EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6217 ISSN 1307-5543 – ejpam.com Published by New York Business Global Novel Bright Soliton and Kink Wave Solutions of Nonlinear Evolution Equation Najla A. Mohammed1, Naveed Shahid2, Tahira Sumbal Shaikh3,∗, Nauman Ahmed2,∗, Farwa Ghafoor3, Muhammad Zafarullah Baber4,5, Ilyas Khan6,7,8, Wei Sin Koh9 1 Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah, Saudi Arabia 2 Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan 3 Department of Mathematics, Lahore College for Women University, Lahore, Pakistan 4 Department of Mathematics, Shanghai University and Newtouch Center for Mathematics of Shanghai University, Shanghai 200444, China 5 Department of Mathematics and Statistics, The University of Lahore Sargodha Campus, Sargodha, Pakistan 6 Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai, Tamil Nadu, India 7 Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, Jordan 8 Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah, Saudi Arabia 9 INTI International University, Persiaran Perdana BBN Putra Nilai, 71800 Nilai, Negeri Sembilan, Malaysia Abstract. This article focuses on the generalized improved Boussinesq (GIB) equation. The GIB equation models nonlinear phenomena in various physical aspects such as shallow water waves, quantum fluid dynamics, and more. The new extended direct algebraic method (NEDAM) is applied to obtain exact solutions for this nonlinear model. Using this approach, we derive kink, anti- kink, solitons, and solitary wave solutions, along with bright, dark, and mixed-form solitons. New families of exponential, hyperbolic, and periodic wave solutions with arbitrary parameters are also constructed. The graphical representations provide deeper insight into the dynamics of nonlinear systems. The obtained results have many potential applications in optical fiber communications, fluid dynamics, and other fields involving wave propagation. 2020 Mathematics Subject Classifications: 35Q51, 35C07, 35R11 Key Words and Phrases: Solitary wave, kink, anti-kink, soliton, dark and bright solitons ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6217 Email addresses: tahira.sumbal@lcwu.edu.pk (T. S. Shaikh), nauman.ahmd01@gmail.com (N. Ahmed), i.said@mu.edu.sa (I. Khan) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 2 of 24 1. Introduction Partial differential equations (PDEs) had a central place among various settings of math- ematics from the mid-nineteenth century onwards, especially in Riemann’s work. Partial differential equations (PDEs) are highly praised due to their practicality in many fields such as acoustics, heat transfer, electro and magnetics, fluid dynamics, elasticity, and quantum mechanics. While these areas may at first glance seem very different, they can all be described in similar mathematical terms via PDEs. Numerous techniques have been developed to examine different facets of solutions and physical phenomena associated with nonlinear wave equations due to their significant mathematical features and broad range of applications. Consequently, studying nonlinear differential equations and determining their precise solutions is a contemporary field of study. This connects with work that began in the 1930s, which involves linking PDEs with the theory of singular integral op- erators. This has emerged as a fundamental issue in harmonic analysis, particularly with the establishment and generalization of the Calderon-Zygmund theory. The major method for the solution of nonlinear PDEs is Lie group analysis [1]. The mathematics of how to identify them has also been studied (lump exact solutions [2, 3] As NLPDEs are widely known to appear in bioscience, physics, chemistry, quantum me- chanics, fluid dynamics and multiple engineering domains, we aim to detect accurate and convenient solutions in these cases [4–6]. Numerous phenomena, including turbulence and shock wave flow in viscous fluids, have been demonstrated to be modeled by the Burgers’ equation. The two-dimensional Burgers’ equations were solved analytically by Fletcher using the Hopf-Cole transformation. Numerous numerical strategies have been constructed to solve this system of equations, such as implicit finite-difference schemes, explicit-implicit methods and techniques based on cubic