2(y,z)+f1(z,t)) + 1 + f2(y, z) ( ∂ ∂tf1(z, t) ) α2 − 3y ( ∂ ∂zf1(z, t) ) + 3 (∫ ( ∂ ∂zf2(y, z) ) dy ) 2α + Λ(z, t) . (37) Set 2: For [ς1, ς2, ς3, ς4] = [1,−1, 2i, 0] and [ϑ1, ϑ2, ϑ3, ϑ4] = [2 + i, 2 − i, 0, 0], Eq.(31) reduces to Γ(℘) = sin(℘) e2℘. (38) Additionally, the remaining parameters can be obtained as follows ω = 2κ3α3, ε1 = 0, γ1 = 15κα, (39) where κ, ε0 are free-chosen parameters. Taking the obtained results into account in Eqs.(32) and (38), one achieves φ(℘) = (15κα+ 2ε0) sin(℘) + ε0 cos(℘) 2 sin(℘) + cos(℘) . (40) In light of the recent result along with Eq.(28), the solution for Eq.(20) is obtained as φ(ξ, t) = (15κα+ 2ε0) sin ( 2α3κ3t+ κξ ) + ε0 cos ( 2α3κ3t+ κξ ) 2 sin(2α3κ3t+ κξ) + cos(2α3κ3t+ κξ) . (41) Hence, using Eqs.(41), (18), and (19) in Eq.(3), a non-soliton solution for Eq.(1) can be established in the following manner u(x, y, z, t) = (15κα+ 2ε0) sin ( 2α3κ3t+ κξ ) + ε0 cos ( 2α3κ3t+ κξ ) 2 sin(2α3κ3t+ κξ) + cos(2α3κ3t+ κξ) + f2(y, z) ( ∂ ∂tf1(z, t) ) α2 − 3y ( ∂ ∂zf1(z, t) ) + 3 (∫ ( ∂ ∂zf2(y, z) ) dy ) 2α + Λ(z, t) , (42) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 8 of 22 where ξ = 2α3κ3t+ κ (αx+ f2(y, z) + f1(z, t)). Set 3: For [ς1, ς2, ς3, ς4] = [1, 1, 1, 0] and [ϑ1, ϑ2, ϑ3, ϑ4] = [1, 2, 0, 0], Eq.(31) reduces to Γ(℘) = e℘ + e2℘. (43) Moreover, the rest of the parameters can be attained as ω = −κ3α3 2 , ε1 = −3κα, γ1 = 0, (44) where κ, ε0 are free-chosen parameters. Taking the obtained results into account in Eqs.(32) and (43), one gets φ(℘) = (−6κα+ ε0) e ℘ − 3κα+ ε0 1 + e℘ . (45) In light of the recent result along with Eq.(28), the solution for Eq.(20) is obtained as φ(ξ, t) = (−6κα+ ε0) e − 1 2 α3κ3t+κξ − 3κα+ ε0 1 + e− 1 2 α3κ3t+κξ . (46) Consequently, using Eqs.(46), (18) and (19) in Eq.(3), a non-soliton solution for Eq.(1) can be established in the following manner u(x, y, z, t) = (−6κα+ ε0) e −α3κ3t 2 +κ(αx+f2(y,z)+f1(z,t)) − 3κα+ ε0 1 + e− α3κ3t 2 +κ(αx+f2(y,z)+f1(z,t)) + f2(y, z) ( ∂ ∂tf1(z, t) ) α2 − 3y ( ∂ ∂zf1(z, t) ) + 3 (∫ ( ∂ ∂zf2(y, z) ) dy ) 2α + Λ(z, t) . (47) Set 4: For [ς1, ς2, ς3, ς4] = [2, 0, 1, 1] and [ϑ1, ϑ2, ϑ3, ϑ4] = [2 + i, 0, 2i, 0], Eq.(31) reduces to Γ(℘) = e2ξ sec(ξ) . (48) Additionally, the remaining parameters can be obtained as follows ω = 2κ3α3, ε1 = 3κα, γ1 = 0 , (49) where κ, ε0 are free-chosen parameters. Taking the obtained results into account in Eqs.(32) and (48), we obtain φ(℘) = 3ακ tan(℘) + 6κα+ ε0. (50) In light of the recent result along with Eq.(28), the solution for Eq.(20) is obtained as φ(ξ, t) = 3ακ tan ( 2α3κ3t+ κξ ) + 6κα+ ε0. (51) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 9 of 22 Hence, using Eqs.(51), (18) and (19) in Eq.(3), a non-soliton solution for Eq.(1) is obtained as u(x, y, z, t) = 3ακ tan ( 2α3κ3t+ κ (αx+ f2(y, z) + f1(z, t)) ) + f2(y, z) ( ∂ ∂tf1(z, t) ) α2 − 3y ( ∂ ∂zf1(z, t) ) + 3 (∫ ( ∂ ∂zf2(y, z) ) dy ) 2α + Λ(z, t) . (52) Set 5: For [ς1, ς2, ς3, ς4] = [1,−1, 2i, 0] and [ϑ1, ϑ2, ϑ3, ϑ4] = [1 + i, 1 − i, 0, 0], Eq.(31) reduces to Γ(℘) = e℘ sin(℘). (53) Moreover, the rest of the parameters can be attained as ω = 2κ3α3, ε1 = −3κα, γ1 = 0, (54) where κ, ε0 are free-chosen parameters. Taking the obtained results into account in Eqs.