EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6235 ISSN 1307-5543 – ejpam.com Published by New York Business Global Exact Solutions of the Damped Telegrapher’s Equation with Harmonic Potential via the Generalized First Integral Method Muhammad Noman Qureshi1,∗, Atif Hassan Soori1, Zeshan Haider1, Waqar Azeem Khan2, Zohaib Arshad2 1 Department of Mathematics, Faculty of Basic and Applied Sciences, Air University, PAF Complex E-9, Islamabad 44000, Pakistan 4 Department of Mathematics, Faculty of Basic and Applied Sciences, Mohi-Ud-Din Islamic University, , AJK, 12010, Pakistan Abstract. This paper aims to develop exact analytical solutions for the telegrapher’s equation incorporating both damping and harmonic potential by employing the generalized first integral method. This approach extends the classical first integral technique through the use of Laurent polynomials, enhancing its ability to address complex nonlinear structures. The telegrapher’s equation, a fundamental model in applied mathematics and physics, describes wave propagation influenced by both dispersive and damping effects, with applications across various engineering and physical systems. By applying suitable transformations, the original nonlinear partial differential equation is reduced to an ordinary differential form. The generalized method is then utilized to derive exact solutions under different parametric conditions. These solutions offer valuable ana- lytical insight into how damping influences wave amplitude, speed, and qualitative behavior. In particular, the method effectively captures the modulation in wave attenuation and propagation characteristics caused by dissipative effects. The key outcome of this study is the demonstration that the generalized first integral method serves as a robust and versatile analytical tool for solv- ing nonlinear damped wave models, where conventional methods often encounter limitations. Its strength lies in simplifying complex nonlinear systems while preserving essential physical effects, providing precise analytical descriptions of wave behavior. Additionally, three-dimensional graph- ical visualizations of the obtained solutions offer a detailed understanding of the system’s spatial and temporal dynamics. This work contributes to the ongoing advancement of analytical tech- niques for nonlinear evolution equations and establishes a foundation for extending this method to other complex dynamical systems involving damping, dissipation, and external potentials. 2020 Mathematics Subject Classifications: 35F20, 35A01, 35R35, 35G20, 34A36 Key Words and Phrases: Telegrapher’s equation, Generalized first Integral method, Laurent polynomials, Wave transformation, Autonomous system ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6235 Email addresses: 211900@students.au.edu.pk (M. Noman), atif.soori@au.edu.pk (A. Soori) 211904@students.au.edu.pk, (Z. Haider), waqar qau85@gmail.com, (W. Azeem), zohaibsalam.pcs@gmail.com, (Z. Arshad) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 2 of 16 1. Introduction Nonlinear partial differential equations play a crucial role in accurately modeling com- plex physical, biological, and engineering systems where linear models often fail to capture intricate dynamics. These equations are fundamental in describing a wide range of phe- nomena, including fluid dynamics, heat transfer, nonlinear optics, population dynamics, and financial markets, where interactions among variables are inherently nonlinear. The importance of nonlinear PDEs is underscored by their ability to describe real-world pro- cesses with greater fidelity, accounting for effects such as turbulence, shock waves, pattern formation, and chaotic behavior. In recent years, fractional calculus has emerged as a powerful tool in modeling complex dynamical systems, offering more accurate descrip- tions of memory and hereditary properties inherent in various physical and engineering processes [1], [2], [3]. Coupled with advances in numerical methods, including finite differ- ence schemes and spectral techniques, these developments have significantly enhanced the analysis and solution of nonlinear partial differential equations [4], [5]. The incorporation of fractional derivatives and novel computational approaches has expanded the capabil- ity to capture anomalous diffusion, viscoelasticity, and chaotic behaviors across multiple scientific domains. Recent studies highlight the growing interdisciplinary interest and pro- vide a strong foundation for further exploration