Bi-directional solitons of dual-mode Gardner equation derived from ideal fluid model

Results in Physics 57 (2024) 107337 Available online 9 January 2024 2211-3797/© 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Results in Physics journal homepage: www.elsevier.com/locate/rinp Bi-directional solitons of dual-mode Gardner equation derived from ideal fluid model Sadia Sadiq a, Ahmad Javid a, Muhammad Bilal Riazb,c,∗, Ghada Ali Basendwahd, Nauman Raza e,f aSchool of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan bIT4Innovations, VSB – Technical University of Ostrava, Ostrava, Czech Republic cDepartment of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon dDepartment of Mathematics, King Abdulaziz University P.O. Box 80203, Jeddah 21589, Saudi Arabia eDepartment of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan fDepartment of Mathematics, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey ARTICLE INFO Keywords: Tanh/coth Tan/cot Dual-mode waves Solitons ABSTRACT In this article, the main goal is to find the exact solution to the dual-mode Gardner equation derived from the ideal fluid model. To achieve this, two distinct methods will be employed: the tan ∕ cot method and the tanh ∕ coth method. Using these methods, the problem can be seen through different angles, providing a full understanding of how the equation works and the various solutions it provides. Furthermore, the periodic, kink and singular solutions will be illustrated through the use of 3D and 2D graphs. This visualization will clearly show how the phase velocity parameter affects the solutions, improving our understanding of wave dynamics within the dual-mode Gardner equation. The findings from this study help in understanding how both solitons and periodic solutions influence wave behavior in shallow water and how light travels through optical fibers. Introduction Surface waves in shallow water have been the subject of extensive research due to their remarkable features. These waves are often described using mathematical models, and one of the fundamental equations used for this purpose is the Korteweg–de Vries (KdV) equation. The KdV equation is notable for its well-defined solutions and plays a crucial role in understanding the dynamics of shallow-water waves [1]. The original creation of this equation can be linked to its development as a single-direction, nonlinear wave equation [1]. This result, derived through an asymptotic expansion of the fundamental wave motion within the shallow water Euler equations which is closely related to the study of small waves in shallow water with a free surface. In this context, the significance of two parameters becomes evident: 𝛼which measures wave size relative to water depth and 𝛽which examines the square of water depth concerning wave length. In this context, the equation addresses two critical factors: linear dispersion which involves the spreading and changing of waves, and nonlinear steepening where waves become taller in shallow water. This equation is essential for understanding how solitary waves behave in these conditions and for studying the