Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe

Saad Althobaiti

doi:10.1515/phys-2024-0033

Article Open Access

Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe

Saad Althobaiti

Published/Copyright: June 11, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

This research article delves into the intricate domain of nonlinear wave dynamics within the framework of a Murnaghan hyperelastic circular pipe. Thus, the current study makes use of some powerful analytical approaches to examine the propagation of nonlinear elastic waves on a Murnaghan hyperelastic circular pipe. The work is exceptional since it allows for the incorporation of double dispersion terms and material nonlinearity in the controlling nonlinear mode. The study entails a thorough examination of the propagation and interaction of solitons within the Murnaghan hyperelastic medium, providing insights into the distinctive nonlinear wave phenomena manifested by circular pipe configurations. Theoretical insights are substantiated by numerical simulations, presenting a comprehensive understanding of the dynamic responses within these elastic structures. In the end, graphical representations of some of the derived solutions have been provided for clarification. In addition, the reported solutions in the study help researchers working in modern fields of engineering and materials science to obtain valuable insights that can inform the design, analysis, and optimization of materials and structures in contemporary applications.

Keywords: elastic structures; hyperelastic material; nonlinear waves; Murnaghan rod; propagation and interaction

1 Introduction

The investigation of the propagation and application of waves on nonlinear structures within elastodynamics represents a compelling and multifaceted research endeavor. In the realm of elastodynamics, where material responses are nonlinear and complex, understanding wave propagation becomes crucial for a wide array of applications. In elastodynamics, the propagation of waves on nonlinear structures is a fascinating subject with enormous implications. In fact, a wide range of modern disciplines, including seismology, oceanology, civil and mechanical engineering, communication, and solid mechanics, to name a few, are greatly inspired by the study and analysis of wave dispersion. Furthermore, the concept is crucial to preventing fatal earthquakes and building damage. The investigation of solitons and nonlinear wave dynamics in hyperelastic materials has evolved from a rich history rooted in the study of nonlinear partial differential equations (PDEs) and their solutions. The theoretical foundation for solitons dates back to the nineteenth century, with contributions from luminaries such as John Scott Russell. The Murnaghan hyperelastic model, developed in the early twentieth century, laid the groundwork for understanding the elastic behavior of materials under deformation. In fact, this field of study would be incomplete without acknowledging the contributions of eminent scientists like Newton, Hooks, Love, Rayleigh, Stoneley, and Murnaghan, to name a few. For more information, read the well-known books by Achenbach [1] and Ewing et al. [2] on the dispersion of waves in elastic media. The dispersion of waves in elastic media is a complex and fascinating phenomenon, particularly when considering its interplay with solitons. Solitons, as self-reinforcing solitary waves that maintain their shape and energy over long distances, exhibit unique behaviors in elastic media. Unlike traditional waves that disperse and spread out over time, solitons resist dispersion due to a delicate balance between nonlinear and dispersive effects. In elastic media, the interaction between the elasticity of the medium and the nonlinear characteristics of solitons leads to intriguing wave dynamics. The elasticity of the medium governs how the soliton propagates and influences its stability, while the nonlinearity prevents the wave from dispersing as it travels. Understanding the dispersion of waves in elastic media is not only crucial for unraveling the fundamental principles of wave behavior but also holds practical significance in fields such as seismology, acoustics, and materials science, where the interaction of solitons with elastic media can have profound implications for signal transmission and material integrity. Nonetheless, the investigation of waves in hyperelastic media, more specifically, the Murnaghan’s circular pipe, is of special significance in this work [3–5]. The material nonlinearity that governs wave propagation in hyperelastic structure is taken into account by the corresponding Cauchy–Green strain tensor using Murnaghan submission [6]; for clear expressions of the associated nonlinear strain and Murnaghan potential tensors in rectangular coordinates, respectively, see the study of Usman and Zaman [7]. Over the years, advancements in mathematical methods and computational techniques have enabled a deeper exploration of soliton solutions and nonlinear wave dynamics. This article builds upon this historical trajectory, aiming to contribute novel insights into the interplay of solitons and Murnaghan hyperelastic circular pipes, thereby expanding our understanding of nonlinear wave behaviors in elastic structures. Furthermore, a wide range of nonlinear models can be found in the general field of mathematical physics, including areas like optics, quantum, and plasmas, to name a few. As a result, a number of researchers have thoroughly investigated the field and developed a variety of analytical techniques – see previous studies [8–12] and references therein – in an effort to obtain precise analytical solutions. It is important to keep in mind, though, that nonlinear models are typically challenging to solve analytically. The obtained analytical solutions, in fact, provide a more profound comprehension of the fundamental dynamics of the governing nonlinear evolution equations.