of nonlinear wave equations with damping and complex potentials. Despite their significance, solving nonlinear PDEs poses a sub- stantial challenge due to the absence of universal solution methods. As a result, various analytical techniques such as perturbation methods, similarity transformations, and vari- ational approaches are often employed. These are complemented by numerical techniques like the finite element method, finite difference schemes, and spectral methods. However, many traditional analytical methods suffer from limitations such as the requirement for small parameters, approximate series expansions, or difficulties in handling damping and potential terms in complex equations. utt + (α+ β)ut + αβ(u) = c2uxx, (1) commonly referred to as the telegrapher’s equation with damping and harmonic potential, describes a system where wave propagation is influenced by both temporal damping and spatial diffusion. Here, utt represents the time acceleration, (α + β)ut is a damping term reflecting system resistance or friction, and αβu introduces a harmonic potential. The term c2uxx corresponds to wave propagation, where c denotes the wave speed. This equation serves as a core model in describing physical processes such as signal transmission, mechanical vibrations, and heat conduction, particularly in contexts where both wave-like behavior and energy dissipation are significant [6], [7]. The parameters α, β, and c have distinct physical interpretations that govern the dynamics of the system: • Damping coefficients α and β: These parameters characterize the dissipative effects within the system. The term (α+β)ut models damping forces proportional to velocity, representing resistance or frictional effects that gradually reduce the wave amplitude over time. Meanwhile, αβu acts as a harmonic potential, providing a M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 3 of 16 restoring force proportional to the displacement, which can model elastic or reactive effects in the medium. Physically, α and β may correspond to different mechanisms of energy loss or gain, such as electrical resistance in transmission lines, viscous damping in mechanical vibrations, or absorption effects in wave propagation through lossy media. • Wave speed c:under The parameter c represents the characteristic speed of wave propagation through the medium in the absence of damping and potential effects. It governs the dispersive term c2uxx, modeling spatial diffusion or wave transmission. In physical contexts, c may represent the speed of signal transmission in telegraph lines, the speed of sound in vibrating strings, or diffusion rates in thermal and fluid systems. Together, these parameters control the interplay between wave propagation, dissipation, and restoring forces, shaping solution characteristics such as amplitude attenuation, wave speed modulation, and oscillatory behavior. In recent years, there has been considerable interest in developing both analytical and numerical methods for solving damped wave equations and related nonlinear models. Techniques such as the exp-function method, tanh-coth method, sine-Gordon expansion method, and homotopy analysis method have been applied to obtain approximate and exact traveling wave solutions in various contexts [8], [9], [10], [11]. Additionally, advanced numerical algorithms and hybrid analytical- numerical techniques have been explored to tackle increasingly complex models incorpo- rating variable coefficients, memory effects, and fractional derivatives. These developments have significantly broadened the range of solvable models, although challenges remain in deriving exact solutions for systems involving both nonlinearities and damping. Tradi- tional methods for solving the telegrapher’s equation, especially in the presence of damp- ing, often face difficulties due to their reliance on complex or approximate procedures [12], [13], [14], [15], [16], [17], [18], [19]. Many of these techniques are limited in their ability to derive exact, closed-form solutions, especially when the equation involves nonlinear terms and damping effects. The Generalized First Integral Method provides a more effective approach to overcoming these limitations. This method extends the classical first integral technique by introducing Laurent polynomials instead of traditional polynomials, allow- ing for a broader class of solutions. Additionally, it facilitates a wave transformation that reduces the PDE to an ordinary differential equation, significantly simplifying the solution