properties of periodic waves, helping us ∗Corresponding author. E-mail addresses: [email protected] (S. Sadiq), [email protected] (A. Javid), [email protected] (M.B. Riaz), [email protected] (G.A. Basendwah), [email protected] (N. Raza). to understand the regular patterns and oscillations seen in various wave phenomena. Now, here’s the important part: this equation changes a lot when 𝛼and 𝛽have very different sizes. In other words, when the size of the waves is different from the depth of the water, It effects how these waves move and behave notably. Equations that deal with higher-order effects, such as the higher-order KdV [2] equations, follow the same rule. The derivation of this equation can be associated with (2 +1)-dimensional Gardner equation [3], making clear how the sizes of 𝛼and 𝛽are related: 𝛽=𝛼2and 𝛾=𝛼3. In our research, it is important to note that we are studying a specific situation with a flat bottom. This involves setting a particular parameter 𝛿= 0. This choice is significant because it simplifies our analysis. Our study revolves around the Gardner equation, which was initially derived in a previous work [4]. 𝑢𝑡+𝑢𝑥+3 2𝛼 𝑢𝑢𝑥−3 8𝛼2𝑢2𝑢𝑥+𝛽(1−3𝜏 6)𝑢𝑥𝑥𝑥 = 0.(1) This equation is fundamental to the analysis and is employed to investigate the behavior of certain non-dimensional parameters: 𝛼=𝑎∕𝐻, wavelength parameter 𝛽=𝐻2∕𝐿2and the bond number 𝜏=𝑇∕𝜌𝑔𝐻2. These non-dimensional parameters are essential to our research because https://doi.org/10.1016/j.rinp.2024.107337 Received 1 December 2023; Received in revised form 21 December 2023; Accepted 7 January 2024 Results in Physics 57 (2024) 107337 2 S. Sadiq et al. they help us to understand the behavior of the system under consideration. In this context, 𝑔represents the gravitational acceleration, 𝐻 signifies the average upstream depth, 𝑎and 𝐿denote characteristic values for wave amplitude and wavelength. We also have 𝑇as the coefficient for surface tension and 𝜌representing the density of water. In our research, we plan to use the approach introduced by Korsunsky [5] to create a new dual-mode model for the equation mentioned in Eq. (1). 𝑢𝑡𝑡 −𝑠2𝑢𝑥𝑥 +( 𝜕 𝜕𝑡 −𝜂𝑠 𝜕 𝜕𝑥 )𝑁(𝑢, 𝑢𝑥,…)+( 𝜕 𝜕𝑡 −𝜇𝑠 𝜕 𝜕𝑥 )𝐿(𝑢𝑥𝑥, 𝑢𝑥𝑥𝑥,…)= 0.(2) In this equation, the function denoted as 𝑢(𝑥, 𝑡)represents a field function defined over the domain −∞ < 𝑥, 𝑡 < ∞. Additionally, we have constraints on two parameters: 𝜂which characterizes nonlinearity and falls within the range of |𝜂|≤1and 𝜇representing dispersion and also constrained within |𝜇|≤1. Furthermore, 𝑠denotes the phase velocity. Building upon the approach outlined in Eq. (2), we transform Eq. (1) into a dual-mode model which is as follows; 𝑢𝑡𝑡 −𝑠2𝑢𝑥𝑥 +(𝜕 𝜕𝑡 −𝜂𝑠 𝜕 𝜕𝑥 )(3 2𝛼𝑢𝑢𝑥−3 8𝛼2𝑢2𝑢𝑥) +(𝜕 𝜕𝑡 −𝜇𝑠 𝜕 𝜕𝑥 )(𝛽(1 − 3𝜏) 6𝑢𝑥𝑥𝑥 +𝑢𝑥)= 0.