(32) and (53), it reads φ(℘) = −3ακ cot(℘)− 3κα+ ε0. (55) In light of the recent result along with Eq.(28), the solution for Eq.(20) is obtained as φ(ξ, t) = −3ακ cot ( 2α3κ3t+ κξ ) − 3κα+ ε0. (56) Therefore, by substituting Eqs.(56), (18), and (19) into (3), we derive a non-soliton solution for Eq.(1) as follows u(x, y, z, t) = −3ακ cot ( 2α3κ3t+ κ (αx+ f2(y, z) + f1(z, t)) ) + f2(y, z) ( ∂ ∂tf1(z, t) ) α2 − 3y ( ∂ ∂zf1(z, t) ) + 3 (∫ ( ∂ ∂zf2(y, z) ) dy ) 2α + Λ(z, t) . (57) Remarks 3.1. All the proposed solutions in this work have been validated using Maple by substituting them back into the original equation. 4. Dynamical analysis of the proposed solutions In this section of the article, we will examine the dynamic behaviors of the solutions obtained in the last section. ■ First of all, we investigate the solution given in Eq.(17). Employing a diverse selection of three arbitrary continuous functions φ, f1,Λ in the formulation of this solution enables the generation of various unique non-traveling exact solutions to Eq.(1). Some of the S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 10 of 22 derived non-traveling exact solutions from the solution (17) are listed below. (a): Through the insertion of φ(ξ, t) = e− ξ2+t2 10 sin(ξ2 + t2), (58a) f1(z, t) = cos(z)e− t2+z2 20 + sin(2t)(1 + z2)0.1, (58b) Λ(z, t) = 5 tanh(sin(z) + cos(t)), (58c) in Eq.(17), we have u1(x, y, z, t) = 5 tanh(cos(t) + sin(z)) + sin (∆) exp ( −∆ 10 ) − 3 2α ( −e 1 20(−t2−z2) sin(z)− 1 10 ze 1 20(−t2−z2) cos(z) + 0.2z sin(2t) (z2 + 1)0.9 ) y, (59) where ∆ = ( e (−t2−z2) 20 cos(z) + ( z2 + 1 )0.1 sin(2t) + αx )2 + t2. The solution u1(x, y, z, t) represents a complex, oscillatory medium where wave behavior is modulated by the spatial coordinates x, y, z and the temporal component t. The term φ introduces a smooth decay and oscillation that interacts with the influences of surrounding fields as represented by f1(z, t) and Λ(z, t). Several dynamics of this solution for different values of α are displayed in Fig.1. Figure 1: 3D plot of u1(1, 1, z, t) for different values of α . (b): By incorporating φ(ξ, t) = sin(ξ2 + t) + cos(2t), (60a) f1(z, t) = sin(z)e−t2/5 + 0.5 cos(3z − t), (60b) Λ(z, t) = 0.5 sin(z) + cos(t) + e−(z 2+t2)/10. (60c) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 11 of 22 in Eq.(17), we have u2(x, y, z, t) = sin (( e− t2 5 sin(z) + 0.5 cos(t− 3z) + αx )2 + t ) − 3y ( e− t2 5 cos(z) + 1.5 sin(t− 3z) ) 2α + e−(t 2+z2)/10 + cos(t) + cos(2t) + 0.5 sin(z). (61) The recent obtained solution combines oscillatory components to describe a dynamic wave- like field in three-dimensional space. It represents the interaction of spatial oscillations with temporal changes, incorporating the effects of a scalar field dependent on altitude z and time t. The bounded nature of the solution ensures physical constraints are main- tained, potentially resembling wave behavior in fluid dynamics or electromagnetic fields. Several dynamics of this solution for different values of α are displayed in Fig.2. Figure 2: 3D plot of u2(x, 0.5, z, 0.5) for different values of α . (c): Through inserting φ(ξ, t) = sin(ξ) + cos(2t) + e−(ξ2+t2), (62a) f1(z, t) = 0.5z2 + 3 sin(z + t) + 1, (62b) Λ(z, t) = 2 log(1 + z2) + √ 1 + t2, (62c) in Eq.