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Soliton-like solutions for a nonlinear doubly dispersive equation in an elastic Murnaghan's rod via Hirota's bilinear method

  • Baboucarr Ceesay , Siegfried Macías , Muhammad Z. Baber , Nauman Ahmed , Alejandro Román-Loera and Jorge E. Macías-Díaz EMAIL logo
Published/Copyright: September 10, 2025
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Open Physics
From the journal Open Physics Volume 23 Issue 1

Abstract

The purpose of this work is to study the physical phenomena of the doubly dispersive model that controls chaotic wave movement in the elastic Murnaghan’s rod. The method of Hirota bilinear transformation is employed to derive various forms of solitary wave solutions, such as multiple waves, periodic lump waves, periodic cross-kink waves, homoclinic breather waves, dark soliton, and mixed waves. In order to see their physical behavior, we use the Mathematica software with selected values of the model parameters to depict their graphical behavior.

Keywords: doubly dispersive equation; Hirota bilinear transformation; periodic cross-kink; lump solutions; homoclinic breathers; multiwaves

1 Introduction

Nonlinear partial differential equations (NLPDEs) are used to characterize numerous physical phenomena that arise in real-world contexts [1,2]. They have many applications in scientific domains such as predator–prey systems with diffusion [3], fluid dynamics [4], plasma physics [5], nonlinear fiber optics [6], biological membrane [7], chaos theory for dynamical systems [8], ion acoustic [9], communication system [10], nonlinear elastic solids [11], and many other scientific disciplines. In that context, studies to derive traveling wave solutions and soliton solutions of NLPDE have been carried out by many investigators. That has led to the emergence of many techniques, such as the sine and cosine method [12], the simple equation method [13], the ϕ 6 -expansion method [14], the Ricatti equation mapping method [15], Hirota’s bilinear method [16], the Sardar sub-equation technique [17], the Jacobi elliptic function expansion (JEFE) method [18], the sinh-Gordon equation expansion method and its extended form [19,20], the G ′ G 2 -expansion technique [21], Darboux’s transformation approach [22], the inverse scattering technique [23], the Bäcklund transformation approach [24], Piccard’s iterative method [25], the homotopy analysis method [26], and many others.

The objective of this work is to employ the Hirota bilinear approach to obtain some novel exact traveling-wave solutions and soliton solutions for a doubly dispersive equation (DDE), which is obtained from nonlinear two-directional long-wave models for longitudinal waves in nonlinear dynamic elasticity. This model is expressed as [27]

(1.1) Q t t − 1 ν ( A Q ) x x − δ 2 1 ν ( q γ Q 2 + ν λ 2 Q t t − ( d a λ 2 Q x ) x ) x x = 0 ,

where Q = Q ( x , t ) is the function of the wave strain, x is the spatial variable, t is the time, ν is the density, δ is a relatively small parameter, q = B A and q = M A represent the scale factors, and the Poisson parameter is given by a = A 2 ( 1 + λ ) . The DDE is a significant nonlinear evolution model commonly employed to describe wave motion in elastic media with spatial inhomogeneities, such as Murnaghan’s rod [28]. This equation is significant in various engineering fields that deal with nonlinear elastic solids, especially in seismology, acoustics, and materials engineering. The solitary strain waves derived from the DDE have applications in long-distance power transmission, non-destructive testing (NDT) methods, and vibration paving of stiff materials.

The DDE has second-order time and space derivatives that are important in analyzing one-way or two-way traveling waves, such as shock waves, tsunamis, solitons, and wave structures. It is found to be of the same form as the Boussinesq equation in shallow water wave theory [29–32]. Solitons, which are derived from delayed differential equations, are significant in various fields. For example, they are used in seismology to describe the process of energy transfer in the course of earthquakes. In acoustics, they assist in the development of sound wave technologies. Soliton waves are also important in studying long-distance energy transfer without dissipation. Furthermore, solitary waves are essential for NDT, which is used to inspect materials such as pipelines and structural components without causing damage. The DDE also serves as a foundation for exploring nonlinear mechanics and condensed matter physics. The general form of the DDE after rescaling is (see [33])

(1.2) Q t t − k 1 Q x x − ( k 2 Q 2 + k 3 Q t t + k 4 Q x x ) x x = 0 .

This equation encompasses the appropriate balance between the second time and spatial derivatives with the help of both nonlinear and dispersive constituents. The symbols k 1 , k 2 , k 3 , and k 4 are non-dimensional parameters, which are specific to the material and influence wave propagation in the medium.

There are many reports in the literature that study DDE solutions. For example, using the sine-Gordon expansion technique, Yel [34] obtained traveling-wave solutions of the DDE in nonlinear dynamic elasticity. On the other hand, Cattani et al. [27] adapted the extended sinh-Gordon expansion approach and the modified exp ( − ϕ ( ξ ) ) expansion approach to obtain singular topological and non-topological soliton-type and periodic wave solutions for the DDE. Ahmed et al. [35] implemented the improved modified extended tanh-function technique on the DDE and generated novel solutions, such as bright and dark solitons, Jacobi elliptic solutions and Weiss elliptic solutions. Alharthi [36] considered DDE in order to explain the phenomena of wave propagation utilizing the extended generalized exponential rational function technique and the Jacobi elliptic function technique on the contour sets, thus obtaining new solutions. Meanwhile, Rehman et al. [37] used the Sardar sub-equation method (SSEM) to find various wave solutions to the DDE for Murnaghan’s rod. Some works by Ozisik et al. [38] were devoted to investigating the extended Kudryashov technique together with the Bernoulli–Riccati technique to find the solution of the DDE. In that way, they reached topological, non-topological and singular solitons, periodic, and rational solutions.

On the other hand, Ibrahim et al. [39] derived optical solitons of the DDE via the Sardar sub-equation. Those solutions describe shallow water flow in a vertically vibrating container with a free surface of small depth. Younas et al. [40] studied the DDE in Murnaghan’s rod and derived several soliton solutions through the new extended direct algebraic method new extended direct algebraic method and the generalized Kudryashov method GKM. The DDE for Murnaghan’s rod was solved by Silambarasan et al. [30] using the F-expansion method. They obtained periodic wave solutions as well as non-topological, singular, and compound solitons. Abourabia and Eldreeny [41] addressed the DDE for strain waves in cylindrical rods by using the ecommutative hyper-complex algebraic method. Their results exhibited the characteristics of solitary waves and their stability was analyzed by using phase portraits. Also, Asjad et al. [42] gave new solutions in the form of traveling waves for the DDE via the direct algebraic extended method. In such way, they produced a number of different types of solutions. Dusunceli et al. [43] applied the improved Bernoulli sub-equation function method (IBSEFM) to the solution of the DDE and calculated some traveling-wave solutions.