(3) Scholars extended Korsunsky’s work on the KdV equation to explore dual-mode equations [6]. Wazwaz, for instance, addressed the dualmode modified KdV and dual-mode higher-order KdV equations [6]. In a comparable manner, H. M. Jaradat, M. Alquran and M. I. Siyam conducted an insightful study centered on the dual-mode Kuramoto– Sivashinsky equation [7]. Furthermore, D. M. Ambrose and A. L. Mazzucato have undertaken the task of investigating global solutions for the dual-mode Kuramoto–Sivashinsky equation, with a particular emphasis on identifying exact solutions [8]. A. M. Wazwaz’s contributions extended to discover multiple kink solutions within the dual-mode Sharma–Tasso–Olver equation and the dual-mode Fourth-Order Burgers equation [9]. In the field of nonlinear evolution equations, various research approaches have been developed, each offering distinct techniques for determining exact solutions [10]. These methods are important for dealing with the complicated aspects of these equations. Some of the methods, as previously discussed, include 𝐺′∕𝐺-expansion method [11], tanh expansion approach [12], Kudryshov technique [13], extended F-expansion procedure [14] and rational sin∕cos strategy [15] among others. Various scholars [16,17] have widely used these strategies to address the numerous issues offered by nonlinear evolution equations. Moreover, researchers have explored additional effective methods that have proven successful in this context. These methods encompass the inverse scattering transform [18], Hirota bilinear method [19], auxiliary equation technique [20], Darboux transformation [21], Painlevé analysis [22] and the Backlund–Darboux transformation [23]. These many methodologies, when combined, give complete information for exploring and solving nonlinear evolution equations, considerably contributing to our comprehensive understanding of their solutions and behavior [24–26]. This article is based on the combined knowledge and approaches created by well-known researchers in the fields of dual-mode equations and nonlinear evolution equations, all of which contribute to our understanding of the dual-mode Gardner equation derived from ideal fluid model. To the best of our knowledge, no previous study on dual-mode equations has attempted to solve this model; thus, in this paper, we set out to do so. We use a variety of techniques, such as tanh∕coth [27] and tan∕cot [28] with the main goal of identifying exact solutions. The article is organized into four sections: In the Section ‘‘Introduction’’, a brief history of dual-mode equations and the Gardner equation is presented. The Section ‘‘Strategy’’ delves into the methodology employed in the paper. The Section ‘‘Results and graphical analysis’’ focuses on the discussion and graphical representations of the findings. The final section concludes the paper. Strategy We will outline a comprehensive methodology consisting of four key stages for addressing a general nonlinear PDE represented as: 𝑄(𝑢, 𝑢𝑥, 𝑢𝑡𝑡, 𝑢𝑥𝑡, 𝑢𝑥𝑥,…..)=0.(4) Here, the equation 𝑄comprises polynomials involving the function 𝑢(𝑥, 𝑡), its partial derivatives and nonlinear terms. The procedural steps are structured as follows: •Step 1: Introduce a transformation; 𝑢(𝑥, 𝑡) = 𝑓(𝜉), 𝜉 =𝑘(𝑥−𝑐𝑡).(5) where 𝑐and 𝑘represents the wave speed and wave number respectively. This substitution reformulates Eq. (4) into an ordinary differential equation (ODE): 𝑅(𝑓, 𝑓′, 𝑓′′, 𝑓 ′′′,…) = 0.