However, the goal of the current study is to investigate wave propagation on a nonlinear Murnaghan hyperelastic circular pipe by utilizing a few analytical techniques known by the names of the exponential function ansatz method [13], the Kudryashov method [14], and the generalized Riccati equation method [15]. This is actually because a thorough investigation of the propagation of nonlinear waves is necessary; nonlinearity is what drives scholars to focus so intently on studying wave phenomena in linear elastic media. The nonlinearity of waves in elastic media, particularly in the context of solitons, unveils a rich and fascinating interplay between the material properties and the dynamic behavior of waves. In elastic media, waves are subject to nonlinear effects due to the intricate relationship between stress, strain, and wave amplitude. The nonlinearity in elastic media gives rise to intriguing phenomena such as wave steepening. Therefore, the current study delves into the intricate dynamics of waves as they traverse through nonlinear structures, shedding light on the unique behaviors that emerge in response to varying conditions. The incorporation of Murnaghan hyperelasticity introduces a unique facet to the investigation, as it allows for a more comprehensive understanding of the material’s response to nonlinear wave propagation. Through advanced theoretical frameworks and mathematical models, we aim to unravel the complexities inherent in the interaction between waves and nonlinear structures. The implications of this research extend to diverse fields, including structural engineering, materials science, and seismic analysis. By comprehensively exploring the propagation of waves on nonlinear structures, this study contributes valuable insights that can inform the design, analysis, and optimization of materials and structures in engineering applications, ultimately advancing our understanding of elastodynamics in complex and practical contexts. The techniques used will undoubtedly produce some interesting exact analytical solutions, which will then be graphically examined for further insight. Furthermore, these results bear practical implications for the design and analysis of elastic structures in diverse engineering applications, highlighting the potential for leveraging nonlinear wave dynamics for innovative engineering solutions. Besides, one may find several contemporary analytical approaches for solving nonlinear equations in the open literature, refer previous studies [16–30] and references therein; refer also the studies of Baleanu et al. [31], Tuan et al. [32], and Baleanu et al. [33] for some new findings in the analysis of nonlinear mathematical models with applications in medicine. Finally, this study is further organized as follows: the model of interest is expressed in Section 2, the three techniques of concern are given in Section 3, the deployment of the deployed methods is explained in Section 4, the graphical representations of the solutions and discussion are provided in Section 5, and some concluding thoughts are provided in Section 6.

2 Predominant model

To investigate the propagation of nonlinear elastic waves over an isotropic homogeneous Murnaghan hyperelastic circular pipe, we assume the double dispersion equation of motion in a compressible cylindrical pipe of the following form [3–5]:

(1) ∂ 2 u ∂ t 2 + α 1 ∂ 4 u ∂ x 2 ∂ t 2 − α 2 ∂ 2 u ∂ x 2 − α 3 ∂ 2 ( u ) 2 ∂ x 2 − α 4 ∂ 4 u ∂ x 2 = 0 ,

where u = u ( x , t ) is the displacement of the wave in x , the spatial length of the pipe, and t is the temporal variable. In addition, x lies in the interval − R ∕ 2 ≤ x ≤ R ∕ 2 , where R is the radius of the cross-sectional area of the pipe over the interval 0 ≤ r ≤ R . Furthermore, α i for i = 1 , 2 , … , 4 are overtly expressed as follows:

(2) α 1 = ν ( 1 − ν ) 2 R 2 , α 2 = c 2 , α 3 = β 2 ρ , α 4 = ν c 2 2 R 2 ,

where ν is the Poisson’s ratio, c = E ∕ ρ is the linear shear speed, ρ is the density, and β is the Murnaghan hyperelastic material constant, taking the following expression:

β = 3 E + 2 l ( 1 − 2 ν ) 3 + 4 m ( 1 + ν ) 2 ( 1 − 2 ν ) + 6 n ν 3 ,

where E is the Young’s modulus and ( l , m , n ) are the Murnaghan moduli.

In this context, nonlinear elastic waves refer to deformations or disturbances in the material that do not follow a linear relationship between stress and strain, often becoming more pronounced as the amplitude of the waves increases. The term isotropic homogeneous Murnaghan hyperelastic specifies the material properties, indicating that the circular pipe exhibits isotropy (having the same mechanical properties in all directions) and homogeneity (uniform properties throughout). The Murnaghan hyperelasticity refers to a specific material model that accounts for nonlinear elastic behavior. Therefore, the current study aims to analyze Eq. (2) using capable analytical approaches, as there is limited literature on the governing nonlinear model. This will help to understand the model’s physical relevance and characterize the behavior of waves in hyperelastic structures; refer previous studies [3–5] for the derivation of the governing model. Equally, refer also the studies of Liu and Xu [34], Samsonov and Sokurinskaya [35], and Shi and Yan [36] for more insights on the dynamics of waves in nonlinear media, together with some of their applications.

3 Analytical methods

Let us consider a generalized PDE of the following form

(3) R ( r , r x , r t , r x x , r t t , r t x x , … ) = 0 .

Suppose some wave transformation, as

(4) r ( x , t ) = V ( χ ) , χ = k x − θ t ,

where k and θ are nonzero constants. By invoking Eq. (4) into Eq. (3), we obtain a reduced ordinary differential equation of the polynomial form:

(5) K ( V ( χ ) , V ′ ( χ ) , V ″ ( χ ) , V ‴ ( χ ) , … ) = 0 .

Furthermore, the solution of Eq. (5) is constructed based on the analytical approaches that are delineated below.

3.1 Exponential function ansatz

After numerous searches, the exponential function ansatz for the governing model was discovered to permit the following projected solution ansatz [13]:

(6) V ( χ ) = A 0 + A 1 Φ ( χ ) ( 1 + Φ ( χ ) ) 2 ,

where A 0 and A 1 are constants (not all equal to zero) to be determined, and

Φ ( χ ) = exp ( χ ) , χ = k x − θ t .

It is a powerful mathematical technique employed in the quest to derive soliton solutions in various nonlinear systems. The exponential function ansatz has found application in diverse fields, from fluid dynamics to optical fibers, offering a versatile and efficient method to obtain insightful solutions in nonlinear systems. This analytical tool not only aids in understanding the fundamental principles behind soliton dynamics but also paves the way for exploring novel applications in science and engineering.

3.2 Kudryashov method

Kudryashov provides the predicted solution for Eq. (5) as follows [14].