process. Unlike perturbative or numerical techniques, Generalized First Integral Method allows for the direct construction of exact traveling wave solutions. In this study, we conduct an analytical investigation of the telegrapher’s equation with damping and harmonic potential using the Generalized First Integral Method. Through an appropriate wave transformation, we reduce the equation to an form and systemati- cally derive exact solutions under varying conditions. These solutions offer insight into how damping influences wave behavior, including attenuation, distortion, and propagation speed variation. The Generalized First Integral Method not only simplifies the analytical process but also deepens our understanding of dissipative wave phenomena. Given its applicability to problems in signal transmission, mechanical systems, and thermal con- M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 4 of 16 duction, this method serves as a valuable tool in both theoretical research and practical modeling within applied mathematics. The structure of the paper is as follows. In Section 2, we provide a concise overview of the Generalized First Integral Method. Section 3 applies this method to derive exact solutions of the telegrapher’s equation with damping and harmonic potential. Section 4 presents a detailed discussion of the results, including physical interpretations and stability consid- erations. Finally, Section 5 offers graphical representations of the obtained solutions to illustrate their spatial and temporal behaviors. 2. Structural Layout of the Generalized First Integral Method Step 1. Consider a general nonlinear PDE in the form F (u, ux, ut, uxx, uxt, ...) = 0. (2) To find the travelling wave solutions to Equation (2), we introduce the wave variable ξ = x− ct, (3) u(x, t) = u(ξ), (4) Differentiating equation (3) and (4), according to (2) we have ux = u ′ (ξ), (5) ut = −cu′ (ξ), (6) uxx = u ′′ (ξ), (7) uxt = −cu′′ (ξ), (8) and so on for the other derivatives. Using equation (5-8) the PDE (2) is converted to an Ordinary differential equation of the form F (u, u ′ ,−cu′ , u ′′ ,−cu′′ , ...) = 0. (9) where u = u(ξ) is an unknown function, F is a polynomial in the variable u and its derivatives. Step 2. Suppose the solution of ODE (9) can be written as follows: U(x, t) = f(ξ), (10) and furthermore, we introduce a new independent variable{ X(ξ) = f(ξ) Y (ξ) = f ′(ξ) (11) M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 5 of 16 Step 3. Under the conditions of Step 2, Equation (9) can be converted to a system of nonlinear ODEs as follows { X ′(ξ) = Y (ξ) Y ′(ξ) = F (X(ξ), Y (ξ)) (12) If we are able to find the integrals for Equation (12), the general solutions to these equa- tions can be determined directly. However, in most cases, even finding a single first integral is challenging. This difficulty arises because, for a given autonomous system, there is no systematic theory that explicitly guides us on how to find its first integrals, nor is there a clear method for identifying what these first integrals are. To address this, we will apply the Division Theorem for Laurent polynomials to obtain one first integral for Equation (13) and (14) [20], which will reduce Equation (9) to a first or second order, integrable ordinary differential equation (ODE). The exact solution to Equation (9) can then be obtained by solving this reduced equation. Division theorem for Laurent Polynomials Suppose P (ω, z) and Q(ω, z) are two Laurent polynomials in C(ω, z) and P (ω, z) is ir- reducible in C[ω, z]. If Q(ω, z) vanishes at any zero point of P (ω, z) then there exist a Laurent polynomial G(ω, z) in C[ω, z] such that Q(ω, z)= P (ω, z) ·G(ω, z). 3. Employing a Generalized First Integral Method for Solving the Telegrapher’s equation with damping and harmonic potential Consider the wave variable, ξ = x+ t, (13) and the function transformation as U(x, t) = f(ξ), (14) Differentiating equations (13) and (14) according to (1), we have Ut = f ′ (ξ), Utt = f ′′ (ξ), Ux = f ′ (ξ), Uxx = f ′′ (ξ), Now, equation(1) becomes f ′′(ξ) = ( −(α+ β) 1− c2 ) f ′(ξ)− ( αβ 1− c2 ) f(ξ). (15) Equation (15) is the required ODE of order two, let { X(ξ) = f(ξ) Y (ξ) = f ′(ξ) (16){ X ′(ξ) = f ′(ξ) Y ′(ξ) = f ′′(ξ) M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 6 of 16 X ′(ξ) = Y (ξ) Y ′(ξ) = ( −(α+ β) 1− c2 ) Y − ( αβ 1− c2 ) X Thus, equation (15) is equivalent to the two dimensional autonomous system X ′(ξ) = Y (ξ) Y ′(ξ) = ( −(α+ β) 1− c2 ) Y − ( αβ 1− c2 ) X (17) let X(ξ) and Y (ξ) be non-trivial