spline functions. Soliman em- ployed partial differential equation similarity reductions to develop a method for tackling the Burgers’ issue. The two-dimensional Burgers equation has been solved using high- order accurate techniques. More recently, it is being proposed to achieve precise complex solutions for nonlinear partial differential equations using symbolic computing and direct algebraic approaches [7]. Long waves, such as solitons, move in stable packets at a steady speed without disintegrating. Solitons, sometimes called shallow-water waves, are waves that break apart only after colliding with other solitons. Solitons’ robustness and use- ful applications in physics have attracted the interest of mathematicians, engineers and physicists due to their special property. Solitons appear as partial differential equations that are nonlinear. The notion of solitons is credited to the Scottish naval architect John Scott Russell. He saw a ”Great Translation Wave” in the Great Britain Canal’s shallow waters in 1834. By constructing a path where the wave could hold its shape and travel a great distance, he aimed to demonstrate the resiliency of the wave. The mathematical community did not think highly of Russell’s ideas, and Array, a researcher, rejected them. In his book ”Tides and Waves,” published in 1845, Array set a hypothesis on long waves and stressed the relationship between height and amplitude and wave speed. This idea denies the existence of the single waves that Russell described [8]. In physics and mathematics, solitary waves, or solitons, are wave packets that propagate N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 3 of 24 at the same rate without changing their shape. They use this wave as an example of self- reinforcing. This happens when, in the medium, nonlinear and dispersive effects counteract themselves and cancel each other. Many soliton solutions are obtained for a significant number of weakly nonlinear dispersive partial differential equations that describe physical systems [9–15]. After observing the phenomenon for the first time in Scotland’s Union Canal in 1834 and reproducing it in a wave tank, John Scott Russell (1808–1882) named it the ”Wave of Translation” [16–20]. A soliton, within optics, signifies an optical field that persists unchanged as it travels, owing to the delicate equilibrium between nonlinear dispersion and linear scattering effects within the medium [21–24]. In recent times, multiple special techniques have been proposed to ensure accurate solu- tions of nonlinear differential equations. Such methods are the Exp-method, the inverse scattering algorithm [25], the Hirota bilinear method [26], the modified extended tanh func- tion method [27], the homogeneous balancing method [28], the sine-cosine method [29], tanh function method [30], variational iteration method [31]. A direct analytical resolu- tion can be obtained for the nonlinear partial differential equations (NLPDEs) using many methods as the modified simplest equation scheme [32], Kudryashov’s method [33], (G’/G) expansion method [34, 35]. Apart from these approaches, a more recent method called the UM scheme is used to analyze exact solutions of NLPDEs in the form of solitary waves. This method yields accurate traveling waves, which are explicit solutions.These explicit so- lutions have rational and polynomial function solutions, similar to waves. Soliton, elliptic, and solitary waves are various polynomial solutions; periodic and soliton rational solu- tions are different types of rational solutions. The conformable time-fractionalnonlinear Schrödinger equation (NLSE) and other nonlinear lattice equations (NLEEs) have been studied by several researchers using the UM method. A nonlinear partial differential equation is solved by the new extended direct algebraic method (NEDAM) [36]. The sine- cosine/ sinh-cosh, generalized exponential rational function method, Φ6-model expansion method are also helpful for obtaining the bright, dark, kink, periodic solitons [37–41]. Numerous techniques have been established to produce the exact solutions of nonlinear partial differential equations. Most of these methods are unable to handle the complexity and nonlinearity of the equations, particularly, nonlinear GIBq equation, or give limited types of