(7) V ( χ ) = A 0 + ∑ j = 1 M A j Φ j ( χ ) ,

where A 0 , A j , j = 1 , 2 , … , M , are constants (not all equal to zero) to be determined and M is a positive integer determined by a homogeneous balancing method. Also, Φ ( χ ) in Eq. (7) satisfies the differential equation

(8) Φ ′ ( χ ) = Φ 2 ( χ ) − Φ ( χ ) ,

and yields the following solution

(9) Φ ( χ ) = 1 1 + r exp ( χ ) ,

where r is a free arbitrary constant, which is referred to as the Kudryashov index. Indeed, the Kudryashov index plays a significant role in getting singular solitons for r < 0 and stable (continuous) solutions for r > 0 .

The Kudryashov method is renowned for its simplicity and applicability to various nonlinear equations, making it a valuable tool in the study of soliton dynamics. The derived soliton solutions offer insights into the behavior of nonlinear waves, contributing to the broader understanding of nonlinear phenomena in diverse fields such as physics, engineering, and mathematical modeling.

3.3 Generalized Riccati equation method

Generalized Riccati equation technique provides the predicted solution for Eq. (5) as [15]:

(10) U ( χ ) = A 0 + ∑ j = 1 M ( A j Φ j ( χ ) + B j Φ − j ( χ ) ) ,

where A 0 , A j , B j , j = 1 , 2 , … , M , are constants (not all equal to zero) to be determined, while M is equally to be determined using the homogeneous balancing method. In addition, Φ ( χ ) in Eq. (10) satisfies the differential equation

(11) Φ ′ ( χ ) = r 1 Φ 2 ( χ ) + r 2 Φ ( χ ) + r 3 ,

where r 1 , r 2 , and r 3 are variable constants (real). Therefore, Eq. (11) admits 27 different solutions; moreover, for brevity, we refer the reader(s) to Alharbi et al. [15] for more details.

Therefore, making use of the predicted solution and its necessary derivative, as given in each of the aforementioned approaches alongside Eqs (8) and (11) as the case maybe, with regard to the second and third approaches, respectively, into Eq. (5) yields a polynomials in Φ ( χ ) . Next, upon collecting the coefficients of these polynomials and afterward setting each to zero, one obtains a set of over-determined algebraic equations for A 0 , A j , B j ( j = 1 , 2 , … , M ) with the aid of a computer program; Mathematica software is thus used in this study. Moreover, the resultant algebraic equations are then solved for A 0 , A j , B j ( j = 1 , 2 , … , M ) , for the acquisition of the overall solution of Eq. (3). This method provides a systematic and efficient approach to analyzing nonlinear wave phenomena and its application has been successfully extended to various fields such as fluid dynamics, optical fibers, and plasma physics. By offering a means to generate soliton solutions in a wide array of nonlinear systems, the generalized Riccati equation method contributes significantly to the understanding and exploration of complex wave dynamics in diverse scientific disciplines.

4 Implementation

To apply the mentioned methodologies to the governing model, we utilize the wave transformation in Eq. (4) into Eq. (1) to obtain the following nonlinear ordinary differential equation:

(12) ( θ 2 − α 2 k 2 ) V ″ ( χ ) − k 2 ( α 4 k 2 − α 1 θ 2 ) V ′ ′ ′ ′ ( χ ) − 2 α 3 k 2 ( V ( χ ) V ″ ( χ ) + V ′ ( χ ) 2 ) = 0 .

After integration, it yields

(13) ( θ 2 − α 2 k 2 ) V ′ ( χ ) − k 2 ( α 4 k 2 − α 1 θ 2 ) V ‴ ( χ ) − 2 α 3 k 2 V ( χ ) V ′ ( χ ) + W = 0 ,

where W is an integration constant that will be explicitly defined later.

Further, upon balancing the highest linear and nonlinear terms in the latter equation using balance principle, that is

V ‴ ( χ ) and V ( χ ) V ′ ( χ ) ,

in the latter equation, that is,

M + 3 = M + ( M + 1 ) ,

which subsequently gives

M = 2 .

In fact, the acquisition of M plays a vital part in the revelation of the predicted solution form.

4.1 Exponential function ansatz

For this, we substitute the exponential function ansatz expressed in Eq. (6) into Eq. (13) and obtain the following algebraic system of equations:

α 4 ( − A 1 ) k 4 − 2 α 3 A 0 A 1 k 2 + α 1 A 1 k 2 θ 2 − α 2 A 1 k 2 + A 1 θ 2 = 0 ,

4 α 3 A 0 A 1 k 2 − 4 α 3 A 1 2 k 2 + 26 α 4 A 1 k 4 − 26 α 1 A 1 k 2 θ 2 + 2 α 2 A 1 k 2 − 2 A 1 θ 2 = 0 ,

12 α 3 A 0 A 1 k 2 + 12 α 3 A 1 2 k 2 − 66 α 4 A 1 k 4 + 66 α 1 A 1 k 2 θ 2 + 6 α 2 A 1 k 2 − 6 A 1 θ 2 = 0 .

After solving the above system, we obtain the value of all coefficients as

(14) A 0 = θ 2 − α 4 k 4 + α 1 k 2 θ 2 − α 2 k 2 2 α 3 k 2 , A 1 = − 6 ( α 1 θ 2 − α 4 k 2 ) α 3 .

