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Solitonic wave solutions of a Hamiltonian nonlinear atom chain model through the Hirota bilinear transformation method

  • Baboucarr Ceesay , Muhammad Z. Baber , Nauman Ahmed , Siegfried Macías , Jorge E. Macías-Díaz EMAIL logo and María G. Medina-Guevara
Published/Copyright: May 9, 2025
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Open Physics
From the journal Open Physics Volume 23 Issue 1

Abstract

This study employs the Hirota bilinear transformation method to investigate solitary and soliton solutions resulting from different symmetry wave functions associated with nonlinear atom chain models. These models are complex dynamic systems that have an impact on a variety of scientific areas. In this work, we derive exact solutions for several symmetry wave functions. Our results reveal a wide range of wave structures, including anti-kink waves (resulting from the interaction of exponential wave functions), M-shaped waves with single and double kinks (generated by combining M-shaped and kink wave functions), and periodic multiple lump waves (produced by the interaction of multi-lump and periodic wave functions). Solitary waves and numerous periodic lump waves have also been reported as a result of various multi-wave function combinations. In the present report, we also derive periodic bright lump waves, periodic cross-kink waves, and dark lump strip waves caused by interactions between exponential, trigonometric, and hyperbolic functions. Additional solutions include solitary waves from mixed wave functions and bright breather waves obtained from homoclinic breather functions. For illustration purposes, we provide three-dimensional visualizations of these solutions as well as accompanying contour plots.

Keywords: nonlinear atomic chain model; solitary wave solutions; soliton solutions; Hirota bilinear transformation method

1 Introduction

Nonlinear partial differential equations (NLPDEs) are one of the primary approaches in the modeling and analysis of numerous complex processes [1–3]. These processes include, but are not limited to, plasma physics [4], nonlinear optics [5], chemical kinetics [6], physics of condensed matter [7], and even certain aspects of biology [8]. The reason for this is that NLPDEs are able to catch the dynamics of complex systems with various independent variables. While linear equations possess the property of being relatively simple, NLPDEs embrace the complexities existing within the natural and artificial systems. In this context, one avenue of investigation in NLPDEs is the search for exact solutions which can be used to interpret some physical reality. To that end, many different analytical and numerical approaches have been suggested. Some of the most notable methods involve the modified exp-function method [9], the modified generalized Kudryashov method [10], the extended Jacobi elliptic function rational expansion method [11], Riccati’s equation expansion method [12], the sine-Gordon expansion method [13], the G ′ G -expansion technique [14], the F -expansion approach [15], the Sardar sub-equation scheme [16], the Darboux transformation technique [17], the bilinear convolutional neural network method [18], the natural transform decomposition method [19], the modified fractional homotopy technique [20], the Laplace decomposition method [21], the Adomian decomposition method [22], the perturbation method [23], and the bilinear residual network method [24]. These approaches are useful for integrable classes of some NLPDEs. However, some of these methods do not provide a generalized approach that can be used for any nonlinear problem. Therefore, there is always an urge to modify the existing methods or come up with new ones that can help in solving more difficult equations.

Among the multiple existing techniques, the Hirota bilinear transformation technique has received special emphasis in view of its application on many complex integrable NLPDEs. This technique is used to convert NLPDEs to bilinear equations, which are easier to deal with. This method allows us to derive soliton solutions, which play an important role in most systems. In the present work, the Hirota bilinear method is used to study solitary and soliton waves in a nonlinear atom chain (NAC) model. It is worth recalling that soliton theory, which is primarily concerned with solitary waves and their propagation, has been quickly established as one of the cornerstones of modern nonlinear physics and mathematics. Solitary waves are a class of waves that can propagate in certain media without changes due to dispersion. These waves propagate and keep their speed for a long time, which is an optimal condition for these waves to be used in describing localized physical processes, such as energy propagation in crystals [25], strain in crystal [26], as well as wave motion in a solid state [27]. Solitary waves have been studied by scientists in many contexts, including aqueous flows, light in fibers, and mechanical waves. For example, some reports investigate solitonic rogue waves induced by the modulation instability in a split-ring-resonator-based left-handed coplanar waveguide [28], ( 2 + 1 ) -dimensional variable coefficient equations in the investigation of oceanic solitons and localized wave solutions [29] and the effects of higher-order dispersion on solitary waves and modulation instability in a monomode fiber [30]. In addition, some reports investigate rogue wave dynamics in barotropic relaxing media [31], N -rotating loop-soliton solutions of the coupled integrable dispersionless equation [32], traveling wave-guide channels of a coupled integrable dispersionless system [33], and their scattering behaviors [34]. In other situations, solitons serve to study the stress, strain, and material responses of structures under different conditions [35–38].