(6) 𝑅constitutes a polynomial involving 𝑓(𝜉)as well as its ordinary derivatives (′=𝑑∕𝑑𝜉). •Step 2: Two different methods will be used to solve Eq. (6). ∙The tanh method allows for the application of the assumption, 𝑓(𝜉) = 𝑎0+𝑎1tanh𝑀(𝑘𝜉).(7) Parameters 𝑎0, 𝑎1and 𝑐are to be determined. An analogous equation for the coth method can be formulated as follows: 𝑓(𝜉) = 𝑎0+𝑎1coth𝑀(𝑘𝜉).(8) ∙The tan method permits the utilization of the stated assumption; 𝑓(𝜉) = 𝑎0+𝑎1tan𝑀(𝑘𝜉).(9) Also, representing the cot method in a similar manner. 𝑓(𝜉) = 𝑎0+𝑎1cot𝑀(𝑘𝜉).(10) •Step 3: The balancing principle is utilized to determine the value of M, taking into account the influence of both nonlinear and the highest-order derivative term in Eq. (6). •Step 4: Upon substituting Eqs. (7)–(10) into Eq. (6), we derive polynomials that encompass terms involving powers of tan𝑖(𝑘𝜉), cot𝑖(𝑘𝜉),coth𝑖(𝑘𝜉)and tanh𝑖(𝑘𝜉). Collecting the coefficients of these polynomials and equating them to zero results in a set of algebraic equations. Using Maple to solve this system with distinct methods yields unique exact solutions for each approach. Tan/Cot method In this section, the methods discussed earlier will be employed to derive novel exact solutions for the dual-mode Gardner equation, as presented in Eq. (3). A concise overview of these techniques was provided in the preceding section. Assuming that the function 𝑢(𝑥, 𝑡)in Eq. (3) is a real-valued function, the following assumption is made: 𝑢(𝑥, 𝑡) = 𝑓(𝜉), 𝜉 =𝑘(𝑥−𝑐𝑡).(11) The transformation is referred to as the traveling wave transformation. By substituting this expression into Eq. (3), the subsequent ODE is obtained: (−3 2𝜂 𝑠𝛼 𝑘2−3 2𝛼 𝑘2𝑐)(𝑓′)2+(3 4𝛼2𝑘2𝑐+3 4𝜂 𝑠𝛼2𝑘2)𝑓(𝑓′)2 +(𝑘2𝑐2−𝜇 𝑠𝑘2−𝑠2𝑘2−𝑘2𝑐)𝑓′′ (12) +(−3 2𝜂 𝑠𝛼 𝑘2−3 2𝛼 𝑘2𝑐)𝑓𝑓′′ +(3 8𝛼2𝑘2𝑐+3 8𝜂 𝑠𝛼2𝑘2)𝑓2𝑓′′ +(−1 6𝛽 𝑘4𝑐−1 6𝜇 𝑠𝛽 𝑘4+1 2𝜇 𝑠𝛽 𝑘4𝜏+1 2𝛽 𝑘4𝑐𝜏)𝑓′′′′ = 0. Results in Physics 57 (2024) 107337 3 S. Sadiq et al. Fig. 1(a). Left, right and dual waves for 𝑢1(𝑥, 𝑡)when 𝑠= 1, 𝜏 = 1, 𝑘 = 1, 𝛼 = 1 and 𝛽= 1. Fig. 1(b). 2D visualization of above Fig (a). Upon performing integration on this differential equation and assuming that the constants of integration are set to zero, we obtain the following simplified form: 1 8(𝛼2𝜂 𝑠 +𝛼2𝑐)𝑓3+3 4(−𝛼 𝜂 𝑠 −𝛼 𝑐)𝑓2+(𝑐2−𝜇 𝑠 −𝑠2−𝑐)𝑓(13) +1 3𝑘2(3𝛽 𝜇 𝑠𝜏 + 3 𝛽 𝑐𝜏 −𝛽 𝜇 𝑠 − 4𝛽 𝑐)𝑓′′ = 0. By equating the leading nonlinear term with the dispersive term in Eq. (13), it is determined that 𝑀equals 1. Substituting this value of 𝑀into Eq. (9) yields the following result: 𝑢(𝑥, 𝑡) = 𝑎0+𝑎1tan(𝑘(𝑥−𝑐𝑡)).(14) Upon substituting this assumption into Eq. (3) and isolating the coefficients corresponding to tan𝑖(𝑘(𝑥−𝑐𝑡)) for 𝑖= 0,2,4,6, solving the resulting system provides the following values for unknown parameters: Case 1: 𝜂≠𝜇 𝑐= ±𝑠, 𝑎0=2 𝛼,(15) 𝑎1=2 𝛼(√−9𝛽𝜏 − 3𝛽 3𝛽𝑘2𝜏−𝛽𝑘2− 3 𝑘), 𝑘2(3𝜏− 1) − 3 >0. As a result, this equation yields a periodic solution, 𝑢1(𝑥, 𝑡) = 2 𝛼(1 + √−9𝛽𝜏 + 3𝛽 3𝑘2𝛽𝜏 −𝑘2𝛽+ 3 𝑘tan(𝑘(𝑠𝑡 +𝑥))).(16) Similarly for cot function we have following singular solution; 𝑢2(𝑥, 𝑡) = 2 𝛼(1 + √−9𝛽𝜏 + 3𝛽 3𝑘2𝛽𝜏 −𝑘2𝛽+ 3 𝑘cot(𝑘(𝑠𝑡 +𝑥))).