Therefore, on substituting the above values in Eq. (14) alongside utilizing Eq. (6), one obtains the solution for the governing model as follows:

(15) u ( x , t ) = θ 2 − α 4 k 4 + α 1 k 2 θ 2 − α 2 k 2 2 α 3 k 2 − 6 ( α 1 θ 2 − α 4 k 2 ) α 3 ( 1 + exp ( k x − θ t ) ) 2 exp ( k x − θ t ) .

4.2 Kudryashov method

Here, we deploy the already explained Kudryashov method to construct an exact solution for the governing model. Thus, since M has been found to be 2, that is, M = 2 , the predicted solution takes the following form from Eq. (7):

(16) U ( χ ) = A 0 + A 1 Φ ( χ ) + A 2 Φ ( χ ) 2 .

In addition, we substitute Eq. (16) into Eq. (13) and act as detailed in Section 3. Moreover, the resulting algebraic equations are obtained as follows:

24 α 1 A 2 k 2 θ 2 − 24 α 4 A 2 k 4 − 4 α 3 A 2 2 k 2 = 0 , α 4 A 1 k 4 − α 1 A 1 k 2 θ 2 + α 2 A 1 k 2 + 2 α 3 A 0 A 1 k 2 − A 1 θ 2 = 0 , 8 α 4 A 2 k 4 − 7 α 4 A 1 k 4 + 7 α 1 A 1 k 2 θ 2 − 8 α 1 A 2 k 2 θ 2 − α 2 A 1 k 2 + 2 α 2 A 2 k 2 + 2 α 3 A 1 2 k 2 − 2 α 3 A 0 A 1 k 2 + 4 α 3 A 0 A 2 k 2 + A 1 θ 2 − 2 A 2 θ 2 = 0 , 12 α 4 A 1 k 4 − 38 α 4 A 2 k 4 − 12 α 1 A 1 k 2 θ 2 + 38 α 1 A 2 k 2 θ 2 − 2 α 2 A 2 k 2 − 2 α 3 A 1 2 k 2 − 4 α 3 A 0 A 2 k 2 + 6 α 3 A 1 A 2 k 2 + 2 A 2 θ 2 = 0 , 54 α 4 A 2 k 4 − 6 α 4 A 1 k 4 + 6 α 1 A 1 k 2 θ 2 − 54 α 1 A 2 k 2 θ 2 + 4 α 3 A 2 2 k 2 − 6 α 3 A 1 A 2 k 2 = 0 .

After solving the above system, we obtain the value of all coefficients as

(17) A 0 = θ 2 − α 4 k 4 + α 1 k 2 θ 2 − α 2 k 2 2 α 3 k 2 , A 1 = − 6 ( α 1 θ 2 − α 4 k 2 ) α 3 , A 2 = 6 ( α 1 θ 2 − α 4 k 2 ) α 3 , W = 0 .

Thus, on putting these values in Eq. (17) alongside utilizing Eq. (9) into Eq. (16), one obtains the exact analytical solution for the model as follows:

(18) u ( x , t ) = θ 2 − α 4 k 4 + α 1 k 2 θ 2 − α 2 k 2 2 α 3 k 2 − 6 ( α 1 θ 2 − α 4 k 2 ) α 3 ( 1 + r exp ( k x − θ t ) ) + 6 ( α 1 θ 2 − α 4 k 2 ) α 3 ( 1 + r exp ( k x − θ t ) ) 2 , = θ 2 − α 4 k 4 + α 1 k 2 θ 2 − α 2 k 2 2 α 3 k 2 − 6 r ( α 1 θ 2 − α 4 k 2 ) α 3 ( 1 + r exp ( k x − θ t ) ) 2 exp ( k x − θ t ) ,

where r is a free, arbitrary constant. In this work, we refer to r as the Kudryashov index. The Kudryashov technique yields a solution that is more general than the exponential function ansatz in Eq. (15). When the Kudryashov index is set to unity ( r = 1 ), the foregoing solution is reduced to Eq. (15).

4.3 Generalized Riccati equation technique

As M = 2 , the expected solution form follows Eq. (10)

(19) V ( χ ) = A 0 + A 1 Φ ( χ ) + A 2 Φ 2 ( χ ) + B 1 Φ − 1 ( χ ) + B 2 Φ − 2 ( χ ) .

Substituting this into Eq. (13) yields a connected system of algebraic equations, which we eliminated for the sake of simplicity. Furthermore, whereas the generalized Riccati equation approach provides 27 various solution types [15], including periodic, hyperbolic, and rational function solutions, our current study focuses solely on the acquisition of hyperbolic solutions, specifically solitonic solutions. Thus, the resulting algebraic equations are as follows:

24 α 4 B 2 k 4 p 3 3 − 24 α 1 B 2 k 2 p 3 3 θ 2 + 4 α 3 B 2 2 k 2 p 3 = 0 , 40 α 4 B 2 k 4 p 1 p 3 2 + 12 α 4 B 1 k 4 p 2 p 3 2 + 38 α 4 B 2 k 4 p 2 2 p 3 = 0 , 12 α 4 A 1 k 4 p 1 2 p 2 − 38 α 4 A 2 k 4 p 1 p 2 2 − 40 α 4 A 2 k 4 p 1 2 p 3 = 0 , 2 α 4 A 1 k 4 p 1 p 3 2 − 6 α 4 A 2 k 4 p 2 p 3 2 = 0 , − 24 α 4 A 2 k 4 p 1 3 + 24 α 1 A 2 k 2 p 1 3 θ 2 − 4 α 3 A 2 2 k 2 p 1 = 0 , 6 α 1 A 1 k 2 p 1 3 θ 2 − 6 α 4 A 1 k 4 p 1 3 − 54 α 4 A 2 k 4 p 1 2 p 2 + 54 α 1 A 2 k 2 p 1 2 p 2 θ 2 − 6 α 3 A 1 A 2 k 2 p 1 − 4 α 3 A 2 2 k 2 p 2 = 0 , 6 α 4 B 1 k 4 p 3 3 + 54 α 4 B 2 k 4 p 2 p 3 2 − 6 α 1 B 1 k 2 p 3 3 θ 2 − 54 α 1 B 2 k 2 p 2 p 3 2 θ 2 + 4 α 3 B 2 2 k 2 p 2 + 6 α 3 B 1 B 2 k 2 p 3 = 0 , ⋮