(17), we have u3(x, y, z, t) = e−t2−(3 sin(t+z)+αx+0.5z2+1) 2 + sin ( 3 sin(t+ z) + αx+ 0.5z2 + 1 ) − 3y(3 cos(t+ z) + z) 2α + cos(2t) + 2 log ( z2 + 1 ) + √ t2 + 1. (63) The formula represents a dynamic system where the spatial variables x, y, and z are modulated by oscillatory components combined with temporal evolution. This functional form may describe phenomena like waves or heat distribution in a medium, illustrating how bounded interactions influence a state variable over time. Several dynamics of this S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 12 of 22 Figure 3: 3D plot of u3(x, 0.5, z, 0.2) for different values of α . solution for different values of α are displayed in Fig.3. (d): Through the insertion of φ(ξ, t) = sin(ξ)e−0.1t2 , (64a) f1(z, t) = sin(0.5t) cos(0.5t) + z2/20, (64b) Λ(z, t) = (1 + z/10) sin (√ z2 + t2 ) , (64c) in Eq.(17), we have u4(x, y, z, t) = sin ( sin(0.5t) cos(0.5z) + αx+ z2 20 ) e−0.1t2 − 3y ( z 10 − 0.5 sin(0.5t) sin(0.5z) ) 2α + (1 + z/10) sin (√ z2 + t2 ) . (65) This solution exhibits a dynamic system where the interaction between multiple dimen- sions generates intricate oscillatory behavior. The contributions from φ create spatial wave patterns, while f1 and Λ modulate these patterns over time and spatial dimensions, reflecting how different physical phenomena can influence the overall system. Several dy- namics of this solution for different values of α are displayed in Fig.4. (e): By incorporating φ(ξ, t) = e−x2 sin(2πt) + cos(πx), (66a) f1(z, t) = sin(z) cos(ωt) + 0.5z, (66b) Λ(z, t) = t sin(2πz), (66c) in Eq.(17), we have u5(x, y, z, t) = cos(π(cos(t) sin(z) + αx+ 0.5z)) + sin(2πt)e−(cos(t) sin(z)+αx+0.5z)2 − 3y(cos(t) cos(z) + 0.5) 2α + t sin(2πz). (67) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 13 of 22 Figure 4: 3D plot of u4(x, 0.5, z, 0.2) for different values of α . This solution is used to explain a three-dimensional wave phenomenon where the oscil- lations and propagation characteristics are influenced by the spatial variables x, y, z and time t. The contributions from φ, f1, and Λ highlight interactions within the system, showcasing complex patterns and modulations over time while remaining bounded in the specified range, giving insight into various physical systems such as fluid dynamics or wave propagation. Several dynamics of this solution for different values of α are displayed in Fig.5. Figure 5: 3D plot of u5(1.5, 2.5, z, t) for different values of α . (f): Through inserting φ(ξ, t) = log(1 + x2) cos(2πt), (68a) f1(z, t) = sin(z2) + cos(3t) 1 + z2 , (68b) Λ(z, t) = (z2 + 2) cos(0.5πt), (68c) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 14 of 22 in Eq.