(17) Here is a graphical representation of 𝑢1(𝑥, 𝑡), depicting the individual left and right waves, as well as a dual-modes graph. Next, 3D and 2D graphs depicting the variation in dual-mode behavior as the phase velocity is altered. Case 2: 𝜇=𝜂 After substituting this assumption into the PDE with 𝜇=𝜂and collecting the coefficients associated with the terms involving tan𝑖(𝑘(𝑥− 𝑐𝑡)) for values of 𝑖including 0, 2, 4 and 6. Subsequently, by solving the resulting system, which yields the following set of values for unknown Fig. 2(a). 3D graph of dual waves for 𝑢1(𝑥, 𝑡)when 𝛽= 1, 𝜏 = 1, 𝑘 = 1, 𝛼 = 1 and 𝑠= 3 𝑎𝑛𝑑 5. Fig. 2(b). 2D graph of dual waves for 𝑢1(𝑥, 𝑡)for same parametric values as used in Fig. 2(a). parameters: 𝑎0=2 𝛼, 𝑎1= 2√−2𝛽𝜏 +2 3𝛽𝑘 𝛼, 𝛼 ≠0,− 2𝛽𝜏 +2 3𝛽 > 0,(18) 𝑐= ± 1 12 √36(𝜏−1 3)2𝑘4𝛽2− 144(𝜏−1 3)𝑘2(𝜂𝑠 +5 4)𝛽+ 144𝑠2+ 360𝜂𝑠 + 225 +5 4+1 12 (−6𝜏+ 2)𝑘2𝛽. where (36(𝜏−1 3)2𝑘4𝛽2− 144(𝜏−1 3)𝑘2(𝜂𝑠 +5 4)𝛽+ 144𝑠2+ 360𝜂𝑠 + 225) >0. As a result, this leads to the periodic solution. 𝑢3(𝑥, 𝑡) = 2 𝛼+ 2√−2𝛽𝜏 +2 3𝛽𝑘 𝛼 Results in Physics 57 (2024) 107337 4 S. Sadiq et al. Fig. 3(a). Left, right and dual waves for 𝑢4(𝑥, 𝑡)when 𝑠= 1, 𝜏 = −1, 𝑘 = 1, 𝛼 = 1, 𝜂 = 0.1and 𝛽= 0.5. Fig. 3(b). 2D visualization of 𝑢4(𝑥, 𝑡)for same parametric values as in Fig. 3(a). Fig. 4(a). Periodic solution of dual waves for 𝑢4(𝑥, 𝑡)when 𝛽= 0.5, 𝜏 = −1, 𝑘 = 1, 𝛼 = 1, 𝜂 = 0.1and 𝑠= 3 𝑎𝑛𝑑 5. × tan(𝑘(1 12 √36(𝜏−1 3)2𝑘4𝛽2− 144(𝜏−1 3)𝑘2(𝜂𝑠 +5 4)𝛽+ 144𝑠2+ 360𝜂𝑠 + 225 (19) +5 4+1 12 (−6𝜏+ 2)𝑘2𝛽)𝑡+𝑥). Likewise, when considering the cot function, the following singular solution emerges: 𝑢4(𝑥, 𝑡) = 2 𝛼+ 2√−2𝛽𝜏 +2 3𝛽𝑘 𝛼 × cot(𝑘(1 12 √36(𝜏−1 3)2𝑘4𝛽2− 144(𝜏−1 3)𝑘2(𝜂𝑠 +5 4)𝛽+ 144𝑠2+ 360𝜂𝑠 + 225 (20) +5 4+1 12 (−6𝜏+ 2)𝑘2𝛽)𝑡+𝑥). Now, showing 2D and 3D graphical representations of the function 𝑢4(𝑥, 𝑡). Following this, 3D and 2D plots are presented to illustrate the changes in dual-mode behavior in 𝑢3(𝑥, 𝑡)as the phase velocity is modified (see Fig. 4(b)). Tanh/Coth method In the analysis, the tanh∕coth approach is used to address the equation. By setting the higher-order nonlinear term equal to the dispersive Fig. 4(b). 2D graphical representation of above Fig. 4(a). term in Eq. (13), the value of 𝑀= 1 is determined. This value is then substituted into Eq. (7) to obtain the resulting outcome. 𝑢(𝑥, 𝑡) = 𝑎0+𝑎1tanh(𝑘(𝑥−𝑐𝑡)).(21) After substituting this assumption into Eq. (3) and collecting the coefficients related to tanh𝑖(𝑘(𝑥−𝑐𝑡)) for 𝑖= 0,2,4and 6the resulting system is solved, providing the following values for unknown parameters: Case 1: 𝜂≠𝜇 𝑐= ±𝑠, 𝑎0=2 𝛼,(22) 𝑎1=2 𝛼√−−9𝛽𝜏 + 3𝛽 3𝛽𝑘2𝜏−𝛽𝑘2+ 3 𝑘, where 𝜏 < 1 3and 1 − 36𝑘2>0,or 𝜏 > 1 3and 1 − 36𝑘2<0. As a result, the obtained values for the parameters, when substituted into Eq. (21), gives a kink wave solution. 𝑢5(𝑥, 𝑡) = 2 𝛼(1 + √−−9𝛽𝜏 + 3𝛽 3𝛽𝑘2𝜏−𝛽𝑘2+ 3 𝑘tanh(𝑘(𝑠𝑡 +𝑥))).(23) Similarly, for 𝜂≠𝜇, we obtain the singular solution using the coth method as outlined earlier in Eq. (8). 𝑢6(𝑥, 𝑡) = 2 𝛼(1 + √−−9𝛽𝜏 + 3𝛽 3𝛽𝑘2𝜏−𝛽𝑘2+ 3 𝑘coth(𝑘(𝑠𝑡 +𝑥))).(24) Displaying 2D and 3D plots representing the left, right and dual-mode graphs of 𝑢6(𝑥, 𝑡). Results in Physics 57 (2024) 107337 5 S. Sadiq et al. Fig. 5(a). Left, right and dual waves for 𝑢6(𝑥, 𝑡)when 𝑠= 1, 𝜏 = 1, 𝑘 = 1, 𝛼 = 1 and 𝛽= 1. Fig. 5(b). 