As a result, solving the acquired system of algebraic solutions yields the following solution sets:

Set I:

(20) A 0 = α 4 k 4 ( − r 2 2 ) − 8 α 4 k 4 r 1 r 3 − α 2 k 2 + α 1 k 2 r 2 2 θ 2 + 8 α 1 k 2 r 1 r 3 θ 2 + θ 2 2 α 3 k 2 , A 1 = 6 r 1 r 2 ( α 1 θ 2 − α 4 k 2 ) α 3 , A 2 = 6 r 1 2 ( α 1 θ 2 − α 4 k 2 ) α 3 , B 1 = B 2 = W = 0 .

Hence, for r 2 2 − 4 r 1 r 3 > 0 , and r 1 r 2 ≠ 0 , or ( r 1 r 3 ≠ 0 ) , we obtain the following solitonic solutions:

1) Dark solitonic solution

(21) u 1 ( x , t ) = A 0 + A 1 Φ 1 ( χ ) + A 2 Φ 1 2 ( χ ) ,

where

Φ 1 ( χ ) = − 1 2 r 1 r 2 + r 4 tanh r 4 2 χ , r 4 = r 2 2 − 4 r 1 r 3 , χ = k x − θ t .

2) Dark-bright solitonic solution

(22) u 2 ( x , t ) = A 0 + A 1 Φ 2 ( χ ) + A 2 Φ 2 2 ( χ ) ,

where

Φ 2 ( χ ) = − 1 2 r 1 ( r 2 + r 4 ( tanh ( r 4 χ ) ± i sech ( r 4 χ ) ) ) , r 4 = r 2 2 − 4 r 1 r 3 , χ = k x − θ t .

3) Dark-singular solitonic solution

(23) u 3 ( x , t ) = A 0 + A 1 Φ 3 ( χ ) + A 2 Φ 3 2 ( χ ) ,

where

Φ 3 ( χ ) = − 1 4 r 1 2 r 2 + r 4 tanh r 4 4 χ ± coth r 4 4 χ , r 4 = r 2 2 − 4 r 1 r 3 , χ = k x − θ t .

Set II:

(24) A 0 = α 4 k 4 ( − r 2 2 ) − 8 α 4 k 4 r 1 r 3 − α 2 k 2 + α 1 k 2 r 2 2 θ 2 + 8 α 1 k 2 r 1 r 3 θ 2 + θ 2 2 α 3 k 2 , B 1 = 6 r 2 r 3 ( α 1 θ 2 − α 4 k 2 ) α 3 , B 2 = 6 r 3 2 ( α 1 θ 2 − α 4 k 2 ) α 3 , A 1 = A 2 = W = 0 .

As preceed, for r 2 2 − 4 r 1 r 3 > 0 , and r 1 r 2 ≠ 0 , or ( r 1 r 3 ≠ 0 ) , we obtain the following solitonic solutions:

1) Dark solitonic solution

(25) u 4 ( x , t ) = A 0 + B 1 Φ 4 − 1 ( χ ) + B 2 Φ 4 − 2 ( χ ) ,

where

Φ 4 ( χ ) = − 1 2 r 1 r 2 + r 4 tanh r 4 2 χ , r 4 = r 2 2 − 4 r 1 r 3 , χ = k x − θ t .

2) Dark-bright solitonic solution

(26) u 5 ( x , t ) = A 0 + B 1 Φ 5 − 1 ( χ ) + B 2 Φ 5 − 2 ( χ ) ,

where

Φ 5 ( χ ) = − 1 2 r 1 ( r 2 + r 4 ( tanh ( r 4 χ ) ± i sech ( r 4 χ ) ) ) , r 4 = r 2 2 − 4 r 1 r 3 , χ = k x − θ t .

3) Dark-singular solitonic solution

(27) u 6 ( x , t ) = A 0 + B 1 Φ 6 − 1 ( χ ) + B 2 Φ 6 − 2 ( χ ) ,

where

Φ 6 ( χ ) = − 1 4 r 1 2 r 2 + r 4 tanh r 4 4 χ ± coth r 4 4 χ , r 4 = r 2 2 − 4 r 1 r 3 , χ = k x − θ t .

5 Graphic representations and discussion

The current work employs three analytical methodologies to investigate the propagation of hyperelastic waves on a nonlinear Murnaghan circular pipe using the solitary wave solutions methodology. The first technique yielded the same solution as the Kudryashov method [14] for r = 1 ; but the generalized Riccati equation method [15] yielded numerous solitonic solutions. We have selected to graphically explore some of these solutions, including the exponential function solution in Eq. (18) and the dark-bright solitonic solution in Eq. (22), which are the most ubiquitous among the examined solitonic solutions [10]. In fact, we depict these solutions in Figures 1, 2, 3, 4, 5, 6, 7, 8 and 9, by simply assuming all the involving parameters to be unity, except for

θ = 0.25 , p 1 = 1 , p 3 = 1 , p 2 = 2.5 .

dark-bright solitonic solution provides a visual representation of the complex dynamics inherent in a nonlinear wave system. It has distinctive structures within the wave profile. The graph provides insights into the stability, coherence, and collision dynamics of solitons within the nonlinear wave context. Analyzing the dark-bright solitonic solution graph helps researchers and scientists understand the intricate nature of nonlinear wave phenomena and their implications in various fields, such as optics, fluid dynamics, and plasma physics.