(17), we have u6(x, y, z, t) = log (cos(3t) + sin ( z2 ) z2 + 1 + αx )2 + 1  cos(2πt) − 3y ( 2z cos(z2) z2+1 − 2z(cos(3t)+sin(z2)) (z2+1)2 ) 2α + ( z2 + 2 ) cos((0.5πt). (69) In this solution, we capture complex oscillations with logarithmic variability influenced by both spatial position and temporal cycles. This indicates various potential states of energy or information flowing in a medium, as the higher-order interactions from the com- positional functions suggest a rich interplay of forces or fields in a transient environment. Several dynamics of this solution for different values of α are displayed in Fig.6. Figure 6: 3D plot of u6(x, 0.5, z, 0.2) for different values of α . ■ Here, we will explore several non-traveling exact solutions obtained from the solution (27), as follows. (a): By inserting g(t) = cos(2πt) + 0.5 sin(5πt), f1(z, t) = 1 + 0.5 cos(3z + t), (70a) f2(z, t) = e−(0.1z)2 sin(4t), Λ(z, t) = 1− e−0.05(z+t) cos(2z), (70b) in Eq.(27), one gets u7(x, y, z, t) = y(0.5 cos(t+ 3z) + 1)(0.5 sin(5πt) + cos(2πt))e−αx−e−0.01z2 sin(4t) × ( 0.03e−0.01z2z sin(4t) α − 2.25 sin(t+ 3z) α(0.5 cos(t+ 3z) + 1) ) + 1− e−0.05(t+z) cos(2z). (71) The solution u7(x, y, z, t) models the dynamic system influenced by spatial, temporal, and oscillatory factors. The interplay between the various components represents a wave-like S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 15 of 22 phenomenon in three dimensions, where f1 and f2 contribute spatial complexity while g and Λ introduce time-dependent modulation. The resulting solution encapsulates the behavior of a waveform that evolves in time and space, bounded within specified limits. (b): By incorporating g(t) = sin(t) + 0.5t2, f1(z, t) = e−0.1z2 cos(0.5t+ z), (72a) f2(z, t) = (1 + cos(z + t 2 ))2,Λ(z, t) = sin(zt) + 0.5z, (72b) in Eq.(27), it reads u8(x, y, z, t) = y ( 0.5t2 + sin(t) ) cos(0.5t+ z) e−αx−(cos(0.5t+z)+1)2−0.1z2 · ( −3 sin(0.5t+ z)− 0.6z cos(0.5t+ z) 2α sec(0.5t+ z) + 3 sin(0.5t+ z) (cos(0.5t+ z) + 1) α ) + sin(tz) + 0.5z. (73) The solution u8(x, y, z, t) represents a dynamical system influenced by various spatial and temporal factors. The components f1, f2, and Λ introduce oscillatory behavior and modulated growth dependent on the dimensions z and time t, while g contributes a smooth, evolving amplitude profile. The derivatives incorporate interactions resulting in shifts of equilibrium state affected by the spatial dimensions and oscillations. (c): By incorporating g(t) = sin2(t) + 0.1, f1(z, t) = e−(z2+t2) cos(2πt) + 0.5, (74a) f2(z, t) = 2e−0.4z2 + cos(πz),Λ(z, t) = sin(0.5z + t) + 1, (74b) in Eq.(27), we obtain u9(x, y, z, t) = sin(t+ 0.5z) + 1 + y ( sin2(t) + 0.1 ) ( e−t2−z2 cos(2πt) + 0.5 ) − 3ze−t2−z2 cos(2πt) α ( e−t2−z2 cos(2πt) + 0.5 ) − 3 ( −1.6e−0.4z2z − π sin(πz) ) 2α  e−αx−2e−0.4z2−cos(πz). (75) This solution models the dynamic behavior of a physical system that exhibits oscillatory and wave-like characteristics in three-dimensional space. The components of the solution contribute to the spatial and temporal modulation of the system. The presence of expo- nential and trigonometric terms suggests a phenomenon where waves propagate through a medium, influenced by both spatial coordinates and time. (d): Through the insertion of g(t) = sin(t2) + 0.5 cos ( t 3 ) , f1(z, t) = e− 1 2 z2 sin(2z + t), (76a) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 16 of 22 f2(z, t) = 0.3z2 + 0.5 sin(2t) + 0.2 cos(3z), Λ(z, t) = 5 sin(z) + 3 cos(t) + z2 10 , (76b) in Eq.