Usually, the behavior of solitary waves is governed by NLPDEs, which are generally complex and often require advance techniques to solve them. What sets these equations apart from their linear counterparts is that they are frequently intractable. Instead, in order to obtain meaningful results, we need sophisticated analytical methods, such as the Hirota bilinear transformation approach or numerical approximations. In many industrial and scientific applications, systems are frequently affected by severe circumstances like high temperatures, high pressures, or fast deformation. The nonlinearities of these equations are essential for capturing those behaviors. Consequently, bettering the design and safety of mechanical systems requires an understanding of nonlinear equations. One area where NLPDEs have proven particularly valuable is in the modeling of NACs. These chains are widely used in the study of materials science and solid mechanics, as they provide insights into the behavior of materials at the atomic level [38–41]. NACs can model a variety of phenomena, including thermal conductivity, wave propagation, defect formation, and material deformation under stress [38,39]. By studying these chains, researchers may obtain a deeper understanding of how materials behave under extreme conditions, leading to the development of high-performance materials for various applications in science, engineering, and technology.

It is also important to point out that wave propagation and solitons in NACs have attracted attention in the last couple of years. In fact, several techniques have been designed to study them rigorously. For instance, Foroutan et al. [42] employed the extended trial equation technique and the improved Bernoulli sub-ordinary differential equation method to solve an NAC problem. They obtained trigonometric, soliton, hyperbolic, and kink solutions. These solutions have applications in generalized momentum for long-range interactions and Hamilton’s equations. In order to analyze NACs, Junaid-U-Rehman et al. [46] made use of the Lie symmetry technique. They generated traveling-wave solutions and utilized the multiplier method to discover the conservation laws. Their results show the dynamics and patterns of the system. Shakeel et al. [47] applied the generalized auxiliary equation method to construct bright, periodic, singular, W-type, and M-type soliton solutions for the NAC model. Among many other reports available in the literature, these solutions form an important aspect in material and structural nanoscience.

On the other hand, the Hirota bilinear transformation technique has made significant contributions to our comprehension of the complex wave solutions produced by various NLPDEs. Notably, a number of researchers have used this technique in recent years to provide exact solutions to a range of NLPDEs. For instance, Ceesay et al. [48] employed the modified regularized long-wave equations to analyze the impact of different M-shaped waves in coastal environments. In order to obtain bilinear equations for NLPDEs with indeterminate coefficients, Yang and Wei [49] also applied Hirota’s bilinear approach. By applying the tanh-coth and Hirota bilinear methods, Wazwaz [50] was able to find many solutions to the Sawada–Kotera–Kadomtsev–Petviashvili model. Furthermore, Rizvi et al. [51] used the Hirota bilinear approach to investigate saturated ferromagnets. Khan and Wazwaz [52] also employed this technique using ansatz functions to derive numerous novel soliton solutions and multi-breather solutions for the Calogero–Bogoyavlenskii–Schiff (CBS) model, whereas Wang et al. [53] adopted a study of plasma and fluid dynamics for the generalized (3 + 1) Kadomtsev–Petviashvili model. Zhao et al. [54] also employed the bilinear approach to produce various lump, mixed lump-stripe solutions using the asymmetric (2 + 1) Nizhnik–Novikov–Veselov model. Using this method, Ren et al. [55] investigated the CBS-type system in the extended (2 + 1) dimensions. In turn, Rizvi and colleagues [56] investigated the coupled Higgs equations using the Hirota bilinear transformation method.