$Figure 1 Graphical representation of Eq. (18) portrayed using a 3D plot when r = 1 r=1 , 0 ≤ t ≤ 5 0\le t\le 5 , ‒ 10 ≤ x ≤ 10 . ‒10\le x\le 10.$

Figure 1

Graphical representation of Eq. (18) portrayed using a 3D plot when r = 1 , 0 ≤ t ≤ 5 , ‒ 10 ≤ x ≤ 10 .

$Figure 2 Graphical representation of Eq. (18) portrayed using a 2D plot when r = 1 r=1 , − 10 ≤ x ≤ 10 . -10\le x\le 10.$

Figure 2

Graphical representation of Eq. (18) portrayed using a 2D plot when r = 1 , − 10 ≤ x ≤ 10 .

$Figure 3 Variational influence of the Kudryashov index r r on the exponential solution determined in Eq. (18) portrayed using a 2D plot when r = 1 r=1 , − 10 ≤ x ≤ 10 . -10\le x\le 10.$

Figure 3

Variational influence of the Kudryashov index r on the exponential solution determined in Eq. (18) portrayed using a 2D plot when r = 1 , − 10 ≤ x ≤ 10 .

$Figure 4 Depiction of the exponential solution determined in Eq. (18) portrayed using a contour plot when r = 1 r=1 , − 10 ≤ x ≤ 10 . -10\le x\le 10.$

Figure 4

Depiction of the exponential solution determined in Eq. (18) portrayed using a contour plot when r = 1 , − 10 ≤ x ≤ 10 .

$Figure 5 Graphical representation of the dark-bright solitonic solution determined in Eq. (22) portrayed using a 3D plot when r = 1 r=1 , 0 ≤ t ≤ 5 0\le t\le 5 , − 10 ≤ x ≤ 10 . -10\le x\le 10.$

Figure 5

Graphical representation of the dark-bright solitonic solution determined in Eq. (22) portrayed using a 3D plot when r = 1 , 0 ≤ t ≤ 5 , − 10 ≤ x ≤ 10 .

$Figure 6 Graphical representation of the dark-bright solitonic solution determined in Eq. (22) portrayed using a 2D plot when r = 1 r=1 , − 10 ≤ x ≤ 10 . -10\le x\le 10.$

Figure 6

Graphical representation of the dark-bright solitonic solution determined in Eq. (22) portrayed using a 2D plot when r = 1 , − 10 ≤ x ≤ 10 .

$Figure 7 Graphical representation of the dark-bright solitonic solution determined in Eq. (22) portrayed using a 3D plot when r = 1 r=1 , 0 ≤ t ≤ 5 0\le t\le 5 , − 10 ≤ x ≤ 10 . -10\le x\le 10.$

Figure 7

Graphical representation of the dark-bright solitonic solution determined in Eq. (22) portrayed using a 3D plot when r = 1 , 0 ≤ t ≤ 5 , − 10 ≤ x ≤ 10 .

$Figure 8 Graphical representation of the dark-bright solitonic solution determined in Eq. (22) portrayed using a 2D plot when r = 1 r=1 , − 10 ≤ x ≤ 10 . -10\le x\le 10.$

Figure 8

Graphical representation of the dark-bright solitonic solution determined in Eq. (22) portrayed using a 2D plot when r = 1 , − 10 ≤ x ≤ 10 .

$Figure 9 Depiction of the exponential solution determined in Eq. (22) portrayed using a contour plot when r = 1 r=1 , − 10 ≤ x ≤ 10 . -10\le x\le 10.$

Figure 9

Depiction of the exponential solution determined in Eq. (22) portrayed using a contour plot when r = 1 , − 10 ≤ x ≤ 10 .

In particular, the Kudryashov method gives a Kink-typed solution as shown in Figure 1 via a 3D plot. Figure 2 depicts the comparable 2D plot. Figure 2 illustrates how temporal variation affects the propagation of hyperelastic waves in the Murnaghan pipe under discussion. Furthermore, Figure 3 depicts the response of the solution Eq. (18) with respect to the variation of the Kudryashov index r ; in fact, a decrease in the evolution of the nonlinear waves in the media with respect to the increase in the Kudryashov index. Kink-typed solution represents a localized disturbance or a sudden change in a physical quantity that propagates without dispersing its energy. The graph typically illustrates the evolution of the Kink-typed solution over space or time, showcasing its distinctive characteristics. In particular, the Kink-typed solution graph reveals the solitary nature of the wave, where a localized disturbance maintains its shape and moves through the medium without changing its form; see also Figure 4 for the corresponding contour depiction of acquired exponential solution in Eq. (18), giving much insights about the inner details of the bell-typed solution.