solutions of equation (16) and let Q(X,Y ) = β∑ k=α ak(X)Y k is an irreducible laurent polynomial in C[X(ξ), Y (ξ)] such that q[X(ξ), Y (ξ)] = β∑ k=α ak(X)Y k = 0 (18) where ak(X)(k = 0, 1, 2, ..., ) are Laurent polynomials of X. Above equation is said to be the first integral of Eq (16). By applying division theorem, a laurent polynomial [g(X) + h(X)Y ] exist in the complex domain C[X,Y ] such that, by chain rule dq dξ = β∑ k=α a′k(X)Y k+1 + β∑ k=α kak(X)Y k · ( −(α+ β) 1− c2 ) + β∑ k=α kak(X)Y k−1 · ( −αβ 1− c2 X ) (19) [g(X) + h(X)Y ] [ β∑ k=α ak(X)Y k ] = [ β∑ k=α a′k(X)Y k+1 ] + [ β∑ k=α kak(X)Y k ] · ( −(α+ β) 1− c2 ) + [ β∑ k=α kak(X)Y k−1 ] · ( −αβ 1− c2 X ) (20) now for α = 0 and β = 1 equation (17) becomes [g(X) + h(X)Y ] { a0(X)Y 0 + a1(X)Y 1 } = { a′0(X)Y 1 + a′1(X)Y 2 } + { a1(X)Y 1 } · ( −(α+ β) 1− c2 ) + { a1(X)Y 0 } · ( −αβ 1− c2 X ) (21) M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 7 of 16 comparing coefficients of Y 2, Y 1, Y 0 Y 2; h(X)a1(X) = a ′ 1(X). (22) Y 1; h(X)ao(X) + g(X)a1(X) = a ′ (0)(X)− a1(X) α+ β 1− c2 . (23) Y 0; g(X)ao(X) = −a1(X)( αβ 1− c2 X). (24) from equation (21) we suppose a Laurent polynomial h(X) = 1 X , so equation (21) =⇒ 1 X = a ′ o(X) ao(X) , Integration gives ao(X) = C1(X). (25) equation (22) =⇒ a0(X) X + C1Xg(X) = a′0(X)− C1X · [ α+ β 1− c2 ] . (26) C1Xg(X) + C1X · [ α+ β 1− c2 ] = a′0(X)− a0(X) X . (27) C1g(X) + C1 · [ α+ β 1− c2 ] = d dX ( a0(X) X ) . (28) Integrating (27) C1 ∫ g(X)dX + C1X · α+ β 1− c2 + C2 = a0(X) X , C1X ∫ g(X)dX + C1X 2 · α+ β 1− c2 + C2X = a0(X). (29) For equation (21) and (22), equation (23) =⇒[ C1X ∫ g(X) dX + C1X 2 · α+ β 1− c2 + C2X ] g(X) = −C1X 2 ( αβ 1− c2 ) [ g(X) ∫ g(X) dX +Xg(X) · α+ β 1− c2 +Ag(X) ] = −X · αβ 1− c2 (30) M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 8 of 16 where A = C2 C1 In applying the Generalized First Integral Method, the function g(X) is assumed as a constant B to simplify the integration process, with its value subsequently determined from the consistency condition of the reduced equation, expressed in terms of the system parameters α, β, and c, so equation(29) =⇒ B2X +BX · [α+ β 1− c2 ] +BA = −X[ αβ 1− c2 ]. (31) compairing Power of X on both sides of (30) X1; B2 +B · [ α+β 1−c2 ] = −αβ 1−c2 . X0; AB = 0. Here A = 0 because B ̸= 0 Now, =⇒ B2 +B · [ α+β 1−c2 ] = −αβ 1−c2 , Quadratic in B B = −(α+β) 1−c2 ± √ (α+β)2−4(αβ)(1−c2) (1−c2)2 2 . (32) g(X) = B where B is the solution of above quadratic equation. Now equation (23) =⇒ a0(X)g(X) = −C1X ( αβ 1− c2 ) X a0(X) = −C1X 2 ( αβ 1−c2 ) B , Now, for α = 0, β = 1 equation (17) =⇒ a0(X) + a1(X)Y = 0, Y = −a0(X) a1(X) , Y = (αβ)X B(1− c2) , dξ dX = 1 (αβ)X B(1− c2) , M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 9 of 16 Integration gives ξ = B(1− c2) αβ lnX +K, ξ −K = B(1− c2) αβ lnX, αβ(ξ −K) B(1− c2) = lnX =⇒ X = f(ξ) = u = exp ( αβ(ξ −K) B(1− c2) ) . (33) Where ξ = x+ t and B = −(α+ β) 1− c2 ± √√√√(α+ β)2 − 4αβ(1− c2) (1− c2)2 2 . Equation (32) is the required solution of Telegrapher’s equation with damping and har- monic potential. This solution best describes underdamped or overdamped wave propaga- tion depending on the discriminant (α+ β)2 − 4αβ(1− c2): • If the discriminant is positive, the system is overdamped, leading to non-oscillatory exponential decay. • If the discriminant is zero, the system is critically damped. • If the discriminant is negative, the system exhibits underdamped oscillations, where waves persist but are modulated by exponential decay. M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 10 of 16 Figure 1: Three-dimensional plot of the exact solution u1(x, t) showing rapid exponential growth as t → 0+ and x → 0. The solution represents a highly localized pulse that sharply increases in amplitude. Physically, this may correspond to a resonance or instability triggered by specific combinations of damping parameters α and β, where energy is not dissipated fast enough to suppress the growth. Figure 2: Plot of the exact solution u2(x, t) demonstrating smooth exponential growth as t → 0, with wave attenuation visible along the spatial axis. This solution reflects a scenario where the damping cannot fully prevent energy concentration near the origin. The result can model physical systems with weak dissipation or external energy input, such as electrical pulses or signals in low-resistance media. M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 11 of 16 Figure 3: Plot of u3(x, t), exhibiting rapid spatial decay with increasing x and temporal flattening as t increases. The