2D visualization of 𝑢6(𝑥, 𝑡)for same parametric values as in Fig. 5(a). Fig. 6(a). Dual wave singular solution for 𝑢6(𝑥, 𝑡)when 𝛽= 1, 𝜏 = 1, 𝑘 = 1, 𝛼 = 1, 𝑠= 3 and 5. Fig. 6(b). 2D visualization of above Fig. 6(a). Next, presenting 3D and 2D graphs depicting the variation in dual mode behavior as the value of phase velocity is altered. Case 2: 𝜇=𝜂: Upon substituting this assumption into PDE with 𝜇=𝜂and then collecting the coefficients of the terms containing tanh𝑖(𝑘(𝑥−𝑐𝑡)) for 𝑖= 0,2,4and 6, proceeding to solve the resulting system, leading to the subsequent set of solutions for the unknown parameters: 𝑎0=2 𝛼, 𝑎1=2 3 𝑘 𝛼√−18𝛽𝜏 + 6𝛽, 𝛼 ≠0,(−18𝛽𝜏 + 6𝛽)>0,(25) 𝑐= ± 1 12 √36(𝜏−1 3)2𝑘4𝛽2+ 144(𝜂𝑠 +5 4)(𝜏−1 3)𝑘2𝛽+ 144𝑠2+ 360𝜂𝑠 + 225 +5 4+1 12 (6𝜏− 2)𝑘2𝛽. where (36(𝜏−1 3)2𝑘4𝛽2+ 144(𝜂𝑠 +5 4)(𝜏−1 3)𝑘2𝛽+ 144𝑠2+ 360𝜂𝑠 + 225) >0. Consequently, this leads to the formation of kink wave solution, 𝑢7(𝑥, 𝑡) = 2 𝛼+2 3 𝑘 𝛼√−18𝛽𝜏 + 6𝛽 × tanh(𝑘(1 12 √36(𝜏−1 3)2𝑘4𝛽2+ 144(𝜂𝑠 +5 4)(𝜏−1 3)𝑘2𝛽+ 144𝑠2+ 360𝜂𝑠 + 225 (26) +5 4+1 12 (6𝜏− 2)𝑘2𝛽)𝑡+𝑥). However, employing the coth method as outlined previously in Eq. (8) and substituting 𝜇=𝜂leads to a singular solution. 𝑢8(𝑥, 𝑡) = 2 𝛼+2 3 𝑘 𝛼√−18𝛽𝜏 + 6𝛽 × coth(𝑘(1 12 √36(𝜏−1 3)2𝑘4𝛽2+ 144(𝜂𝑠 +5 4)(𝜏−1 3)𝑘2𝛽+ 144𝑠2+ 360𝜂𝑠 + 225 (27) +5 4+1 12 (6𝜏− 2)𝑘2𝛽)𝑡+𝑥). Next, presenting 2D and 3D plots illustrating the left, right, and dualmode representation of 𝑢7(𝑥, 𝑡)(see Fig. 7(b)). In the following, 3D and 2D graphical representations are shown to capture how the dual mode behavior evolves with changes in the phase velocity. Results and graphical analysis It is important to understand that the symbol 𝑐assumes two distinct values within this context, signifying that nonlinear equation under examination displays a unique dual-mode motion involving dual waves. Specifically, left-wave and right-wave exhibit simultaneous propagation. In this study, we focused on two distinct methods on dual mode Gardner equation derived from ideal fluid model: tan∕cot method and tanh∕coth method. In Figs. 1(a) and (b), we depict the left, right and dual-mode graphs of 𝑢1(𝑥, 𝑡).Figs. 2(a) and (b) illustrate the effect of the phase velocity parameter 𝑠. Notably, as we increase the value of the phase velocity parameter, the left and right waves converge, moving closer to each Results in Physics 57 (2024) 107337 6 S. Sadiq et al. Fig. 7(a). Left, right and dual waves for 𝑢7(𝑥, 𝑡)when 𝑠= 1, 𝜏 = −1, 𝑘 = 1, 𝛼 = 1, 𝜂 = 1 and 𝛽= 1. Fig. 7(b). 2D profile of above kink waves as shown in Fig. 7(a). Fig. 8(a). Dual wave kink solution for 𝑢7(𝑥, 𝑡)when 𝛽= 1, 𝜏 = −1, 𝑘 = 1, 𝛼 = 1, 𝜂 = 1 and 𝑠= 3 and 5. Fig. 8(b). 2D representation of above Fig. 8(a). other. In the second case, when the dispersive parameter equals the nonlinearity parameter while employing the tan∕cot method, we obtain two types of solutions, 𝑢3(𝑥, 𝑡)and 𝑢4(𝑥, 𝑡). We have generated 3D and 2D graphs of 𝑢4(𝑥, 𝑡)to visualize the impact of the phase velocity in Figs. 3(a) and (b). As we increase the value of 𝑠, the waves again draw closer to each other. Subsequently, when applying the tanh∕coth method, we explore a scenario where the nonlinearity and dispersive parameters are not equal. We have plotted 2D and 3D graphs of 𝑢6(𝑥, 𝑡)in Figs. 