Figures 5 and 6, as well as Figures 7 and 8, show the evolution of the hyperelastic wave in the Murnaghan pipe by illustrating the real and imaginary sections of the dark-bright solitonic solution. The 3D plots in Figures 5 and 7 show kink-typed shapes, whereas Figures 6 and 8 show that the evolution of the hyperelastic wave in the media increases with propagation time; see also Figure 9 for the corresponding contour depiction of acquired exponential solution in Eq. (22), giving much insights about the inner details of the Kink-typed solution. Furthermore, interested readers might refer previous studies [16–30] for some of the most recent proposed and implemented analytical methodologies for the analysis of wave propagation in modern nonlinear media. In addition, the acquired solitonic solutions in the study stand out to augment the findings reported by Yel [3], Abourabia and Eldreeny [4], and Rehman et al. [5] to mention a few, with regard to the retrieval of solitonic solutions for nonlinear waves in hyperelastic materials [6]. In addition, the reported solutions in this study will help scientists working in structural engineering, materials science, and seismic analysis to obtain valuable insights that can inform the design, analysis, and optimization of materials and structures in engineering applications.

6 Conclusion

The scenario of wave propagation on a nonlinear Murnaghan hyperelastic circular pipe was investigated in this study. Double dispersion terms were taken into account when modeling the structure using the nonlinear Murnaghan potential with nonlinear strain tensors. In order to create some significant model solutions that describe the dynamic behavior of nonlinear waves on the governing circular pipe, three analytical techniques were used. The elucidation of soliton solutions and nonlinear wave dynamics in this unique structural configuration enhances our understanding of material behavior and informs future developments in elastic structures and wave propagation studies. In actuality, analytical solutions were obtained, including some solitonic and exponential solutions. Although additional solutions were also found by the use of the generalized Riccati equation method, it was decided not to be discussed here because solitonic solutions are more useful for researching pulse propagation in optical media. Finally, we have illustrated a few of the acquired solutions and examined how the Kudryashov index affects the nonlinear waves. The findings contribute to our understanding of the complex interplay between material elasticity and wave dynamics, offering potential applications in diverse fields such as materials science, seismic analysis, structural engineering, and nonlinear wave research. As we continue to unravel the complexities of nonlinear phenomena, this study serves as a stepping stone for future investigations and applications in the broader domain of hyperelastic materials and their dynamic behaviors. Furthermore, these results bear practical implications for the design and analysis of elastic structures in diverse engineering applications, highlighting the potential for leveraging nonlinear wave dynamics for innovative engineering solutions.

Acknowledgments

The author extends his appreciation to Taif University, Saudi Arabia, for supporting this work through project number TU-DSPP-2024-66.

Funding information: The research was funded by Taif University, Saudi Arabia, Project No. TU-DSPP-2024-66.
Author contributions: The author has accepted responsi- bility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author states no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

[1] Achenbach JD. Wave propagation in elastic solids, eight impression; Amsterdam, The Netherlands: Elsevier; 1999. Search in Google Scholar

[2] Ewing WM, Jardetzky WS, Press F, Beiser A. Elastic waves in layered media. New York: McgrawHill; 1957. Search in Google Scholar

[3] Yel G. New wave patterns to the doubly dispersive equation in nonlinear dynamic elasticity. Pramana J Phy. 2020;94:79. Search in Google Scholar

[4] Abourabia AM, Eldreeny YA. A soliton solution of the DD-Equation of the Murnaghanas rod via the commutative hyper complex analysis. Partial Diff Equ Appl Math. 2022;6:100420. Search in Google Scholar

[5] Rehman SU, Seadawy AR, Rizvi STR, Ahmed S, Althobaiti S. Investigation of double dispersive waves in nonlinear elastic inhomogeneous Murnaghan’s rod. Modern Phys Lett B. 2022;36(8):2150628. Search in Google Scholar

[6] Rushchitsky JJ, Sinchilo SV. On two-dimensional nonlinear wave equations for the Murnaghan model. Int Appl Mech. 2013;49(5):512–20. Search in Google Scholar

[7] Usman M, Zaman FD. Lie symmetry analysis and conservation laws of non-linear (2+1) elastic wave equation. Arabian J Math. 2023;12:265–76. 10.1007/s40065-022-00392-y. Search in Google Scholar

[8] Nuruddeen RI, Nass AM. Exact solitary wave solution for the fractional and classical GEW-Burgers equations: an application of Kudryashov method. J Taibah University Sci. 2018;12:309–14. Search in Google Scholar

[9] Alrashed R. et al. Collective variables approach to the vector-coupled system of Chen-Lee-Liu equation. Chaos Solitons Fractals. 2022;161:112315. Search in Google Scholar

[10] Wazwaz A. A variety of soliton solutions for the Boussinesq-Burgers equation and the higher-order Boussinesq-Burgers equation. Filomat, 2017;31(3):831–40. Search in Google Scholar

[11] Bokhari AH, Kara AH, Zaman FD. Exact solutions of some general nonlinear wave equations in elasticity. Nonlinear Dyn. 2007;48(1):49–54. Search in Google Scholar

[12] Obaidullah U, Jamal S. A computational procedure for exact solutions of Burgers’ hierarchy of nonlinear partial differential equations. J Appl Math Comp. 2021;65(1):541–51. Search in Google Scholar

[13] Seadawy AR, Nuruddeen RI, Aboodh KS, Zakariya YF. On the exponential solutions to three extracts from extended fifth-order KdV equation. J King Saud Uni-Sci. 2020;32:765–9. Search in Google Scholar

[14] Mubaraki AM, Nuruddeen RI, Ali KK, Gomez-Aguilar JF. Additional solitonic and other analytical solutions for the higher-order Boussinesq-Burgers equation. Optical Quantum Electr. 2024;56:2. Search in Google Scholar