It is worthwhile mentioning that the Hirota bilinear approach extends beyond the traditional study of solitons, and gives margin to study other number of interesting wave forms (including rogue, singular, kink, anti-kink, breather, M-shaped, and lump waves). This fact demonstrates the method’s broad application. These solutions (which are not just theoretical in nature) have real-world applications in fields like biomembrane, fluid dynamics, wave optical pulse, shallow water modeling, etc. Furthermore, given their significant advancements in our knowledge of waves and soliton solutions and their practical applications, these examples show the applicability of the Hirota bilinear technique across fields. In the present study, we departed from the Hirota bilinear transformation method and applied it to an NCA model. After examining earlier research conducted by different investigators who employed the NCA model, they reveal that the Hirota bilinear transformation techniques have not been applied to this system. The purpose of this study is to fill this vacuum in the research literature. In terms of originality, the present work derives novel solutions of an NAC model. Indeed, as some of the reviewers pointed out, there are already some solutions for particular NAC models reported in the literature. However, the investigation of NAC models with an associated Hamiltonian is a problem for which exact solutions have not been derived. Additionally, the Hirota bilinear transformation method has not been employed in those contexts. In that sense, the present work is one of the first in the literature, which tackles the derivation of analytical solutions for NAC models using the methodology proposed by Hirota.

2 Preliminaries

2.1 Physical model

The purpose of this subsection is to provide an overview of the NAC model under investigation. In our physical model, we will assume only one degree of freedom. The Hamiltonian of the system under study is provided by [42]

(1) H = ∑ n 1 2 m L ˙ n 2 + ∑ l ≠ n M ( ∣ L n − L l ∣ ) ,

where M ( ∣ L n − L l ∣ ) is the nonlinear potential, m is the mass of atom, and the dot represents the first derivative with respect to time. All functions depend only on the temporal variable, and the inner-most sum at the right-hand side of this equation takes into account only the nearest neighbors, that is, l = n ± 1 . Moreover, in what follows, we will consider the following potential function:

(2) M ( k n l ) = 1 12 k n l 2 [ 6 α i + 4 β i k n l + γ i k n l 2 ] .

Here the relative displacement between the l th and the n th atoms is denoted by k n l . The unique particle-to-particle interactions are displayed by index i . Recall that Hamilton’s equations are given by

(3) ∂ H ∂ L n = − Q ˙ n , ∂ H ∂ Q n = ∂ L n ∂ t = L ˙ n .

Applying Hamilton’s equations to Eqs (1) and (2), we derive the approximate equation of motion

(4) d 2 L n d t 2 = α 1 ( L n + 1 − 2 L n + L n − 1 ) − α 2 ( L n + 2 − 2 L n + L n − 2 ) + β 1 [ ( L n + 1 − L n ) 2 − ( L n − L n − 1 ) 2 ] + β 2 [ ( L n + 2 − L n ) 2 − ( L n − L n − 2 ) 2 ] + γ 1 [ ( L n + 1 − L n ) 2 − ( L n − L n − 1 ) 2 ] + γ 2 [ ( L n + 2 − L n ) 2 − ( L n − L n − 2 ) 2 ] ,

where Q n denotes the generalized momentum in Hamilton’s equations. It is important to mention here that, in addition to chain atom models with long-range interactions [42], the present model is an extension of various other models studied in the literature. Some of them are models in which authors investigated the existence of dynamic solitary wave solutions [43], the presence of conservation laws using Lie analysis [44,45].

If the inter-atom spacing ε is sufficiently small to attain the continuum limit, then we can let ε n → x . As a result, we obtain the expansions

(5) L n ± 1 = L ± ε L x + 1 2 ! ε 2 L x x ± 1 3 ! ε 3 L x x x + 1 4 ! ε 4 L x x x x ± 1 5 ! ε 5 L x x x x x + …

and

(6) L n ± 2 = L ± 2 ε L x + 4 2 ! ε 2 L x x ± 8 3 ! ε 3 L x x x + 16 4 ! ε 4 L x x x x ± 32 5 ! ε 5 L x x x x x + … .

Hence, Eq. (4) can be expressed as the NLPDE given by

(7) L t t − a 2 L x x − b L x L x x − c L x 2 L x x − d L x x x x = 0 ,

with the constants

(8) a 2 = ε 2 ( α 1 + 4 α 2 ) m , b = 2 ε 3 ( β 1 + 8 β 2 ) m , c = ε 4 ( γ 1 + 16 γ 2 ) m , d = ε 4 ( α 1 + 16 α 2 ) 12 m .