It is worth pointing out that Alquran and Al-Smadi [33] presented various bidirectional wave solutions to the generalized DDE using the modified rational sine-cosine and sinh-cosh functions in conjunction with the unified method. In this context, Eremeyev and Kolpakov [44] adopted a numerical approach to investigate the wave propagation in elastic inhomogeneous materials. These authors reported on the solitary wave regime and the analysis of its properties. Eremeyev et al. [45] also used the same model to examine harmonic wave propagation in elastic inhomogeneous materials with rheology. In this case, the authors were able to compute analytical solutions to the dispersion relations and wave travel for different materials. In turn, Islam [46] used the modified Khater approach to investigate some wave solutions to the DDE. By examining the system’s dynamics, stability, and bifurcation, this author offered novel analytical perspectives. Khatun et al. [47] investigated the beta space–time fractional DDE employing the ( G ′ G , 1 G ) -expansion method to derive different exact soliton solutions, such as circular, hyperbolic, and rational forms. They also investigated stability via linear analysis and bifurcation theory to uncover varied wave structures applicable in nonlinear elasticity, plasma, acoustics, and seismic research. Khater et al. [48] examined the (2 + 1)-dimensional Zabolotskaya–Khokhlov equation with dissipation by employing the Khater II method, modified Kudryashov method, and He’s variational iteration scheme. The exact and numerical solutions are derived, indicating exact complex dispersive and dissipative wave structures. The obtained results supply valuable information for acoustics, fluid mechanics, and nonlinear optics. Naz et al. [49] appllied the modified exponential method to the fractional-order longitudinal wave equation of a magneto-elastic circular rod with conformable derivative theory. New kink-shaped, bell-shaped, and bright soliton solutions are given in the study, which are of significant physical interest from the graphical analysis. The results are of paramount significance to tsunami modeling, wave dynamics, and fractional-order magneto-elastic systems. However, out of all of the analytical approaches mentioned so far, the Hirota bilinear method is perhaps one of the most valuable tools in soliton research. It has greatly contributed to the understanding of complicated wave structures generated by diverse NLPDEs.

As we will point out in the following section, the extensive applicability of the Hirota bilinear method has been thoroughly demonstrated in the literature. The results obtained through this method go beyond the derivation of classical solitons and includes various other interesting wave shapes, such as rogue, singular, kink, anti-kink, breather, M-shaped, and lump waves. Such solutions are not merely theoretical in nature and can be found in practical usage, as in wave optical pulses, biological membranes, fluids dynamics, modeling of shallow water waves, etc. In summary, these reports exhibit the relevance of the Hirota bilinear method across disciplines, especially in improving our understanding of waves and soliton solutions and their applications in real life. In this work, we will employ the Hirota bilinear transformation method to derive solutions of the DDE described in Eq. (1.2). We have thoroughly reviewed the specialized literature, and, to the best of our knowledge, the Hirota bilinear transformation approach has not yet been applied to the DDE in previous studies. Moreover, wave solutions have not been obtained through this method, and effects of the model parameters were not addressed either. This points out a gap in the research literature which this study seeks to address.

2 Method

It is important to note that various researchers have employed the Hirota bilinear method in recent years to obtain exact solutions to a variety of NLPDEs. For example, Alsallami et al. [50] were able to solve the stochastic fractional Drinfel’d–Sokolov–Wilson equation, Garcia Guirao et al. [51] used the method to study the fourth-order extended (2 + 1)-dimensional Boussinesq equation, and Ceesay et al. [52] were able to study the fluid ionic wave phenomena. In turn, Yang and Wei [53] also utilized Hirota’s bilinear technique to derive bilinear equations with indeterminate coefficients for NLPDEs. In a similar setting, Wazwaz [54] applied both the tanh–coth and Hirota’s bilnear method, and was able to obtain numerous solutions to the problem of Sawada–Kotera–Kadmotsev–Petviashvili. An extensive work on bilinear method was performed by Hereman and Zhuang [55]. Moreover, other investigators have also used the Hirota bilinear method in their research on different models. For instance, Rizvi et al. [56] employed the Hirota bilinear method to study saturated ferromagnets. Wang et al. [57] conducted research of plasma and fluid dynamics for the generalized ( 3 + 1 ) -dimensional Kadomtsev–Petviashvili model, while Khan and Wazwaz [58] also used this method with ansatz functions to derive several new shapes and many-breather solutions. In addition, Zhao et al. [59] studied the asymmetric ( 2 + 1 ) -Nizhnik–Novikov–Veselov model to provide some lump and mixed lump-stripe wave solutions. By employing this approach, Ren et al. [60] explored the Calogero–Bogoyavlenskii–Schiff system in 2 + 1 dimensions. Also, Rizvi et al. [61] investigated the coupled Higgs equations via this technique and Yel et al. [62] studied the Hirota–Maccari system.

In the following, we will derive solutions of the DDE using the Hirota bilinear method. More precisely, we solve Eq. (1.2) under the assumption that there exist solutions satisfying

(2.1) Q ( x , t ) = P ( ϱ ) , ϱ = x − β t .

In this transformation, the parameter β ≠ 0 represents the wave velocity. Using this transformation, the NLPDE becomes an ordinary differential equation (ODE) via the function P of ϱ . Eq. (2.1) is used then to transform Eq. (1.2) into the ODE:

(2.2) − k 2 P 2 + ( − ( β 2 k 3 + k 4 ) ) d 2 P d ϱ 2 + P ( β 2 − k 1 ) = 0 .

We use Eq. (2.2) to determine the different wave structures that arise from (1.2).

First, suppose that the solution of Eq. (2.2) is of the form

(2.3) P ( ϱ ) = m Ω d 2 Ω d ϱ 2 − d Ω d ϱ 2 Ω 2 ,

where Ω = Ω ( ϱ ) represents the class of wave functions considered, and m ≠ 0 is a constant to be determined. Upon substituting Eq. (2.3) into Eq. (2.2), we obtain

(2.4) 0 = Ω 3 ( β 2 k 3 + k 4 ) d 4 Ω d ϱ 4 + ( k 1 − β 2 ) d 2 Ω d ϱ 2 + ( k 2 m − 6 ( β 2 k 3 + k 4 ) ) d Ω d ϱ 4 + 2 Ω ( 6 ( β 2 k 3 + k 4 ) − k 2 m ) d 2 Ω d ϱ 2 d Ω d ϱ 2 + Ω 2 ( β 2 − k 1 ) d Ω d ϱ 2 − 4 ( β 2 k 3 + k 4 ) d 3 Ω d ϱ 3 d Ω d ϱ + ( k 2 m − 3 ( β 2 k 3 + k 4 ) ) d 2 Ω d ϱ 2 2 .

3 Results

We derive various wave structures in the sequel from Eq. (2.4). To that end, we consider various forms of the function Ω ( ϱ ) . For brevity, if P * represents a solution of (2.3), then the corresponding solution of the DDE will be denoted by Q * ( x , t ) = P * ( ϱ ) = P * ( x − β t ) .