(27), we derive u10(x, y, z, t) = y sin(t+ 2z) ( sin ( t2 ) + 0.5 cos ( t 3 )) e−αx−0.5 sin(2t)−0.8z2−0.2 cos(3z) 3e z2 2 csc(t+ 2z) ( 2e− z2 2 cos(t+ 2z)− e− z2 2 z sin(t+ 2z) ) 2α − 3(0.6z − 0.6 sin(3z)) 2α  + 3 cos(t) + z2 10 + 5 sin(z). (77) The derived solution u10(x, y, z, t) describes a wave-like phenomenon that varies in space and time, influenced by factors such as external perturbations, dissipation due to the parameter α, and interactions between spatial components captured in f1 and f2. The oscillatory behavior of the solution may model various physical systems, such as fluid dynamics or heat transfer, where bounded fluctuations are crucial for stability analysis. (e): By substituting the selections of g(t) = sin(t), f1(z, t) = (1 + 0.2 sin(3z + t)), (78a) f2(z, t) = e−0.1z sin(0.5t) cos(1.2z), Λ(z, t) = sin(z + t), (78b) into Eq.(27), we obtain the following solution u11(x, y, z, t) = y sin(t)(0.2 sin(t+ 3z) + 1)( 0.9 cos(t+ 3z) α(0.2 sin(t+ 3z) + 1) − 3 ( −1.2e−0.1z sin(0.5t) sin (1.2z)− 0.1e−0.1z sin(0.5t) cos(1.2z) ) 2α ) × e−αx−e−0.1z sin(0.5t) cos(1.2z) + sin(t+ z). (79) This solution represents a dynamic field that varies in space and time, influenced by the oscillatory characteristics of its components. The dependence on spatial variables x, y, and z indicates that it models a scenario with wave-like behavior influenced by time-varying parameters. The inclusion of smooth, non-singular functions facilitates a bounded and smooth spatial distribution, keeping values within a controlled range, indicating some form of constraint or equilibrium in the physical context. (f): By Considering g(t) = esin(t)+0.5 cos(2t), f1(z, t) = sin(z) + 0.5 cos(z + t) + 1, (80a) f2(z, t) = cos(z) sin(t), Λ(z, t) = 2 sin(z) cos(t), (80b) S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 17 of 22 Figure 7: 3D plots of u7(1, 1, z, t)− u12(1, 1, z, t) with α = 1 . in Eq.(27), we have u12(x, y, z, t) = y(0.5 cos(t+ z) + sin(z) + 1) ( 3 sin(t) sin(z) 2α + 3(cos(z)− 0.5 sin(t+ z)) 2α(0.5 cos(t+ z) + sin(z) + 1) ) e−αx−sin(t) cos(z)+sin(t)+0.5 cos(2t) + 2 cos(t) sin(z). (81) The solution u12(x, y, z, t) models the dynamic system influenced by spatial, temporal, and oscillatory factors. The interplay between the various components represents a wave- like phenomenon in three dimensions, where f1 and f2 contribute spatial complexity while g and Λ introduce time-dependent modulation. The resulting solution encapsulates the behavior of a waveform that evolves in time and space, bounded within specified limits. In Fig.7, we have plotted the 3D dynamics of solutions u7(1, 1, z, t) − u12(1, 1, z, t) along with taking α = 1. ■ Moreover, some solutions derived from the solution (37) are plotted in Fig.8. In this Figure, we have taken α = ε0 = 1, corresponding to: (a) f1(z, t) = sin(z + t), f2(y, z) = cos(y + z),Λ(z, t) = (1 + ez 2+t2)−1, (b) f1(z, t) = tanh(zt), f2(y, z) = sin(z + y),Λ(z, t) = (1 + ez 2+t2)−1, (c) f1(z, t) = cos(zt), f2(y, z) = cos(z + y),Λ(z, t) = tanh(z2 + t2). S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 18 of 22 Figure 8: 3D plot of proposed in Eq.