solutions. Our research explores the efficacy of an innovative algebraic approach in solving nonlinear evolution equations, thereby expanding the toolkit for analytical solu- tions. Through the derivation of precise solutions and examination of their characteristics, we gain a deeper understanding of nonlinear dynamics. This study’s findings have signif- icant implications for understanding intricate processes in applied sciences, providing a robust framework for investigation and prediction. In recent years, there has been a significant increase in interestin solving ordinary differ- ential equations and the study of solitary waves. Nonlinear partial differential equations (NLPDEs) have been utilized to simulate different types of phenomena in various appli- cation domains. This work aims to present a new approach, namely NEDAM, by which we can accurately soliton solutions for the GIBq problem. It can be beneficial for develop- ing different types of soliton solutions and providing effective and rapid simulations. The NEDAM is the extended form of the traditional direct algebraic method. In this extension, N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 4 of 24 an auxiliary equation in a more generic form is involved, a logarithmic nonlinearity scale factor lnB and three arbitrary parameters λ, µ and ν. The addition enables this method to be more effective thanproducing a broader class of analytical solutions having periodic, kinks, anti-kinks and exponential forms. We have studied several GIBq problems where this strategy has proven successful. This method yields a wider variety of solutions and is a robust, powerful and potent technique for investigating NLEEs. It also works with computer algebra systems. Nonlinear evolution equations are pivotal in modeling complex phenomena across various scientific disciplines. While traditional methods like inverse scattering transform and Hi- rota’s bilinear method have been instrumental in solving these equations, they often come with limitations in applicability and complexity. In response, our study leverages the ex- tended direct algebraic method, a novel approach that has shown promise in deriving exact solutions for nonlinear systems. By applying this method to nonlinear evolution equations, we aim to uncover new insights into the dynamics of these systems. Our research not only demonstrates the efficacy of the extended direct algebraic method but also contributes to the existing body of knowledge by providing previously unexplored solutions. This advancement has significant implications for understanding intricate processes in applied sciences, offering a robust framework for investigation and prediction. Through this work, we seek to expand the toolkit for analytical solutions and enhance our understanding of nonlinear dynamics [42, 43]. The new extended direct algebraic method’s applicability to nonlinear partial differential equations (NLPDEs) is notable, yet not all-encompassing. Its effectiveness is pronounced for equations with polynomial nonlinearity and single spatial dimensions. However, equa- tions with highly complex nonlinearity, non-polynomial terms, fractional derivatives, or high-dimensional systems may pose challenges, potentially limiting the method’s efficacy. Recognizing these boundaries enables researchers to judiciously apply this method and explore alternative approaches when confronted with complex NLPDEs. The extended hyperbolic function method’s efficacy can be further contextualized by comparing it with other symbolic calculation methods, such as the Hirota bilinear method. The comparison in Table 1 would highlight the strengths and limitations of each approach, providing a more comprehensive understanding of their applicability to nonlinear partial differential equations. The new extended direct algebraic method is employed in this study due to its efficacy in tackling nonlinear evolution equations. Its systematic approach enables efficient derivation of exact solutions, making it a valuable tool for understanding complex phenomena. Al- though not universally applicable, this method’s flexibility and simplicity render it suitable for a broad range of nonlinear equations. By leveraging its strengths, this research aims to uncover novel insights into the behavior