3.1 Multi-wave solutions

Following the approach used in previous studies [63,64], we will derive multi-wave solutions by using the function

(3.1) Ω ( ϱ ) = h 1 cosh ( z ( g 1 ϱ + g 2 ) ) + h 2 cos ( z ( g 3 ϱ + g 4 ) ) + h 3 cosh ( z ( g 5 ϱ + g 6 ) ) ,

where g i , h j , and z ≠ 0 are the arbitrary constants, for each 1 ≤ i ≤ 6 and 1 ≤ j ≤ 3 . We substitute Eq. (3.1) into (2.4). Next, set the hyperbolic and trigonometric function coefficients equal to zero. After reducing algebraically, we obtain the following two groups of solutions:

Group 1

In this case, we consider the constants

(3.2) g 1 = β 2 − k 1 4 β 2 k 3 z 2 + 4 k 4 z 2 , g 3 = k 1 − β 2 2 β 2 k 3 z 2 + k 4 z 2 , g 5 = β 2 − k 1 4 β 2 k 3 z 2 + 4 k 4 z 2 , m = 6 ( β 2 k 3 + k 4 ) k 2 ,

where k 1 , k 2 , k 3 , k 4 ≠ 0 are the free constant parameters. We substitute these constants into Eq. (3.1). Next, we employ the result in Eq. (2.3) to obtain the solution

(3.3) P 1 M U ( ϱ ) = − 3 A 1 β 2 − k 1 − h 2 k 1 − β 2 sin z g 4 + ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) 2 A 2 + h 3 cosh z g 6 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) 2 − β 2 + k 1 2 k 2 ,

where

(3.4) A 1 = h 1 sinh z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) + h 3 sinh z g 6 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) ,

(3.5) A 2 = h 2 cos z g 4 + ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) + h 1 cosh z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) .

Group 2

In this case, we will consider the constant h 1 = 0 , along with

(3.6) g 3 = − k 1 − β 2 2 β 2 k 3 z 2 + k 4 z 2 , g 5 = − β 2 − k 1 4 β 2 k 3 z 2 + 4 k 4 z 2 , m = 6 ( β 2 k 3 + k 4 ) k 2 ,

where k 1 , k 2 , k 3 , k 4 ≠ 0 are the free parameters. Using these constants in Eq. (3.1), substituting the result then into Eq. (2.3), we readily reach that

(3.7) P 2 M U ( ϱ ) = A 5 + 6 h 2 h 3 ( β 2 − k 1 ) cos z g 4 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) cosh z g 6 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) 2 k 2 h 2 cos z g 4 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) + h 3 cosh z g 6 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) 2 ,

where

(3.8) A 5 = 6 h 3 h 2 − ( β 2 − k 1 ) 2 sin × z g 4 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) × sinh z g 6 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) + 3 h 2 2 ( β 2 − k 1 ) + 3 h 3 2 ( β 2 − k 1 ) .

3.2 Periodic lump solutions

In this case, we follow the approach used in [63,64], and allow

(3.9) Ω ( ϱ ) = ( g 1 ϱ + g 2 ) 2 + ( g 3 ϱ + g 4 ) 2 + cos ( g 5 ϱ + g 6 ) + g 7 ,

where g i , for each 1 ≤ i ≤ 7 , are the arbitrary constants. We substitute Eq. (3.9) into Eq. (2.4). Next, we set ϱ and the trigonometric function coefficients equal to zero. In that way, we obtain solutions for the constant parameter values g 1 = − i g 3 , g 2 = − i g 4 , g 7 = 0 , and

(3.10) g 5 = − k 1 − β 2 2 β 2 k 3 + k 4 , m = 6 ( β 2 k 3 + k 4 ) k 2 ,

where k 1 , k 2 , k 3 , k 4 ≠ 0 are the free parameters. Substituting these values into Eqs. (3.9) and (2.3), we obtain that

(3.11) P 1 P L ( ϱ ) = 3 ( β 2 − k 1 ) sec 2 g 6 − ϱ k 1 − β 2 2 β 2 k 3 + k 4 2 k 2 .

3.3 Periodic cross-kink solutions

In order to derive periodic cross-kink solutions of the DDE, we follow [65,66] and define

(3.12) Ω ( ϱ ) = h 1 exp ( z ( g 3 ϱ + g 4 ) ) + exp ( − z ( g 1 ϱ + g 2 ) ) + h 2 cos ( z ( g 5 ϱ + g 6 ) ) + h 3 cosh ( z ( g 7 ϱ + g 8 ) ) + g 9 ,

where g i , h j , and z ≠ 0 are the arbitrary constants, for each 1 ≤ i ≤ 9 and 1 ≤ j ≤ 3 . We substitute Eq. (3.12) into Eq. (2.4). Next, we set the exponential, hyperbolic, and trigonometric function coefficients equal to zero, to obtain the following groups of solutions.

Group 1

We set h 2 = 0 , g 9 = 0 , and

(3.13) g 1 = − β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) , g 3 = − β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) , g 7 = β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) , m = 6 ( β 2 k 3 + k 4 ) k 2 ,

where k 1 , k 2 , k 3 , k 4 ≠ 0 are the free parameters. After replacing these values into Eq. (3.12), and then into Eq. (2.3), we reach the solution

(3.14) P 1 P C K ( ϱ ) = 3 ( β 2 − k 1 ) ( 2 h 1 e ( g 4 + g 8 ) z + h 3 ) ( h 3 e ( g 2 + g 8 ) z + 2 ) exp − z − g 2 + g 8 + ϱ β 2 − k 1 z 2 ( β 2 k 3 + k 4 ) 2 k 2 A 7 + h 1 exp z g 2 + g 4 − ϱ β 2 − k 1 z 2 ( β 2 k 3 + k 4 ) + 1 2 ,

where

(3.15) A 7 = h 3 exp z g 2 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) × cosh z g 8 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) .

Group 2

In this case, we let h 1 = 0 , g 9 = 0 , and

(3.16) g 1 = − β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) , g 5 = − k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) , g 7 = − β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) , m = 6 ( β 2 k 3 + k 4 ) k 2 ,

where k 1 , k 2 , k 3 , k 4 ≠ 0 are the free parameters. Substituting these values into Eq. (3.12), and then the result into Eq. (2.3), yields the following solution:

(3.17) P 2 P C K ( ϱ ) = 3 ( A 9 + 1 ) 2 ( β 2 − k 1 ) exp z ϱ β 2 − k 1 z 2 ( β 2 k 3 + k 4 ) − 2 g 2 z − A 10 2 2 k 2 A 11 + exp − z g 2 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) 2 .

Here,

(3.18) A 9 = exp z g 2 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) × h 2 cos z g 6 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) + h 3 cosh z g 8 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) ,

(3.19) A 10 = β 2 − k 1 exp − z g 2 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) − h 3 sinh z g 8 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) + h 2 k 1 − β 2 sin z g 6 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 )

and

(3.20) A 11 = h 2 cos z g 6 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) + h 3 cosh z g 8 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) .

3.4 Homoclinic breather solutions

Following previous studies [65,66], homoclinic breather solutions are obtained when

(3.21) Ω ( ϱ ) = h 1 exp ( z ( g 3 ϱ + g 4 ) ) + exp ( − z ( g 1 ϱ + g 2 ) ) + h 2 cos ( z ( g 5 ϱ + g 6 ) ) ,

where g i , h j , and z ≠ 0 are the arbitrary constants, for each 1 ≤ i ≤ 6 and 1 ≤ j ≤ 2 . Proceeding as before, we substitute Eq. (3.12) into Eq. (2.4). Next, we set the exponential and trigonometric function coefficients equal to zero, in order to derive two groups of solutions.