(37) with α = ε0 = 1 along with κ = 0.5, corresponding to: (a) f1(z, t) = sin(z + t), f2(y, z) = cos(y + z),Λ(z, t) = 1 1+ez 2+t2 , (b) f1(z, t) = tanh(zt), f2(y, z) = sin(z + y),Λ(z, t) = 1 1+ez 2+t2 , (c) f1(z, t) = cos(zt), f2(y, z) = cos(z + y),Λ(z, t) = tanh(z2 + t2). 5. Conclusion This paper successfully presents a novel approach for deriving a variety of explicit non-traveling wave solutions to the (3+1)-dimensional nonlinear evolution equation using a modified generalized variable separation technique. The solutions obtained not only encompass a richer diversity of non-travelling forms, including periodic solitary waves and soliton-like structures but also demonstrate the effectiveness of the proposed method in addressing complex nonlinear partial differential equations. Each solution’s validity was rigorously confirmed through symbolic computational methods using Maple, reinforcing the reliability of these findings. The proposed method offers several advantages, including its novelty and versatility in deriving a wide variety of non-traveling wave solutions, such as periodic solitary waves and soliton-like structures, for the (3+1)-dimensional nonlinear evolution equation. Its effectiveness in addressing complex nonlinear partial differential equations is demonstrated through the obtained solutions, which are rigorously validated using symbolic computational tools like Maple. However, the method also presents cer- tain limitations. The complexity of the mathematical transformations and assumptions involved may restrict its accessibility to researchers without a strong mathematical back- ground. Additionally, the approach is specifically tailored to the (3+1)-dimensional non- linear evolution equation, and its applicability to other types of equations remains to be explored. Furthermore, the reliance on symbolic computation tools suggests that the method may be computationally intensive for certain cases. Despite these limitations, the method represents a significant contribution to the field, providing a foundation for future research in nonlinear dynamics and wave phenomena. The results obtained in this article for the equation can be considered innovative findings for the main equation, which have not been presented in previous literature. The findings of this paper provide new insights into non-traveling wave phenomena. These insights establish a foundation for future research in nonlinear dynamics. S. A. M. Alsallami, M. Alkinidri / Eur. J. Pure Appl. Math, 18 (2) (2025), 5987 19 of 22 Acknowledgements The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia for funding this research work through grant number: 25UQU4290491GSSR02. Funding statement This research work was funded by Umm Al-Qura University, Saudi Arabia under grant number: 25UQU4290491GSSR02. References [1] A Al-Mamun, C Lu, SN Ananna, H Ismail, A Bari, and MM Uddin. The influence of fractionality and unconstrained parameters on mathematical and graphical analysis of the time fractional phi-four model. Chaos, Solitons & Fractals, 183:114892, 2024. 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