[15] Alharbi, R., Alshaery AA, Bakodah HO, Nuruddeen RI, Gómez-Aguilar JF. Revisiting (2+1)-dimensional Burgers’ dynamical equations: analytical approach and Reynolds number examination. Phys Scripta. 2023;98(8):085225. Search in Google Scholar

[16] Hosseini K, Bekir A, Ansari R. Exact solutions of nonlinear conformable time-fractional Boussinesq equations using the exp(−θ(ξ)) -expansion method. Opt Quantum Electron. 2017;49:131. Search in Google Scholar

[17] Islam MT, Akbar MA, Azad MAK. A rational (G∕G)-expansion method and its application to modified kdv-Burgers equation and the (2+1)-dimensional Boussineq equation. Nonlinear Stud. 2015;6:1. Search in Google Scholar

[18] Baskonus HM, Sulaiman TA, Bulut H. New solitary wave solutions to the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiffand the Kadomtsev-Petvi-Ashvili hierarchy equations. Indian J Phys. 2017;91:1237–43. Search in Google Scholar

[19] Banaja M, Al Qarni AA, Bakodah HO, Zhou Q, Moshokoa SP, Biswas A. The investigate of optical solitons in cascaded system by improved Adomian decomposition scheme. Int J Light Electron Opt. 2017;130:1107–1144. Search in Google Scholar

[20] Khatun MA, Arefin MA, Akbar MA, Uddin MH. Numerous explicit soliton solutions to the fractional simplified Camassa-Holm equation through two reliable techniques. Ain Shams Eng J. 2023;14(12):102214. Search in Google Scholar

[21] Zaman UHM, Arefin MA, Akbar MA, Uddin MH. Stable and effective traveling wave solutions to the non-linear fractional Gardner and Zakharov-Kuznetsov-Benjamin-Bona-Mahony equations. Partial Diff Equ Appl Math. 2023;7:100509. Search in Google Scholar

[22] Arefin MA, Khatun MA, Uddin MH, Inc M. Investigation of adequate closed form travelling wave solution to the space-time fractional non-linear evolution equations. J Ocean Eng Sci. 2022;7:292–303. Search in Google Scholar

[23] Sadiya U, Arefin MA, Uddin MH. Consistent travelling waves solutions to the non-linear time fractional Klein-Gordon and sine-Gordon equations through extended tanh-function approach. J Taibah University Sci. 2022;16:594–607. Search in Google Scholar

[24] Althobaiti S, Nuruddeen RI, Magaji AY, Gomez-Aguilar JF. New revelations and extensional study on the recent sixth-order 3D Kadomtsev-Petviashvili-Sawada-Kotera-Ramani equation. Opt Quantum Electr. 2024;56:820. Search in Google Scholar

[25] Ali KK, Nuruddeen RI, Yildirim A. On the new extensions to the Benjamin-Ono equation. Comput Methods Differ Equ. 2020;8:424–45. Search in Google Scholar

[26] Ullah MS, Ali MZ, Roshid HO, Hoque MF. Collision phenomena among lump, periodic and stripe soliton solutions to a (2+1)-dimensional. Benjamin-Bona-Mahony-Burgers Model Eur Phys J Plus. 2021;136:370. Search in Google Scholar

[27] Ullah MS, Abdeljabbar A, Roshid HO, Ali MZ. Application of the unified method to solve the Biswas-Arshed model. Results Phys. 2022;42:105946. Search in Google Scholar

[28] Du S, Haq NU, Rahman MU. Novel multiple solitons, their bifurcations and high order breathers for the novel extended Vakhnenko-Parkes equation. Results Phys. 2023;54:107038. Search in Google Scholar

[29] Luo R, Ullah R, Emadifar H, Rahman MU. Bifurcations, chaotic dynamics, sensitivity analysis and some novel optical solitons of the perturbed non-linear Schrodinger equation with Kerr law non-linearity. Results Phys. 2023;54:107133. Search in Google Scholar

[30] Rahman MU, Assiri TA, Saifullah S, Khan MA, Sun M. Some new optical solitary waves solutions of a third order dispersive Schrodinger equation with Kerr nonlinearity using an efficient approach associated with Riccati equation. Opt Quant Electron. 2024;56:646. Search in Google Scholar

[31] Baleanu D, Etemad S, Mohammadi H, Rezapour S. A novel modeling of boundary value problems on the glucose graph. Commun Nonlinear Sci Numer Simul. 2021;100:105844. Search in Google Scholar

[32] Tuan NH, Mohammadi H, Rezapour S. A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Solitons Fractals. 2020;140:110107. Search in Google Scholar

[33] Baleanu D, Jajarmi A, Mohammadi H, Rezaour S. A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos Solitons Fractals. 2020;134:109705. Search in Google Scholar

[34] Liu Y, Xu R. Potential well method for Cauchy problem of generalized double dispersion equations. J Math Anal Appl. 2008;338(2):1169–87Search in Google Scholar

[35] Samsonov AM, Sokurinskaya EV. On the excitation of a longitudinal deformation soliton in a non-linear elastic rod. Sov Phys-Tech Phys. 1988;33(5):989–91. Search in Google Scholar

[36] Shi J, Yan XL. Analytical solutions for the elastic circular rod nonlinear wave, Boussinesq, and dispersive long wave equations. J Appl Math. 2014;2014:487571. Search in Google Scholar

Received: 2024-02-08

Revised: 2024-04-05

Accepted: 2024-05-03

Published Online: 2024-06-11

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

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0033

Keywords for this article

elastic structures; hyperelastic material; nonlinear waves; Murnaghan rod; propagation and interaction

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