Group 1

In this case, we let

(3.22) g 1 = β 2 − k 1 4 β 2 k 3 z 2 + 4 k 4 z 2 , g 3 = β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) , g 5 = − k 1 − β 2 2 β 2 k 3 z 2 + k 4 z 2 , m = 6 ( β 2 k 3 + k 4 ) k 2 ,

where k 1 , k 2 , k 3 , k 4 ≠ 0 are the free constant parameters. Inserting these values into Eq. (3.21), and then the result into Eq. (2.3), we reach the solution

(3.23) P 1 H B ( ϱ ) = 3 ( A 16 − ( β 2 − k 1 ) 2 + A 15 ( β 2 − k 1 ) ) exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) 2 k 2 A 17 + h 1 exp z g 2 + g 4 + ϱ β 2 − k 1 z 2 ( β 2 k 3 + k 4 ) + 1 2 ,

where

(3.24) A 15 = h 2 2 exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) + 4 h 1 exp z g 4 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) + 2 h 2 cos z g 6 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) × h 1 exp z g 2 + g 4 + ϱ β 2 − k 1 z 2 ( β 2 k 3 + k 4 ) + 1 ,

(3.25) A 16 = − 2 h 1 h 2 exp z g 2 + g 4 + ϱ β 2 − k 1 z 2 ( β 2 k 3 + k 4 ) × sin z g 6 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) + 2 h 2 sin z g 6 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) ,

and

(3.26) A 17 = h 2 exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) × cos z g 6 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) .

Group 2

In this group, we let h 1 = 0 ,

(3.27) g 1 = β 2 − k 1 4 β 2 k 3 z 2 + 4 k 4 z 2 , g 5 = − k 1 − β 2 2 β 2 k 3 z 2 + k 4 z 2 , m = 6 ( β 2 k 3 + k 4 ) k 2 ,

where k 1 , k 2 , k 3 , k 4 ≠ 0 are the free constant parameters. Proceeding as in the other cases, we obtain the solution

(3.28) P 2 H B ( ϱ ) 3 h 2 exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) A 21 + 2 ( β 2 − k 1 ) cos z g 6 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) 2 k 2 h 2 exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) cos z g 6 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) + 1 2 ,

where

(3.29) A 21 = h 2 ( β 2 − k 1 ) exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) + 2 − ( β 2 − k 1 ) 2 sin z g 6 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) .

3.5 Interaction of two-exponent solution

In this case, we follow previous studies [66,67] to obtain solutions with the interaction of two exponents. To that end, we let

(3.30) Ω ( ϱ ) = h 1 exp ( z ( g 3 ϱ + g 4 ) ) + h 2 exp ( z ( g 1 ϱ + g 2 ) ) ,

where g i , h j , and z ≠ 0 are the arbitrary constants, for each 1 ≤ i ≤ 4 and 1 ≤ j ≤ 2 . Proceeding as earlier, we substitute Eq. (3.30) into Eq. (2.4). We set the exponential function coefficients equal to zero. In this case, we obtain tone group of solutions by letting

(3.31) g 1 = g 3 + k 1 − β 2 z 2 ( β 2 − k 1 ) ( β 2 k 3 + k 4 ) , m = 6 ( β 2 k 3 + k 4 ) k 2 ,

where k 1 , k 2 , k 3 , k 4 ≠ 0 are the free parameters. Substitute these constants into Eq. (3.30), and then the result into Eq. (2.3) to reach the solution

(3.32) P 1 M I ( ϱ ) = 6 h 1 h 2 ( β 2 − k 1 ) exp z 2 g 3 ϱ + g 2 + g 4 + ϱ ( k 1 − β 2 ) z 2 ( β 2 − k 1 ) ( β 2 k 3 + k 4 ) k 2 h 2 exp z ϱ g 3 + k 1 − β 2 z 2 ( β 2 − k 1 ) ( β 2 k 3 + k 4 ) + g 2 + h 1 e z ( g 3 ϱ + g 4 ) 2 .

3.6 Mixed wave solutions

This type of wave structure function is offered by letting [66,67]

(3.33) Ω ( ϱ ) = h 1 exp ( z ( g 1 ϱ + g 2 ) ) + h 2 exp ( − z ( g 1 ϱ + g 2 ) ) + h 3 sin ( z ( g 3 ϱ + g 4 ) ) + h 4 sinh ( z ( g 5 ϱ + g 6 ) ) ,

where g i , h j , and z ≠ 0 are the arbitrary constants, for each 1 ≤ i ≤ 6 and 1 ≤ j ≤ 4 . We substitute Eq. (3.33) into Eq. (2.4), and set the exponential, hyperbolic, and trigonometric function coefficients equal to zero. Two groups of solutions are derived in this case.

Group 1

In this group, we let

(3.34) g 1 = β 2 − k 1 4 β 2 k 3 z 2 + 4 k 4 z 2 , g 3 = − k 1 − β 2 2 β 2 k 3 z 2 + k 4 z 2 , g 5 = − β 2 − k 1 4 β 2 k 3 z 2 + 4 k 4 z 2 , m = 6 ( β 2 k 3 + k 4 ) k 2 ,

where k 1 , k 2 , k 3 , k 4 ≠ 0 are the free parameters. Substituting these values into Eq. (3.33) and into Eq. (2.3), we have

(3.35) P 1 M I ( ϱ ) = 3 ( A 23 2 ( β 2 − k 1 ) − A 24 2 ) exp − 2 z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) 2 k 2 A 25 + h 1 exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) 2 ,

where

(3.36) A 23 = h 1 exp 2 g 2 z + z ϱ β 2 − k 1 z 2 ( β 2 k 3 + k 4 ) + exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) ⋅ h 3 sin z g 4 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) + h 4 sinh z g 6 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) + h 2 ,

(3.37) A 24 = h 2 β 2 − k 1 − h 1 β 2 − k 1 × exp 2 g 2 z + z ϱ β 2 − k 1 z 2 ( β 2 k 3 + k 4 ) + exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) ⋅ h 3 k 1 − β 2 cos z g 4 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) + h 4 β 2 − k 1 cosh z g 6 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) ,

and

(3.38) A 25 = h 2 exp − z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) + h 3 sin z g 4 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) + h 4 sinh z g 6 − ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) .

Group 2

Let now h 2 = 0 ,

(3.39) g 1 = β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) , g 3 = − k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) , g 5 = β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) , m = 6 ( β 2 k 3 + k 4 ) k 2 ,

where k 1 , k 2 , k 3 , k 4 ≠ 0 are the free constants. Proceeding as before, we reach the solution

(3.40) P 2 M I ( ϱ ) = − 3 A 29 β 2 − k 1 − h 3 k 1 − β 2 cos z g 4 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) 2 A 30 + h 1 exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) 2 − β 2 + k 1 2 k 2 ,

where

(3.41) A 29 = h 1 exp z g 2 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) + h 4 cosh z g 6 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) ,

(3.42) A 30 = h 3 sin z g 4 − ϱ k 1 − β 2 2 z 2 ( β 2 k 3 + k 4 ) + h 4 sinh z g 6 + ϱ β 2 − k 1 2 z 2 ( β 2 k 3 + k 4 ) .