solution characterizes a strongly damped wave where energy dissipates quickly in both time and space. Such behavior is typical in overdamped electrical or thermal systems, where signals are quickly attenuated. Figure 4: Three-dimensional surface plot of the approximate solution u4(x, t) to the damped wave equation utt + (α + β)ut + αβu = c2uxx. The plot illustrates the temporal and spatial evolution of the solution. The exponential growth in u4(x, t) over time, visible from the surface, indicates a scenario where the damping effect is insufficient to counteract the restoring and wave propagation forces. The parameters α, β, and c significantly influence the system’s dynamic response. M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 12 of 16 4. Results and Discussion The provided graphs represent the explicit solutions to the telegrapher’s equation utt + (α+ β)ut + (αβ)u = c2uxx, with the solution expressed as X = f(ξ) = u = exp ( αβ(ξ−K) B(1−c2) ) , and B is the perimeter depending on α, β and c defined by B = −(α+ β) 1− c2 ± √√√√(α+ β)2 − 4αβ(1− c2) (1− c2)2 2 , Each of the four 3D surface plots corresponds to different choices of the parameters α, β, c and K highlighting the diverse behaviors this solution can capture. Figure 1( Graph of u1(x, t) displays a highly singular behavior centered near x = 0 t = 0. The solution grows extremely large in a very narrow region, suggesting that for spe- cific parameter values, the system experiences rapid and significant amplification. This behavior models real-world phenomena where initial conditions or parameter choices lead to instability or catastrophic failure, such as resonance in mechanical systems or blow-up in nonlinear media. Figure 2( Graph of u2(x, t) shows rapid growth along the time axis while remaining relatively flat along the spatial direction. This indicates that the temporal damping is not sufficient to counteract the intrinsic growth, making this solution appropriate to model systems where energy input over time leads to an unstable build-up, such as certain elec- trical circuits or unstable feedback systems. Figure 3( Graph of u3(x, t) portrays a localized sharp peak at an early time that dis- sipates as time progresses. This suggests a pulse-like or shock-like solution that decays naturally due to the damping effects. It captures scenarios like dissipative wave propaga- tion, where an initial disturbance gradually loses energy as it travels, which is crucial in fields like seismology, material sciences, and acoustic wave modeling. Figure 4( Graph of u4(x, t) differs significantly by exhibiting a smooth, controlled in- crease in amplitude without any signs of instability. Here, the solution remains bounded and grows predictably, which corresponds to stable regimes where the damping dominates and prevents any explosive behavior. Such behavior is desirable in engineering applica- tions where stability under oscillations is critical, such as suspension bridges, skyscraper design, and vibration control in aerospace structures. The advantage of this explicit solution structure is profound. It provides a direct and exact way to understand how damping parameters α, β and the wave propagation speed c interact to govern the overall behavior of the system. Analytical solutions like these are extremely valuable because they offer insight into the qualitative behavior of the system without requiring extensive numerical simulations. They allow for a clear understanding of stability conditions, energy dissipation, and the critical thresholds for transition from M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 13 of 16 stability to instability. Furthermore, in practical applications, having an explicit solu- tion enables engineers and scientists to design systems with desired dynamic properties. By adjusting the parameters based on the explicit formula, one can tailor the system’s response to prevent dangerous instabilities or to harness controlled amplification where needed (such as signal boosting). The importance of such solutions extends to various fields, including mechanical and civil engineering, physics, materials science, and control theory. In conclusion, the diverse behaviors captured by these graphs ranging from explosive growth to smooth decay highlight the richness of the dynamics described by the damped wave equation. The explicit solutions serve as powerful tools both for theoretical analysis and for practical design and optimization in complex systems. 