5(a) and (b) which reveal a singular solution. Once more, the effect of the phase velocity is evident as we manipulate its value in Figs. 6(a) and (b). In the final case, we discover two solutions: singular and kink waves. We have constructed graphs of 𝑢7(𝑥, 𝑡)to visualize these solutions through Figs. 8(a) and (b). Throughout these investigations, we observe the consistent impact of the phase velocity by varying the values of the associated constants and all the results are verified and their validity is checked by Maple. The employed methods, tan∕cot and tanh∕coth, offer valuable insights into nonlinear wave behavior. Their results provide a deeper understanding of how parameters affect wave propagation. In our study, we have discovered a range of wave behaviors, including kink waves, singular solutions and periodic patterns. Each of these findings carries its unique significance. For instance, kink waves play a role in fluid dynamics, singular solutions can apply to the study of shock waves and periodic patterns are adaptable in fields like signal processing and communication theory. Conclusion In conclusion, our investigation into the dual-mode Gardner equation within the ideal fluid model has yielded significant insights into nonlinear wave behavior. The utilization of the tan∕cot and tanh∕coth methods has allowed for a detailed exploration of the interactions among left, right and dual-mode waves. The outcomes of our study have discovered a variety of solutions, encompassing periodic, singular and kink waves. Of particular note is the profound impact of phase velocity on wave propagation, an observation that holds substantial implications within the framework of ideal fluid models. These findings not only advance our comprehension of wave dynamics but also show considerable relevance in the field of fluid mechanics, exploring possibilities for practical applications. Our study highlights the effectiveness these mathematical techniques and their validity in solving other nonlinear equations. CRediT authorship contribution statement Sadia Sadiq: Writing – original draft, Validation, Software, Methodology, Conceptualization. Ahmad Javid: Writing – review & editing, Supervision, Formal analysis, Conceptualization. Muhammad Bilal Riaz: Investigation, Conceptualization, Reviewing and editing, Resources, Validations, Funding acquisition. Ghada Results in Physics 57 (2024) 107337 7 S. Sadiq et al. Ali Basendwah: Resources. Nauman Raza: Resources, Validation, Visualization. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability No data was used for the research described in the article. Acknowledgment The author Muhammad Bilal Riaz thankful to Ministry of Education, Youth and Sports of the Czech Republic for their support through the e-INFRA CZ (ID:90254). Ethics approval and consent to participate The authors declare that there is no conflict with publication ethics. References [1] Burde GI, Sergyeyev A. Ordering of two small parameters in the shallow water wave problem. J Phys A Math Theor 2013;46:075501. [2] Karczewska A, Rozmej P. 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