4 Discussion

In this section, the traveling-wave solutions for the DDE are shown graphically. The purpose is to illustrate the soliton and solitary wave solutions obtained in the previous section. The figures show a variety of soliton solutions in three dimensions along with their matching contours and density plots. These solutions include multi-wave, periodic lumps, periodic cross-kink waves, homoclinic breather waves, interaction via two-exponent wave structure, and mixed waves. Each of the figures shows (a) 3D graphs, (b) contour plots, and (c) density plots, and details on the functions and parameters used are presented in the caption of each figure. To start with, Figure 1 shows multiple lump-type waves arising from the multi-wave structure function, while Figure 2 depicts multiple breather waves arising from the multi-wave structure function. In turn, Figure 3 shows periodic bright lump waves derived from the periodic lump wave profile function. Figures 4 and 5 show periodic cross-kink wave profiles. Figures 6 and 7 show bright breather waves obtained from the homoclinic breather wave function. Figure 8 shows a dark soliton derived from the structure function of interaction between double exponents. Finally, Figures 9 and 10 show mixtures of wave shapes in the form of solitary waves, derived from the mixed wave function. Multiple wave solutions, periodic lump waves, and mixed waves are all different physical behaviors in nonlinear systems. Multiple wave solutions involve the collision of several solitary waves, e.g., solitons, that keep their shape and speed even after collision. Periodic lump waves are localized waves that appear periodically in space or time, having lump-like features with a periodic structure. Mixed waves are the coexistence or interaction of multiple wave types such as a soliton on a periodic wave leading to hybrid, complex dynamics. They occur in many physical situations across fluid mechanics, optics, and plasma physics, where nonlinear wave–wave interactions control wave development.

Figure 1 Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 1 M U ( x , t ) {Q}_{1MU}\left(x,t) . The graphs were obtained by selecting parameter values g 2 = 9.7 {g}_{2}=9.7 , g 4 = 8.7 {g}_{4}=8.7 , g 6 = 2.9 {g}_{6}=2.9 , h 1 = 0.6 {h}_{1}=0.6 , h 2 = 5.3 {h}_{2}=5.3 , h 3 = 2.9 {h}_{3}=2.9 , z = 0.04 z=0.04 , k 1 = 2.09 {k}_{1}=2.09 , k 2 = 0.6 {k}_{2}=0.6 , k 3 = 1.2 {k}_{3}=1.2 , k 4 = 0.2 {k}_{4}=0.2 , and β = 0.2 \beta =0.2 .
Figure 1

Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 1 M U ( x , t ) . The graphs were obtained by selecting parameter values g 2 = 9.7 , g 4 = 8.7 , g 6 = 2.9 , h 1 = 0.6 , h 2 = 5.3 , h 3 = 2.9 , z = 0.04 , k 1 = 2.09 , k 2 = 0.6 , k 3 = 1.2 , k 4 = 0.2 , and β = 0.2 .

Figure 2 Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 2 M U ( x , t ) {Q}_{2MU}\left(x,t) . The graphs were obtained by selecting parameter values g 4 = 7.2 {g}_{4}=7.2 , g 6 = 0.3 {g}_{6}=0.3 , h 2 = 1.7 {h}_{2}=1.7 , h 3 = 2.3 {h}_{3}=2.3 , z = 0.38 z=0.38 , k 1 = 9.4 {k}_{1}=9.4 , k 2 = 3.7 {k}_{2}=3.7 , k 3 = 2.3 {k}_{3}=2.3 , k 4 = 1.7 {k}_{4}=1.7 , and β = 1.2 . \beta =1.2.
Figure 2

Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 2 M U ( x , t ) . The graphs were obtained by selecting parameter values g 4 = 7.2 , g 6 = 0.3 , h 2 = 1.7 , h 3 = 2.3 , z = 0.38 , k 1 = 9.4 , k 2 = 3.7 , k 3 = 2.3 , k 4 = 1.7 , and β = 1.2 .

Figure 3 Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 1 L P ( x , t ) {Q}_{1LP}\left(x,t) . The chosen parameter values are as follows: g 6 = 1.549 {g}_{6}=1.549 , k 1 = 2.23 {k}_{1}=2.23 , k 2 = 1.43 {k}_{2}=1.43 , k 3 = 5.28 {k}_{3}=5.28 , k 4 = 0.3 {k}_{4}=0.3 , and β = 8.7 . \beta =8.7.
Figure 3

Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 1 L P ( x , t ) . The chosen parameter values are as follows: g 6 = 1.549 , k 1 = 2.23 , k 2 = 1.43 , k 3 = 5.28 , k 4 = 0.3 , and β = 8.7 .

Figure 4 Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 1 P C K ( x , t ) {Q}_{1PCK}\left(x,t) corresponding to the parameter values g 2 = 8.2 {g}_{2}=8.2 , g 4 = 2.9 {g}_{4}=2.9 , g 8 = 2.1 {g}_{8}=2.1 , h 1 = 6.3 {h}_{1}=6.3 , h 3 = 8.01 {h}_{3}=8.01 , z = 4.1 z=4.1 , k 1 = 5.9 {k}_{1}=5.9 , k 2 = 2.3 {k}_{2}=2.3 , k 3 = 3.0 {k}_{3}=3.0 , k 4 = 1.6 {k}_{4}=1.6 , and β = 0.3 . \beta =0.3.
Figure 4

Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 1 P C K ( x , t ) corresponding to the parameter values g 2 = 8.2 , g 4 = 2.9 , g 8 = 2.1 , h 1 = 6.3 , h 3 = 8.01 , z = 4.1 , k 1 = 5.9 , k 2 = 2.3 , k 3 = 3.0 , k 4 = 1.6 , and β = 0.3 .

Figure 5 Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 2 P C K ( x , t ) {Q}_{2PCK}\left(x,t) corresponding to the parameter values g 2 = 1.1 {g}_{2}=1.1 , g 6 = 4.4 {g}_{6}=4.4 , g 8 = 2.8 {g}_{8}=2.8 , h 2 = 4.4 {h}_{2}=4.4 , h 3 = 2.9 {h}_{3}=2.9 , k 1 = 1.2 {k}_{1}=1.2 , z = 1.4 z=1.4 , k 2 = 3.5 {k}_{2}=3.5 , k 3 = 1.1 {k}_{3}=1.1 , k 4 = 0.5 {k}_{4}=0.5 , and β = 0.3 . \beta =0.3.
Figure 5

Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 2 P C K ( x , t ) corresponding to the parameter values g 2 = 1.1 , g 6 = 4.4 , g 8 = 2.8 , h 2 = 4.4 , h 3 = 2.9 , k 1 = 1.2 , z = 1.4 , k 2 = 3.5 , k 3 = 1.1 , k 4 = 0.5 , and β = 0.3 .