5. Stability Analysis of the Exact Solution To assess the stability of the exact traveling wave solution f(ξ) = exp ( αβ(ξ −K) B(1− c2) ) , we consider a small perturbation of the form, u(ξ, t) = f(ξ) + ϵϕ(ξ, t), where 0 < ϵ≪ 1 and ϕ(ξ, t) denotes the perturbation function. Substituting this perturbed solution into the original equation and linearizing by ne- glecting terms of order ϵ2 and higher, we obtain the linearized equation governing the perturbation dynamics: ϕtt + (α+ β)ϕt + αβϕ = c2ϕxx. Assuming a solution of the form ϕ(ξ, t) = eλtψ(ξ), where λ is the growth rate param- eter, leads to the characteristic equation λ2 + (α+ β)λ+ αβ = 0. The roots of this quadratic equation are λ = −(α+ β)± √ (α+ β)2 − 4αβ 2 . For positive damping coefficients α > 0 and β > 0, both roots have negative real parts. This implies that any small perturbation will decay exponentially over time, confirming the linear stability of the exact traveling wave solution. Hence, the solution obtained through the generalized first integral method is stable and physically relevant for modeling wave propagation under the influence of damping and harmonic potential. M. Noman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6235 14 of 16 6. Conclusions This paper introduces a novel approach to solving partial differential equations ex- actly, by replacing conventional polynomials with Laurent polynomials within the first integral method. This innovation is grounded in the Division Theorem for Laurent poly- nomials in two complex variables, which provides a more flexible and powerful algebraic framework for deriving first integrals. To validate the proposed technique, we revisit and apply the Euclidean Algorithm for Laurent polynomials, demonstrating its effectiveness in simplifying the solution process for nonlinear differential equations. The core contribu- tion of this work is the development of the generalized first integral method, a new tool for obtaining exact solutions to complex dynamical systems, particularly those involving damping effects and external potentials. This method not only facilitates the derivation of precise analytical solutions but also highlights the critical role of algebraic methods, particularly the properties of Laurent polynomials, in advancing the theory of nonlinear evolution equations. By transforming complex systems into integrable first-order ordi- nary differential equations, we were able to uncover exact solutions that provide valuable insights into the dynamic behavior of such systems. The importance of this approach extends beyond the specific case studied here. The generalized first integral method rep- resents a significant advancement in the analytical study of nonlinear systems, offering a systematic and efficient tool for obtaining exact solutions to a wide range of partial differential equations that involve damping or dissipation. The technique is applicable to various fields of applied mathematics, such as fluid dynamics, signal processing, and heat diffusion, where the need for exact solutions to nonlinear models is critical. In conclusion, this paper not only introduces an effective new method for solving nonlinear partial dif- ferential equations but also contributes to the broader analytical framework available for the study of complex dynamical systems. Future research could explore the application of the generalized first integral method to other types of nonlinear differential equations and more complicated systems involving variable coefficients or coupled components. Further- more, extending this method to higher-dimensional problems and systems with spatially or temporally varying damping and potential terms could significantly broaden its appli- cability. Another valuable direction would be integrating this technique with numerical or perturbation-based approaches to develop hybrid analytical-numerical frameworks for problems where exact solutions are difficult to obtain. These prospective extensions would enhance the versatility of the method and promote its adoption in various scientific and engineering disciplines. Acknowledgements The authors Muhammad Noman Qureshi, Dr. Atif Hassan Soori, Zeshan Haider, Dr. Waqar Azeem Khan and Zohaib Arshad would like to thankful the Air Marshal Abdul Moeed Khan, HI(M), Vice Chancellor, Air University, Islamabad, Pakistan. Competing interests The authors declare that they have no competing interests. Data Availability Statement Please contact the authors for data requests. M. Noman et al. / Eur. J. 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