of the equation under investigation, ultimately contributing to the advancement of knowledge in this field [44–47]. Our research is driven by the quest for robust analytical frameworks to tackle nonlinear evolution equations, which underpin complex phenomena in diverse scientific realms. The pursuit of exact solutions is paramount, as they offer profound insights into system be- havior and serve as benchmarks for numerical validation. By harnessing the potential N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 5 of 24 Method Applicability Solution Form Complexity Advantages Limitations Proposed Method NLPDEs with poly- nomial nonlinearity Exact hyper- bolic solu- tions Moderate Simple, effi- cient for cer- tain NLPDEs Limited for highly nonlin- ear equations Hirota Bilinear Method Wide range of NLPDEs Exact bilinear solu- tions High Effective for multi-soliton solutions Requires care- ful variable transformation Extended Tanh Method NLPDEs with poly- nomial nonlinearity Exact tanh solu- tions Moderate Straightforward, suitable for certain NLPDEs Limited for high-degree nonlinearity Other Methods Specific types of NLPDEs Exact solu- tions in various High Powerful tool for certain NLPDEs Requires advanced tech- niques, limited applicability Table 1: Comparison table. of the extended direct algebraic method, we seek to unlock novel solutions and propel advancements in this field, ultimately enriching our understanding of nonlinear dynamics. Nonlinear evolution equations govern a wide range of physical phenomena, including wave dynamics in optics, fluid flow, and plasma physics. The intricate behavior exhibited by these systems, such as soliton formation and chaotic dynamics, stems from the interplay between nonlinearity and dispersion. By investigating exact solutions to these equations, we can gain a deeper understanding of the underlying physics and unlock new avenues for research and applications in fields like materials science and engineering. 1.1. Generalized Improved Boussinesq (GIBE) Equation The Generalized Improved Boussinesq Equation (GIBE) is utt−auxx−uxxtt−buxxt= ( u2 ) xx +cu3+duxxu, (1) x ∈ [a, b], t ∈ [0, T ], T > 0, (2) where u = u(x, t) is a wave profile with x demonstrating the spatial component and t indicating the temporal component, respectively. Equation (1) represents the generalized form of the classical Boussinesq framework that covers the dispersive and highly nonlinear behaviours of wave phenomena. In this equation, term uxxtt represents the dispersion of high order, uxxt reflects the spatiotemporal dissipative behavior, the terms ( u2 ) xx and u3 account for nonlinear effects and the nonlinear interaction term is modeled by the term N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 6 of 24 uxx u. Moreover, the constants a, b, c and d are the parameters representing the linear diffusion coefficient, rate of spatiotemporal dissipation and the rate of nonlinear reactions, respectively. 1.2. Overview of NEDAM Here, we provide a high-level overview of the proposed method. 1. Step 1: Suppose NLPDE R(W,Wt,Wx,Wtt,Wxt,Wxx, . . . ) = 0, (3) where W = W (x, t)symbolizes anunidentified function and Rsignifies a polynomial of W (x, t) with its arguments. 1. Step 2: The real variables x and t are combined to generate the compound variable W (x, t) = ϕ (ζ) , ζ = jx− ρt, (4) where ρ indicates the wave velocity and j indicates the wave number. Eq. 3 becomes an ordinary differential equation (ODE) when the travelingwave transformation Eq.(4) is employed π ( ϕ, ϕ′, ϕ′′, ϕ′′′, . . . ) = 0, (5) where π is a polynomial of ϕ with its derivatives. 1. Step 3: We assume that the following polynomial, which may be used to represent the trial solution to Eq. (5) ϕ (ζ) = N∑ i=0 αiΩ i (ζ) , αN ̸=0, (6) where αi(0 ≤ i ≤ N) are unknown constants and Ω(ζ) is a real-valued functionthat satisfies the auxiliary ODE, Ω′ (ζ) = ln (B) ( µ+ λΩ(ζ) + νΩ2 (ζ) ) , B ̸= 0, 1, (7) where µ, ν and λ are constants. The solution to Eq. (7) can be expressed as follows: 1. When λ2 −4µν<0 andν ̸=0 then, Ω1 (ζ) = − λ 2ν + √ − (λ2 − 4µν) 2ν tanB (√ − (λ2 − 4µν) 2 ζ ) , (8) Ω2 (ζ) = − λ 2ν − √ − (λ2 − 4µν) 2ν cotB (√ − (λ2 − 4µν) 2 ζ ) , (9) N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 7 of 24 Ω3 (ζ)= − λ 2ν + √ − (λ2−4µν) 2ν( tanB (√ − (λ2−4µν)ζ ) ±√ pqsecB (√ − (λ2−4µν)ζ )) , (10) Ω4 (ζ)= − λ 2ν − √ − (λ2−4µν) 2ν( cotB (√ − (λ2−4µν)ζ ) ±√ pqcscB (√ − (λ2−4µν)ζ )) , (11) Ω5 (ζ)= − λ 2ν + √ − (λ2−4µν) 4ν( tanB (√ − (λ2−4µν) 4 ζ ) − cotB (√ − (λ2−4µν) 4 ζ )) . (12) 2. When λ2 − 4µν > 0 and ν ̸= 0 then, Ω6 (ζ) = − λ 2ν − √ (λ2 − 4µν) 2ν tanhB (√ (λ2 − 4µν) 2 ζ ) , (13) Ω7 (ζ) = − λ 2ν − √ (λ2 − 4µν) 2ν cothB (√ (λ2 − 4µν) 2 ζ ) , (14) Ω8 (ζ) = − λ 2ν − √ (λ2 − 4µν) 2ν( tanhB (√ (λ2 − 4µν) ζ ) ± i √ pqsechB (√ (λ2 − 4µν) ζ )) , (15) Ω9 (ζ) = − λ 2ν − √ − (λ2 − 4µν) 2ν( cothB (√ (λ2 − 4µν) ζ ) ±√ pqcschB (√ (λ2 − 4µν) ζ )) , (16) Ω10 (ζ) = − λ 2ν + √ − (λ2 − 4µν) 4ν( tanhB (√ (λ2 − 4µν) 4 ζ ) + cothB (√ (λ2 − 4µν) 4 ζ )) . (17) 3. When µν > 0 and λ = 0 then, Ω11 (ζ) = √ µ ν tanB ( √ µν ζ) , (18) N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 8 of 24 Ω12 (ζ) = − √ µ ν cotB ( √ µν ζ) , (19) Ω13 (ζ) = √ µ ν (tanB (2 √ µν ζ)±√ pqsecB (2 √ µν ζ)) , (20) Ω14 (ζ) = − √ µ ν (cotB (2 √ µν ζ)±√ pqcscB (2 √ µν ζ)) , (21) Ω15 (ζ) = 1 2 √ µ ν ( tanB (√ µν 2 ζ ) − cotB (√ µν 2 ζ )) . (22) 4. When µν < 0 and λ = 0 then, Ω16 (ζ) = − √ −µ ν tanhB (√ −µν ζ ) , (23) Ω17 (ζ) = − √ −µ ν cothB (√ −µν ζ ) , (24) Ω18 (ζ) = − √ −µ ν ( tanhB ( 2 √ −µν ζ ) ± i √ pqsechB ( 2 √ −µν ζ )) , (25) Ω19 (ζ) = − √ −µ ν ( cothB ( 2 √ −µν ζ ) ±√ pqcschB ( 2 √ −µν ζ )) , (26) Ω20 (ζ) = −1 2 √ −µ ν ( tanhB (√ −µν 2 ζ ) − cothB (√ −µν 2 ζ )) . (27) 5. When λ = 0 and ν = µ then, Ω21 (ζ) = tanB(µζ), (28) Ω22 (ζ) = −cotB(µζ), (29) Ω23 (ζ) = tanB (2µζ)±√ pqsecB(2µζ), (30) Ω24 (ζ) = −cotB (2µζ)±√ pqcscB(2µζ), (31) Ω25 (ζ) = 1 2 ( tanB (µ 2 ζ ) − cotB (µ 2 ζ )) . (32) 6. When λ = 0 and ν = −µ then, Ω26 (ζ) = −tanhB(µζ), (33) Ω27 (ζ) = −cothB(µζ), (34) Ω28 (ζ) = −tanhB (2µζ)± i √ pqsechB (2µζ) , (35) Ω29 (ζ) = −cothB (2µζ)±√ pqcschB(2µζ), (36) Ω30 (ζ) = −1 2 ( tanhB (µ 2 ζ ) + cotB (µ 2 ζ )) . (37) N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 9 of 24 7. When λ2 = 4µν then, Ω31 (ζ) = −2µ(λζln(B) + 2) λ2ζln(B ) . (38) 8. When λ = χ, µ = rχ (r ̸= 0) and ν = 0 then, Ω32 (ζ) = Bχζ − r. (39) 9. When λ = ν = 0 then, Ω33 (ζ) = µζln(B) . (40) 10. When λ = µ = 0 then, Ω34 (ζ) = −1 νζln(B) . (41) 11. When λ ̸= 0, µ = 0 then, Ω35 (ζ) = − pλ ν(coshB(λζ)− sinhB (λζ) + p) , (42) Ω36 (ζ) = − λ(sinhB (λζ) + coshB (λζ)) ν(sinhB (λζ) + coshB (λζ) + q) . (43) 12. When λ = χ, ν = rχ (r ̸= 0) and µ = 0 then, Ω37 (ζ) = pBχζ p− rqBχζ . (44) The consequent claims state generalized hyperbolic and trigonometric functions in terms of the previous results. sinhB (ζ) = pBζ − pB−ζ 2 , coshB (ζ) = pBζ + pB−ζ 2 , tanhB (ζ) = pBζ − pB−ζ pBζ + pB−ζ , cothB (ζ) = pBζ + pB−ζ pBζ − pB−ζ , sechB (ζ) = 2 pBζ + pB−ζ , cschB (ζ) = 2 pBζ − pB−ζ , sinB (ζ) = pBiζ − pB−iζ 2 , cosB (ζ) = pBiζ + pB−iζ 2 , tanB (ζ) = −i pBiζ − pB−iζ pBiζ + pB−iζ , cotB (ζ) = i pBiζ + pB−iζ pBiζ − pB−iζ , secB (ζ) = 2 pBiζ + pB−iζ , cscB (ζ) = 2i pBiζ − pB−iζ , where ζ is an independent variable and p, q > 0. N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 10 of 24 1. Step 4: To calculate N for Eq. (6), we apply the homogeneous balancing procedure from Eq. (5). 2. Step 5: To create the strategic equations, just insert Eq. (6) and its derivatives into Eq. (5) and solve for coefficients of the same power of Ω(ζ) to zero. When Mathematica solves the strategic equations, we have to solve the system to obtain the values of the unknowns. 1.3. Exact Wave Solution The above equation (1) can be solved using the wave transformation u(x, t) = Q(ζ), where ζ = x− kt and k ̸= 0. By putting the relevant derivatives of above equation into Eq.(1), we get k2Q′′ (ζ)− aQ′′ (ζ)− k2Q′′′′ (ζ) + bkQ′′′ (ζ) = ( Q2 )′′ (ζ) + cQ3 (ζ) + dQ′′ (ζ)Q (ζ) . Also, we may write above Eq. as k2Q′′ − aQ ′′ − k2Q′′′′ + bkQ′′′ − ( Q2 )′′ − cQ3 − dQ′′Q = 0. (45) Application of NEDAM The new extended direct algebraic method’s applicability to nonlinear partial differential equations (NLPDEs) is notable, yet not all-encompassing. Its effectiveness is pronounced for equations with polynomial nonlinearity and single spatial dimensions. However, equa- tions with highly complex nonlinearity, non-polynomial terms, fractional derivatives, or high-dimensional systems may pose challenges, potentially limiting the method’s efficacy. Recognizing these boundaries enables researchers to judiciously apply this method and explore alternative