Figure 6 Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 1 H B ( x , t ) {Q}_{1HB}\left(x,t) corresponding to the parameter values g 2 = 0.6 {g}_{2}=0.6 , g 4 = 0.7 {g}_{4}=0.7 , g 6 = 0.2 {g}_{6}=0.2 , h 1 = 2.3 {h}_{1}=2.3 , h 2 = 3.4 {h}_{2}=3.4 , z = 6.2 z=6.2 , k 1 = 4.65 {k}_{1}=4.65 , k 2 = 1.8 {k}_{2}=1.8 , k 3 = 0.19 {k}_{3}=0.19 , k 4 = 0.2 {k}_{4}=0.2 , and β = 8.9 . \beta =8.9.
Figure 6

Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 1 H B ( x , t ) corresponding to the parameter values g 2 = 0.6 , g 4 = 0.7 , g 6 = 0.2 , h 1 = 2.3 , h 2 = 3.4 , z = 6.2 , k 1 = 4.65 , k 2 = 1.8 , k 3 = 0.19 , k 4 = 0.2 , and β = 8.9 .

Figure 7 Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 2 H B ( x , t ) {Q}_{2HB}\left(x,t) . The following parameter values were chosen: g 2 = 8.0 {g}_{2}=8.0 , g 6 = 9.4 {g}_{6}=9.4 , h 2 = 3.4 {h}_{2}=3.4 , z = 1.5 z=1.5 , k 1 = 0.57 {k}_{1}=0.57 , k 2 = 0.3 {k}_{2}=0.3 , k 3 = 9.7 {k}_{3}=9.7 , k 4 = 1.1 {k}_{4}=1.1 , and β = 9.3 . \beta =9.3.
Figure 7

Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 2 H B ( x , t ) . The following parameter values were chosen: g 2 = 8.0 , g 6 = 9.4 , h 2 = 3.4 , z = 1.5 , k 1 = 0.57 , k 2 = 0.3 , k 3 = 9.7 , k 4 = 1.1 , and β = 9.3 .

Figure 8 Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 2 H B ( x , t ) {Q}_{2HB}\left(x,t) for the following choice of parameter values: g 2 = 0.9 {g}_{2}=0.9 , g 3 = 3.1 {g}_{3}=3.1 , g 4 = 2.2 {g}_{4}=2.2 , h 1 = 8.1 {h}_{1}=8.1 , h 2 = 2.9 {h}_{2}=2.9 , z = 1.2 z=1.2 , k 1 = 4.91 {k}_{1}=4.91 , k 2 = 1.4 {k}_{2}=1.4 , k 3 = 0.33 {k}_{3}=0.33 , k 4 = 0.5 {k}_{4}=0.5 , and β = 2.2 . \beta =2.2.
Figure 8

Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 2 H B ( x , t ) for the following choice of parameter values: g 2 = 0.9 , g 3 = 3.1 , g 4 = 2.2 , h 1 = 8.1 , h 2 = 2.9 , z = 1.2 , k 1 = 4.91 , k 2 = 1.4 , k 3 = 0.33 , k 4 = 0.5 , and β = 2.2 .

Figure 9 Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 1 M I ( x , t ) {Q}_{1MI}\left(x,t) corresponding to the parameter values g 2 = 8.9 {g}_{2}=8.9 , g 4 = 3.8 {g}_{4}=3.8 , g 6 = 4.2 {g}_{6}=4.2 , h 1 = 1.3 {h}_{1}=1.3 , h 2 = 8.1 {h}_{2}=8.1 , h 3 = 5.4 {h}_{3}=5.4 , h 4 = 7.2 {h}_{4}=7.2 , z = 9.7 z=9.7 , k 1 = 2.1 {k}_{1}=2.1 , k 2 = 1.5 {k}_{2}=1.5 , k 3 = 1.4 {k}_{3}=1.4 , k 4 = 2.9 {k}_{4}=2.9 , and β = 1.8 \beta =1.8 .
Figure 9

Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 1 M I ( x , t ) corresponding to the parameter values g 2 = 8.9 , g 4 = 3.8 , g 6 = 4.2 , h 1 = 1.3 , h 2 = 8.1 , h 3 = 5.4 , h 4 = 7.2 , z = 9.7 , k 1 = 2.1 , k 2 = 1.5 , k 3 = 1.4 , k 4 = 2.9 , and β = 1.8 .

Figure 10 Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 2 M I ( x , t ) {Q}_{2MI}\left(x,t) corresponding to the parameter values g 2 = 4.7 {g}_{2}=4.7 , g 4 = 0.2 {g}_{4}=0.2 , g 6 = 8.7 {g}_{6}=8.7 , h 1 = 1.1 {h}_{1}=1.1 , h 3 = 0.1 {h}_{3}=0.1 , h 4 = 0.1 {h}_{4}=0.1 , z = 6.1 z=6.1 , k 1 = 2.13 {k}_{1}=2.13 , k 2 = 0.2 {k}_{2}=0.2 , k 3 = 0.6 {k}_{3}=0.6 , k 4 = 10.0 {k}_{4}=10.0 , and β = 0.5 \beta =0.5 .
Figure 10

Graphs showing (a) three-dimensional, (b) contour, and (c) density plots of the solution Q 2 M I ( x , t ) corresponding to the parameter values g 2 = 4.7 , g 4 = 0.2 , g 6 = 8.7 , h 1 = 1.1 , h 3 = 0.1 , h 4 = 0.1 , z = 6.1 , k 1 = 2.13 , k 2 = 0.2 , k 3 = 0.6 , k 4 = 10.0 , and β = 0.5 .

It is worth noting that multi-wave solutions depict how several interacting primary wave fronts of diverse contents lead to the emergence of distortion and stress profiles. Periodic lump waves illustrate how energy is being sustained within given intervals, thereby showing that the material is subjected to compressive forces for a while before being released. These wave types are crucial in elucidating the ability of elastic rods to store and transmit energy in a cyclical manner, which is a key aspect in the study of wave motion in solids. Furthermore, periodic cross-kink waves show clear uniform regions that are interleaved with narrower regions of apparent distortion, which may suggest buckling or sudden high straining rates in elastic materials under load. Also, we observe that homoclinic breather waves can persist where the oscillations are confined, suggesting that sufficiently long waves with little energy leakage from the rod can be sustained. These aspects help explain the non-trivial content of nonlinear wave propagation in elastic solids. Dark solitons and mixed wave solutions show that there are many ways in which waves interact in Murnaghan’s rods. In this case, dark solitons are areas of depressions or less deformed medium showing some stabilization of the material, whereas mixed wave solutions showcase the fact that elastic rods are capable of having even simultaneous different types of wave motion propagating within them. These facts highlight the intricacy of wave motions in elastic structures and may prove useful for engineering designs as well as for nonlinear wave propagation research in solids.

The physical interpretation of the model parameters helps improve the practical insight into the wave structures obtained. In particular, the parameter β signifies the velocity of the wave and the speed with which disturbances travel within the elastic Murnaghan’s rod. The amplitude of the solutions, as it is dependent on parameters such as m , h 1 , h 2 , and h 3 , reflects the intensity of strain or deformation the medium undergoes. Material-dependent constants k 1 , k 2 , k 3 , and k 4 control the relative strength of dispersion and nonlinearity, determining the temporal evolution of wave profiles. Additionally, the spatial localization of derived solutions, for example, lump and breather waves, indicates that energy is localized in space, and thus, these waveforms are highly applicable for NDT and energy-guided transmission in elastic media.

The values utilized in the graphical plots were arbitrarily chosen to ensure visibility, clarity, and variability of the wave structures in question. The values are not experimentally obtained but were selected within physically reasonable ranges to characterize the qualitative character of the exact solutions. The main aim was to exhibit richness of solution space and the effect of parameter variation on wave shape, localization, and interaction. Additional work can include tuning these parameters against actual material properties for application-oriented modeling.

5 Comparison

The solutions achieved in this work, obtained through the Hirota bilinear method, are a major improvement over most of the earlier reported outcomes for DDE as in nonlinear elastic rods. Previous studies have used a wide range of analytical methods such as the sine-Gordon expansion technique [34], extended sinh-Gordon and modified exp ( − ϕ ( ξ ) ) techniques [27], enhanced modified extended tanh-function method [35], generalized exponential rational and Jacobi elliptic function methods [36], and SSEMs [37,39], for obtaining exact solutions. The studies yielded various soliton and periodic waveforms such as bright and dark solitons, kink and anti-kink waves, hyperbolic, trigonometric, and elliptic solutions. Although such methods are successful, they tend to produce isolated wave structures and may be restrictive in fully capturing the entire richness of nonlinear wave behavior in elastic media.

In contrast, the Hirota bilinear approach is a better structured and integrated procedure that can build up diverse and composite wave structures, including multi-wave interaction, periodic lumps, cross-kink waves, homoclinic breathers, and mixed solitons. The variety of solution types reflects a better insight into the nonlinear processes that govern wave propagation in elastic rods such as those accommodated by Murnaghan’s theory. The capacity of this method to generate topological and non-topological, localized and periodic, and bright and dark soliton solutions places it well in comparison to other analytical methods. In contrast to recent researches involving coupled nonlinear Schrödinger equations that also utilize the Hirota technique to discover new optical soliton structures [48], the present contribution expands its application to mechanical wave systems and thus presents a broader understanding of intricate wave phenomenon in doubly dispersive and nonlinear elastic systems.

6 Conclusion

In this work, we used the Hirota bilinear transformation approach to derive various wave structures, which are solutions of a DDE in an elastic Murnaghan’s rod. Using the Hirota technique, we were able to obtain traveling-wave solutions in the form of periodic lump waves, periodic cross-kink waves, homoclinic breather waves, dark soliton, and mixed waves. In addition, the complex structures and interactions of these wave structures were illustrated in the form of 3D, contour, and density plots. The graphical representations illustrate that the shape of the wave affects the propagation of the wave itself. This is particularly evident in the density plots and wave interaction patterns. As a comment, this method has the ability to describe the deformation of single solitary waves in an elastic perfect material and can suggest how such systems behave physically. To the best of our knowledge, this is the first study in which the Hirota bilinear method is used in the context of doubly dispersive media, and the results derived in this manuscript may apply to both fundamental and practical investigations, especially for engineering designs involving elastic wave propagation.

Future work might involve the examination of the stability, chaos, and perturbation response of the DDE to determine its robustness in the presence of small disturbances. The inclusion of fractional-order derivatives would enable the modeling of memory and hereditary effects in viscoelastic or incompressible elastic materials. Adding stochastic terms could potentially facilitate the inclusion of uncertainties and random disturbance that are generally found in real environments. The other possibility is the generalization of the model to a variable-coefficient problem, which would be more representative of spatial or temporal inhomogeneities of the material. These developments would not only improve theoretical insight into the DDE but also extend its availability to a wider range of physical systems, such as composite materials, biological tissue, and engineered smart materials.

Acknowledgments

The authors wish to thank the anonymous reviewers for their comments. All of their criticisms and suggestions were followed at litteram, resulting in a substantial improvement on the overall quality of this work.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: Conceptualization, N.A., and J.E.M.D.; methodology, N.A., and; software, B.C., S.M., M.Z.B., N.A., A.R.-L., and J.E.M.-D.; validation, B.C., S.M., M.Z.B., N.A., A.R.-L., and J.E.M.-D.; formal analysis, B.C., S.M., M.Z.B., N.A., and J.E.M.-D.; investigation, N.A., and J.E.M.-D.; resources, B.C., S.M., M.Z.B., N.A., A.R.-L., and J.E.M.-D.; data curation, B.C., S.M., M.Z.B., N.A., A.R.-L., and J.E.M.-D.; writing – original draft preparation, B.C., S.M., M.Z.B., N.A., A.R.-L., and J.E.M.-D.; writing – review and editing, B.C., S.M., M.Z.B., N.A., A.R.-L., and J.E.M.-D.; visualization, B.C.; supervision, N.A., and J.E.M.-D. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: Jorge E. Macías-Díaz, who is the co-author of this article, is a current Editorial Board member of Open Physics. This fact did not affect the peer-review process. The authors state no other conflict of interest.

  4. Data availability statement: The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2025-04-03
Revised: 2025-05-26
Accepted: 2025-06-30
Published Online: 2025-09-10