approaches when confronted with complex NLPDEs. The suggested method is employed for appraising novel solutions to Eq. (1). Applying the homogeneous balance principle, which Eq. (45) yields, n = 2. So, the solution to Eq. (45) is assumed to be formulated as: Q(ζ) = b0 + b1Ω(ζ) + b2Ω 2 (ζ) . (46) By entering Eq. (46) and its derivatives into Eq. (45) and specifying coefficients of comparable powers of Ω(ζ) to zero, we can quickly get the strategic equations. We have Ω′(ζ) = ln(B)(µ+ λΩ(ζ) + νΩ2 (ζ)), B ̸= 0, 1. The system of algebraic equations may be solved by using the softwareMathematica to find the constant values. Family-1. b0= −6λ √ k4ν2 (λ2−4µν) ln4 (B)− 12k2µν2ln2 (B)+k2ν − aν 2ν , N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 11 of 24 b1 = −6 ( k2λνln2(B)+ √ k4ν2ln4(B)(λ2 − 4µν) ) , b2= −6k2ν2ln2(B), b = − 5 √ k4ν2 (λ2−4µν) ln4(B) kνln(B) , c = 0, d = 0. Family-2. b0 = 1 2 ( k2ln2(B)(− ( λ2 + 8µν ) ) + k2 − a ) , b1 = −6k2λνln2 (B) , b2 = −6k2ν2ln2 (B) , b = 0, c = 0, d = 0. The solutions to Eq. (1) for the family 2 can be summarized as follows. 1. For the cases λ2 − 4µν < 0 and ν ̸= 0, the following pair of outcomes that we find for both trigonometric as well as mixed-trigonometric solutions. ϕ1 (x, t) = 1 2ν × { G1) } + 1 2ν { ν ( 3k2ln2 (B) ( λ2 − 4µν ) +k2 − a ) + } 1 2ν { 3k2νln2 (B) ( λ2 − 4µν ) tan2 B (√ (4µν − λ2) 2 ζ )} , (47) where G1 = −6 √ 4µν − λ2 √ k4ν2ln4(B)(λ2 − 4µν)tanB( √ (4µν − λ2) 2 ζ), ϕ2 (x, t) = 1 2ν { {ν [ 3k2ln2 (B) ( λ2−4µν ) + k2 − a ] +G2 } + 1 2ν { 3k2νln2 (B) ( λ2−4µν ) cot2B (√ 4µν − λ2 2 ζ )} , (48) where G2 = 6 √ 4µν − λ2 √ k4ν2ln4(B)(λ2 − 4µν)tanB( √ (4µν − λ2) 2 ζ) (49) ϕ3 (x, t) = 1 2ν { ν [ 3k2ln2 (B) ( λ2−4µν ) + k2 − a ] − 6 √ k4ν2 (λ2−4µν) ln4 (B) √ 4µν − λ2 ( tanB (√ 4µν − λ2 ζ ) ±√ pqsecB (√ 4µν − λ2 ζ )) + 3k2νln2 (B) ( λ2 − 4µν ) × ( tanB (√ 4µν − λ2 ζ ) ±√ pqsecB (√ 4µν − λ2 ζ ))2} , (50) N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 12 of 24 ϕ4 (x, t) = 1 2ν { ν [ 3k2ln2 (B) ( λ2−4µν ) + k2 − a ] + 6 √ k4ν2 (λ2−4µν) ln4 (B) √ 4µν − λ2 ( cotB (√ 4µν − λ2 ζ ) ±√ pqcscB (√ 4µν − λ2 ζ )) + 3k2νln2 (B) ( λ2−4µν ) × (( cotB (√ 4µν − λ2 ζ ) ±√ pqcscB (√ 4µν − λ2 ζ )))2} , (51) ϕ5 (x, t) = 1 8ν { 4ν [ 3k2ln2 (B) ( λ2−4µν ) + k2 − a ] − 3 √ k4ν2ln4 (B) (λ2 − 4µν) 4 √ 4µν − λ2tanB (√ − (λ2 − 4µν) 4 ζ ) − cotB (√ − (λ2 − 4µν) 4 ζ ) + 3k2λνln2 (B)] × (λ2 − 4µν) ( tanB (√ − (λ2 − 4µν) 4 ζ ) − cotB (√ − (λ2 − 4µν) 4 ζ ))2} . (52) 1. For the cases λ2 − 4µν > 0 and ν ̸= 0, several results are discovered as follows. The way to solve the kink is expressed as ϕ6 (x, t) = 1 2ν × { ν [ 3k2ln2 (B) ( λ2−4µν ) + k2 − a ] +G3 −3k2νln2 (B)× ( λ2 − 4µν ) tanh2B (√ (λ2 − 4µν) 2 ζ )} , (53) where G3 = 6 √ k4ν2ln4 (B) (λ2 − 4µν)× √ (λ2 − 4µν)tanhB (√ (λ2 − 4µν) 2 ζ ) , ϕ7 (x, t) = 1 2ν × { ν [ 3k2ln2 (B) ( λ2−4µν ) + k2 − a ] +G4 −3k2νln2 (B)× ( λ2 − 4µν ) coth2B (√ (λ2 − 4µν) 2 ζ )} , (54) 6 √ k4ν2ln4 (B) (λ2 − 4µν)× √ (λ2 − 4µν)cothB (√ (λ2 − 4µν) 2 ζ ) ϕ8 (x, t) = 1 2ν ×{ν [ 3k2ln2 (B) (∆) + k2 − a ] + 6 √ k4ν2ln4 (B) (∆)×√ (∆) ( tanhB (√ (∆) ζ ) ± i √ pqsechB (√ (∆) ζ )) − 3k2νln2 (B) N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 13 of 24 × (∆) ( tanhB (√ (∆) ζ ) ± i √ pqsechB (√ (λ2 − 4µν) ζ ))2 }, (55) ϕ9 (x, t) = 1 2ν ×{ν [ 3k2ln2 (B) (∆) + k2 − a ] + 6 √ k4ν2ln4 (B) (∆)×√ (−∆) ( cothB (√ (∆) ζ ) ±√ pqcschB (√ (∆) ζ )) + 3k2νln2 (B)( λ2 − 4µν ) (( cothB (√ (∆) ζ ) ±√ pqcschB (√ (∆) ζ )))2 }, (56) ϕ10 (x, t) = 1 8ν ×{4ν [ 3k2ln2 (B) (∆) + k2 − a ] − 3 √ k4ν2ln4 (B) (∆)× 4 √ (4µν − λ2) ( tanhB (√ (∆) 4 ζ ) + cothB (√ (∆) 4 ζ )) + 3k2νln2 (B) ( λ2 − 4µν )(( tanhB (√ (λ2 − 4µν) 4 ζ ) + cothB (√ (λ2 − 4µν) 4 ζ )))2 }, (57) where ∆ = λ2 − µν, for the Eq.s (55), (56), (57) . (58) 2. For µν > 0 and λ = 0, ϕ11 (x, t) = {− 12 √ µ √ −k4µν3ln4 (B)tanB (√ µν ζ ) √ ν − 6k2µνln2 (B) ( tan2 B ( √ µν ζ) + 1 ) + 1 2 ( k2 − a ) }, (59) ϕ12 (x, t) = 12 √ µ √ −k4µν3ln4 (B)cotB (√ µν ζ ) √ ν − 6k2µνln2 (B) ( cot2B ( √ µν ζ) + 1 ) + 1 2 ( k2 − a ) . (60) ϕ13 (x, t) = { −12 √ µ √ −k4µν3ln4 (B) ( tanB ( 2 √ µν ζ ) ±√ pqsecB ( 2 √ µν ζ )) √ ν − 6k2µνln2 (B) ( ((tanB (2 √ µν ζ)±√ pqsecB (2 √ µν ζ)))2 + 1 ) + 1 2 ( k2 − a ) }, (61) ϕ14 (x, t) = { 12 √ µ √ −k4µν3ln4 (B) ( cotB ( 2 √ µν ζ ) ±√ pqcscB ( 2 √ µν ζ )) √ ν − 6k2µνln2 (B) ( (cotB (2 √ µν ζ)±√ pqcscB (2 √ µν ζ))2 + 1 ) + 1 2 ( k2 − a ) }, (62) N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 14 of 24 ϕ15 (x, t) = 1 2ν ×{ 12ν √ µ √ −k4µν3ln4 (B) ( cotB (√ µν 2 ζ ) − tanB (√ µν 2 ζ )) √ ν + ν [ −12k2µν2ln2 (B) + k2 − a ] − 3k2µν2ln2 (B) ( tanB (√ µν 2 ζ ) − cotB (√ µν 2 ζ ))2 }. (63) 1. For µν < 0 and λ = 0, ϕ16 (x, t) = {G4 + 6k2µνln2 (B) ( tanh2B (√ −µν ζ ) − 1 ) + 1 2 ( k2 − a ) , (64) where G4 = 12 √ −µ √ −k4µν3ln4 (B)tanhB ( √ −µν ζ) √ ν , ϕ17 (x, t) = G5 + 6k2µνln2 (B) ( coth2B (√ −µν ζ ) − 1 ) + 1 2 ( k2 − a ) , (65) where G5 = { 12 √ −µ √ −k4µν3ln4 (B)cothB ( √ −µν ζ) √ ν , ϕ18 (x, t) = { G6 ( tanhB (2 √ −µν ζ)± i √ pqsechB (2 √ −µν ζ) ) √ ν + 6k2µνln2 (B) (( tanhB ( 2 √ −µν ζ ) ± i √ pqsechB ( 2 √ −µν ζ ))2 − 1 ) + 1 2 ( k2 − a ) }, (66) where G6 = 12 √ −µ √ −k4µν3ln4(B), ϕ19 (x, t) = { 12 √ −µ √ −k4µν3ln4 (B) (Ω) √ ν + 6k2µνln2 (B) ( (Ω)2 − 1 ) + 1 2 ( k2 − a ) }, (67) where Ω = cothB ( 2 √ −µν ζ ) ±√ pqcschB ( 2 √ −µν ζ ) , ϕ20 (x, t) = 1 2ν ×{ G7 ( tanhB (√ −µν 2 ζ ) − cothB (√ −µν 2 ζ )) √ ν + N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 15 of 24 ν [ −12k2µν2ln2 (B) + k2 − a ] + 3k2µν2ln2 (B)G8}. (68) where G7 = 12ν √ −µ √ −k4µν3ln4 (B), G8 = ( tanhB (√ −µν 2 ζ ) − cothB (√ −µν 2 ζ ))2 , 2. For λ = 0 and ν = µ, ϕ21 (x, t) = { G9 − 1 2 k2 [ 12µ2ln2 (B) + 12µ2ln2 (B) tan2 B (µζ)− 1 ] − a 2 } , (69) where G9 = −12 √ −k4µ4ln4 (B)tanB (µζ) , ϕ22 (x, t) = { −1 2 k2 [ 12µ2ln2 (B) + 12µ2ln2 (B) cot2B (µζ)− 1 ] − a 2 } , (70) where G10 = 12 √ −k4µ4ln4 (B)cotB (µζ) , (71) ϕ23 (x, t) = {−12 √ −k4µ4ln4 (B) (tanB (2µζ)±√ pqsecB (2µζ)) −1 2 k2[12µ2ln2 (B) + 12µ2ln2 (B) (tanB (2µζ)±√ pqsecB (2µζ))2 − 1]− a 2 }, (72) ϕ24 (x, t) = {12 √ −k4µ4ln4 (B) (cotB (2µζ)±√ pqcscB (2µζ)) −1 2 k2[12µ2ln2 (B) + 12µ2ln2 (B) (cotB (2µζ)±√ pqcscB(2µζ)) 2 − 1]− a 2 }, (73) ϕ25 (x, t) = 1 2 ×{−12 √ −k4µ4ln4 (B) ( tanB (µ 2 ζ ) − cotB (µ 2 ζ )) −12k2µ2ln2 (B) + k2 − a− 3k2µ2ln2 (B) ( tanB (µ 2 ζ ) − cotB (µ 2 ζ ))2 }. (74) 1. For λ = 0 and ν = −µ, ϕ26 (x, t) = 12 √ k4µ4ln4 (B)tanhB (µζ) + 6k2µ2ln2 (B)− 6k2µ2ln2 (B) tanh2B (µζ) z + 1 2 (k2 − a), (75) N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 16 of 24 ϕ27 (x, t) = 12 √ k4µ4ln4 (B)cothB (µζ) + 6k2µ2ln2 (B)− 6k2µ2ln2 (B) coth2B (µζ) + 1 2 (k2 − a), (76) ϕ28 (x, t) = −12 √ k4µ4ln4 (B) (−tanhB (2µζ)± i √ pqsechB (2µζ)) + 6k2µ2ln2 (B)− 6k2µ2ln2 (B) ( − tanhB (2µζ)± i √ pqsechB(2µζ) )2 + 1 2 (k2 − a), (77) ϕ29 (x, t) = −12 √ k4µ4ln4 (B) (−cothB (2µζ)±√ pqcschB (2µζ)) + 6k2µ2ln2 (B)− 6k2µ2ln2 (B) ( − cothB (2µζ)±√ pqcschB(2µζ) )2 + 1 2 (k2 − a), (78) ϕ30 (x, t) = 1 2 ×{12 √ k4µ4ln4 (B) ( tanhB (µ 2 ζ ) + cotB (µ 2 ζ )) +12k2µ2ln2 (B) + k2 − a− 3k2µ2ln2 (B) ( tanhB (µ 2 ζ ) + cotB (µ 2 ζ ))2 }. (79) 2. For λ2 = 4µν, ϕ31 (x, t) = 1 λ2 ( 6k2µνlnB(λ2 − 4µν)(ln (B) + 4) ) + 1 2 (k2 − a)λ4ζ2 − 1 λ4ζ2 ( 96k2µ2ν2bigg). (80) 3. For λ = χ, µ = rχ (r ̸= 0)and ν = 0, ϕ32 (x, t) = 0. (81) 4. For λ = ν = 0, ϕ33 (x, t) = 0. (82) 5. For λ = µ = 0. ϕ34 (x, t) = { 1 2 ( k2 − a ) − 6k2 ζ2 } . (83) N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 17 of 24 6. For λ ̸= 0, µ = 0, we found the hyperbolic function solutions ϕ35 (x, t) = 3λ √ k4ν2λ2ln4 (B) (−coshB (λζ)− sinhB (λζ) + p) ν ( coshB (λζ)− sinhB (λζ) + p ) + 1 2 (k2 − a) + 6pλ2k2ln2(B) ( coshB(λζ)− sinhB(λζ) ) ( coshB(λζ)− sinhB(λζ) + p )2 ,(84) ϕ36 (x, t) = 1 2 ( 3λ2k2 ln2(B) + k2 − a ) − 3λ2k2 ln2(B) ( sinhB(λζ) + coshB(λζ) )2 ( sinhB(λζ) + coshB(λζ) + q )2 + 42k2νλ2 ln2(B) ( sinhB(λζ) + coshB(λζ) ) ν ( sinhB(λζ) + coshB(λζ) + q ) , (85) ϕ37 (x, t) = 1 2 6 √ k4ν2χ4ln4 (B) ( r (2p− q)Bχζ + p ) r ( rqBχζ − p ) + k2 − a − 6k2prχ2ln2 (B)Bχζ ( r (p− q)Bχζ + p ) ( p− rqBχζ )2 , (86) for all solutions ζ = x − ct. 2. Graphical Representation The traveling wave solutions are presented in Figure 1-9 in three types of diagrams, a contour plot on an arbitrary constants range, a two-dimensional plotline as well and a three-dimensional plotline by using Mathematica. Graphical representations are important because they may help us better understand solutions and analyze complex data. In soliton theory, wave propagation for the GIB model plays a significant role. N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 18 of 24 (a) (b) (c) Figure 1: Visual representation of the Eq. 47, with the suitable parametersµ = 1.2, ν = 1.5, λ = 0.6, a = 0.5, k = 1.2 and B = 0.9. (a) (b) (c) Figure 2: Graphical illustration of the