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2025-0189
Keywords for this article
doubly dispersive equation; Hirota bilinear transformation; periodic cross-kink; lump solutions; homoclinic breathers; multiwaves
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  41. Novel constructed dynamical analytical solutions and conserved quantities of the new (2+1)-dimensional KdV model describing acoustic wave propagation
  42. Tavis–Cummings model in the presence of a deformed field and time-dependent coupling
  43. Spinning dynamics of stress-dependent viscosity of generalized Cross-nonlinear materials affected by gravitationally swirling disk
  44. Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies
  45. Robust control and preservation of quantum steering, nonlocality, and coherence in open atomic systems
  46. Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow
  47. Dynamic analysis, circuit realization, and its synchronization of a new chaotic hyperjerk system
  48. Decoherence of steerability and coherence dynamics induced by nonlinear qubit–cavity interactions
  49. Finite element analysis of turbulent thermal enhancement in grooved channels with flat- and plus-shaped fins
  50. Modulational instability and associated ion-acoustic modulated envelope solitons in a quantum plasma having ion beams
  51. Statistical inference of constant-stress partially accelerated life tests under type II generalized hybrid censored data from Burr III distribution
  52. On solutions of the Dirac equation for 1D hydrogenic atoms or ions
  53. Entropy optimization for chemically reactive magnetized unsteady thin film hybrid nanofluid flow on inclined surface subject to nonlinear mixed convection and variable temperature
  54. Stability analysis, circuit simulation, and color image encryption of a novel four-dimensional hyperchaotic model with hidden and self-excited attractors
  55. A high-accuracy exponential time integration scheme for the Darcy–Forchheimer Williamson fluid flow with temperature-dependent conductivity
  56. Novel analysis of fractional regularized long-wave equation in plasma dynamics
  57. Development of a photoelectrode based on a bismuth(iii) oxyiodide/intercalated iodide-poly(1H-pyrrole) rough spherical nanocomposite for green hydrogen generation
  58. Investigation of solar radiation effects on the energy performance of the (Al2O3–CuO–Cu)/H2O ternary nanofluidic system through a convectively heated cylinder
  59. Quantum resources for a system of two atoms interacting with a deformed field in the presence of intensity-dependent coupling
  60. Studying bifurcations and chaotic dynamics in the generalized hyperelastic-rod wave equation through Hamiltonian mechanics
  61. A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions
  62. Interaction solutions of high-order breathers and lumps for a (3+1)-dimensional conformable fractional potential-YTSF-like model
  63. Hydraulic fracturing radioactive source tracing technology based on hydraulic fracturing tracing mechanics model
  64. Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet
  65. Numerical investigation of mixed convection and viscous dissipation in couple stress nanofluid flow: A merged Adomian decomposition method and Mohand transform
  66. Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves
  67. Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy
  68. Solitons and travelling waves structure for M-fractional Kairat-II equation using three explicit methods
  69. Impact of nanoparticle shapes on the heat transfer properties of Cu and CuO nanofluids flowing over a stretching surface with slip effects: A computational study
  70. Computational simulation of heat transfer and nanofluid flow for two-sided lid-driven square cavity under the influence of magnetic field
  71. Irreversibility analysis of a bioconvective two-phase nanofluid in a Maxwell (non-Newtonian) flow induced by a rotating disk with thermal radiation
  72. Hydrodynamic and sensitivity analysis of a polymeric calendering process for non-Newtonian fluids with temperature-dependent viscosity
  73. Exploring the peakon solitons molecules and solitary wave structure to the nonlinear damped Kortewege–de Vries equation through efficient technique
  74. Modeling and heat transfer analysis of magnetized hybrid micropolar blood-based nanofluid flow in Darcy–Forchheimer porous stenosis narrow arteries
  75. Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating
  76. Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints
  77. Influence of a magnetic field on double-porosity photo-thermoelastic materials under Lord–Shulman theory
  78. Soliton-like solutions for a nonlinear doubly dispersive equation in an elastic Murnaghan's rod via Hirota's bilinear method
  79. Analytical and numerical investigation of exact wave patterns and chaotic dynamics in the extended improved Boussinesq equation
  80. Nonclassical correlation dynamics of Heisenberg XYZ states with (x, y)-spin--orbit interaction, x-magnetic field, and intrinsic decoherence effects
  81. Exact traveling wave and soliton solutions for chemotaxis model and (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation
  82. Unveiling the transformative role of samarium in ZnO: Exploring structural and optical modifications for advanced functional applications
  83. On the derivation of solitary wave solutions for the time-fractional Rosenau equation through two analytical techniques
  84. Analyzing the role of length and radius of MWCNTs in a nanofluid flow influenced by variable thermal conductivity and viscosity considering Marangoni convection
  85. Advanced mathematical analysis of heat and mass transfer in oscillatory micropolar bio-nanofluid flows via peristaltic waves and electroosmotic effects
  86. Exact bound state solutions of the radial Schrödinger equation for the Coulomb potential by conformable Nikiforov–Uvarov approach
  87. Some anisotropic and perfect fluid plane symmetric solutions of Einstein's field equations using killing symmetries
  88. Nonlinear dynamics of the dissipative ion-acoustic solitary waves in anisotropic rotating magnetoplasmas
  89. Curves in multiplicative equiaffine plane
  90. Exact solution of the three-dimensional (3D) Z2 lattice gauge theory
  91. Propagation properties of Airyprime pulses in relaxing nonlinear media
  92. Symbolic computation: Analytical solutions and dynamics of a shallow water wave equation in coastal engineering
  93. Wave propagation in nonlocal piezo-photo-hygrothermoelastic semiconductors subjected to heat and moisture flux
  94. Comparative reaction dynamics in rotating nanofluid systems: Quartic and cubic kinetics under MHD influence
  95. Laplace transform technique and probabilistic analysis-based hypothesis testing in medical and engineering applications
  96. Physical properties of ternary chloro-perovskites KTCl3 (T = Ge, Al) for optoelectronic applications
  97. Gravitational length stretching: Curvature-induced modulation of quantum probability densities
  98. The search for the cosmological cold dark matter axion – A new refined narrow mass window and detection scheme
  99. A comparative study of quantum resources in bipartite Lipkin–Meshkov–Glick model under DM interaction and Zeeman splitting
  100. PbO-doped K2O–BaO–Al2O3–B2O3–TeO2-glasses: Mechanical and shielding efficacy
  101. Nanospherical arsenic(iii) oxoiodide/iodide-intercalated poly(N-methylpyrrole) composite synthesis for broad-spectrum optical detection
  102. Sine power Burr X distribution with estimation and applications in physics and other fields
  103. Numerical modeling of enhanced reactive oxygen plasma in pulsed laser deposition of metal oxide thin films
  104. Dynamical analyses and dispersive soliton solutions to the nonlinear fractional model in stratified fluids
  105. Computation of exact analytical soliton solutions and their dynamics in advanced optical system
  106. An innovative approximation concerning the diffusion and electrical conductivity tensor at critical altitudes within the F-region of ionospheric plasma at low latitudes
  107. An analytical investigation to the (3+1)-dimensional Yu–Toda–Sassa–Fukuyama equation with dynamical analysis: Bifurcation
  108. Swirling-annular-flow-induced instability of a micro shell considering Knudsen number and viscosity effects
  109. Numerical analysis of non-similar convection flows of a two-phase nanofluid past a semi-infinite vertical plate with thermal radiation
  110. MgO NPs reinforced PCL/PVC nanocomposite films with enhanced UV shielding and thermal stability for packaging applications
  111. Optimal conditions for indoor air purification using non-thermal Corona discharge electrostatic precipitator
  112. Investigation of thermal conductivity and Raman spectra for HfAlB, TaAlB, and WAlB based on first-principles calculations
  113. Tunable double plasmon-induced transparency based on monolayer patterned graphene metamaterial
  114. DSC: depth data quality optimization framework for RGBD camouflaged object detection
  115. A new family of Poisson-exponential distributions with applications to cancer data and glass fiber reliability
  116. Numerical investigation of couple stress under slip conditions via modified Adomian decomposition method
  117. Monitoring plateau lake area changes in Yunnan province, southwestern China using medium-resolution remote sensing imagery: applicability of water indices and environmental dependencies
  118. Heterodyne interferometric fiber-optic gyroscope
  119. Exact solutions of Einstein’s field equations via homothetic symmetries of non-static plane symmetric spacetime
  120. A widespread study of discrete entropic model and its distribution along with fluctuations of energy
  121. Empirical model integration for accurate charge carrier mobility simulation in silicon MOSFETs
  122. The influence of scattering correction effect based on optical path distribution on CO2 retrieval
  123. Anisotropic dissociation and spectral response of 1-Bromo-4-chlorobenzene under static directional electric fields
  124. Role of tungsten oxide (WO3) on thermal and optical properties of smart polymer composites
  125. Analysis of iterative deblurring: no explicit noise
  126. 10.1515/phys-2025-0251
  127. Review Article
  128. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  129. Erratum
  130. Erratum to “On Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model (Open Physics 2021;19:679–682)”
  131. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  132. Possible explanation for the neutron lifetime puzzle
  133. Special Issue on Nanomaterial utilization and structural optimization - Part III
  134. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  135. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  136. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  137. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  138. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
  139. Novel exact solitons to the fractional modified mixed-Korteweg--de Vries model with a stability analysis
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