Eq. 48, with the suitable parameters µ = 1.2, ν = −1.5, λ = 1.6, a = 0.5, k = 1.2 and B = 0.9 . (a) (b) (c) Figure 3: Visual representation of the Eq. 53, with the suitable parameters µ = 1.2, ν = −1.5, λ = 1.6, a = 0.5, k = 1.2 and B = 0.9. (a) (b) (c) Figure 4: Visual representation of the Eq. 56, with the suitable parameters µ = 1.2, ν = 1.5, λ = 1.6, a = 0.5, k = 1.2 and B = 0.9. N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 19 of 24 (a) (b) (c) Figure 5: Visual representation of the Eq. 58, with the suitable parameters µ = 1.2, ν = 1.5, λ = 0, a = 0.5, k = 1.2 and B = 0.9. (a) (b) (c) Figure 6: Visual representation of the Eq. 60, with the suitable parametersµ = 1.2, ν = 1.5, λ = 0, a = 0.5, k = 1.2, B = 0.9, p = 0.6 and q = 0.4. (a) (b) (c) Figure 7: Visual representation of the Eq. 68, with the suitable parametersµ = 1.2, ν = 1.5, λ = 0, a = 0.5, k = 1.2, B = 0.9, p = 0.6 and q = 0.4. (a) (b) (c) Figure 8: Visual representation of the Eq. 69, with the suitable parametersµ = 1.2, ν = 1.5, λ = 0, a = 0.5, k = 1.2, B = 0.9, p = 0.6 and q = 0.4. N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 20 of 24 (a) (b) (c) Figure 9: Graphical illustration of the Eq. 73, with the suitable parametersµ = 1.2, ν = 1.5, λ = 0, a = 0.5, k = 1.2, B = 0.9, p = 0.6 and q = 0.4. 3. Conclusions In conclusion, the new extended direct algebraic method has proven to be a potent tool for solving nonlinear evolution equations, offering valuable insights into their complex dynamics. By applying this method, researchers can unlock new exact solutions, shedding light on the intricate behavior of nonlinear systems. Future studies can further harness the potential of this method, exploring its applicability to diverse nonlinear equations and refining its capabilities to drive innovation in relevant fields. This work solved the generalized improved Boussinesq equation (GIBE), which is a math- ematical model that describes nonlinear processes, by applying the new extended direct algebraic method (NEDAM). Using creative methods and systematic investigation, a wide variety of soliton solutions were found, providing insight into the behavior of nonlinear systems. The solutions were classified as bright, dark, complicated and combination forms, as well as single forms like kink, anti-kink and solitary wave solutions among others. The derivation procedure revealed new families of periodic, hyperbolic and exponential wave solutions with arbitrary parameters. For certain parameter values, the results were dis- played using contour plots, 2D line plots and 3D surface plots. These findings significantly advance the field and enhance our understanding of complicated nonlinear systems.The kink wave solutions, bright and dark soliton solutions, and periodic solutions that are ob- tained in this research have vast applications in various realistic physical systems. Some of the fields of application are nonlinear optics, fluid dynamics, quantum systems, etc. The soliton obtained in this study describes the pulse propagation in an optical fiber under the conditions that balance both the nonlinearity and dispersion. Shallow water waves in fluid dynamics. Also, the dynamics of particle density distribution is modeled by the soliton solutions of the quantum system (say), Bose-Einstein condensation. We can conclude that the exact solution generated in this study portrays both the experimental and theoretical analyses across their domains. The investigation of soliton dynamics using both analytical and graphical techniques cre- ates new research opportunities and has implications for several scientific and technical fields. In addition to expanding theoretical knowledge, this work establishes a strong basis for future research on nonlinear processes. N. A. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6217 21 of 24 References [1] G. W. Wang and T. Z. Xu. Invariant analysis and exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation by Lie group analysis. Nonlinear Dynamics, 76:571–580, 2014. [2] Y. H. Yin, S. J. Chen, and X. Lü. Localized characteristics of lump and interaction solutions to two extended Jimbo–Miwa equations. Chinese Physics B, 29(12):120502, 2020. [3] X. Lü, Y. F. Hua, S. J. Chen, and X. F. Tang. 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