It is worthwhile to mention that Eq. (7) may find useful applications in the investigation of one-dimensional chains of atoms when they are studied from a microscopic point of view, in which case, the assumption that ε is sufficiently small is physically justified. In this study, we will apply the Hirota bilinear transformation approach and some ansatz functions to obtain various wave solutions for the NAC model given by Eq. (7).

2.2 Analytical method

At this stage, we recall the steps of the Hirota bilinear transformation method and apply them to solve Eq. (7) analytically. We start by applying the Cole–Hopf transformation, which is a logarithmic transformation for the dependent variable L , and we suppose that the solution to Eq. (7) is given by

(9) L ( x , t ) = R ( ln ( Γ ( x , t ) ) ) x ,

where R is a constant to be determined. To find R , we begin by using the following function: L ( x , t ) = e α i x + ω i t . Here, α i and ω i represent wave numbers and dispersions, respectively. Substituting this equation in Eq. (7) yields the dispersion relation

(10) ω i = α i a 2 + d α i 2 .

As a consequence, it follows that

(11) L ( x , t ) = e α i x + α i a 2 + d α i 2 t .

We then consider the function Γ ( x , t ) = 1 + e α i x + α i a 2 + d α i 2 t , and substitute it in Eq. (9). The result is then substituted in the linear part of Eq. (7), and then we solve for R to obtain R = 12 d b and c = 0 . Hence, the logarithmic transformation becomes

(12) L ( x , t ) = 12 d b ( ln ( Γ ( x , t ) ) ) x .

In this way, we transformed Eq. (7) into the bilinear form

(13) ( D t 2 − a 2 D x 2 − d D x 4 ) Γ ⋅ Γ = 0 ,

using transformation (12). Here the bilinear operator D x u D t v is defined by

(14) D x u D t v Γ ⋅ Λ = ∂ ∂ x − ∂ ∂ x ′ u ∂ ∂ t − ∂ ∂ t ′ v Γ ( x , t ) Λ ( x ′ , t ′ ) ∣ x = x ′ , t = t ′ .

Finally, we can write Eq. (13) as

(15) − a 2 Γ ∂ 2 Γ ∂ x 2 + a 2 ∂ Γ ∂ x 2 − d Γ ∂ 4 Γ ∂ x 4 + 4 d ∂ Γ ∂ x ∂ 3 Γ ∂ x 3 − 3 d ∂ 2 Γ ∂ x 2 2 + Γ ∂ 2 Γ ∂ t 2 − ∂ Γ ∂ t 2 = 0 .

3 Results

Next we will consider various types of forms for the function Γ and substitute them in Eq. (15). Then, we expand, evaluate, collect similar terms, and set them equal to 0. Finally, we solve this system of equations to obtain a number of potential sets of solutions for each case.

3.1 Interaction via double exponents integral equation (IE)

Following the results in the study by Ceesay et al. [57], this wave structure is derived by letting

(16) Γ ( x , t ) = h 1 exp ( l ( r 1 ( λ t + x ) + r 2 ) ) + h 2 exp ( l ( r 3 ( λ t + x ) + r 4 ) ) ,

where r i (for 1 ≤ i ≤ 4 ), h j (for 1 ≤ j ≤ 2 ), and l ≠ 0 are arbitrary constants, and the parameter λ ≠ 0 is the wave speed. Eq. (16) and its first four partial derivatives were substituted in Eq. (15). Then, we reduce the terms algebraically in order to obtain a simplified expression. Assembling similar terms together and equating each expression’s coefficient to zero, the following set of constant values resulted:

Set 1. r 3 = d l 2 r 1 − d λ 2 l 2 − a 2 d l 2 d l 2 .

Substituting this constant in Eq. (16) and the outcome in Eq. (12), we obtain

(17) L 1 I E ( x , t ) = 12 h 1 d l 2 ( λ 2 − a 2 ) e l ( r 1 ( λ t + x ) + r 2 ) h 2 exp l − ( λ t + x ) d l 2 ( λ 2 − a 2 ) d l 2 + r 1 ( λ t + x ) + r 4 + h 1 e l ( r 1 ( λ t + x ) + r 2 ) − d l 2 ( λ 2 − a 2 ) + d l 2 r 1 b l ,

where a , b , d ≠ 0 are free parameters. Figure 1(a) depicts the solution L 1 I E ( x , t ) corresponding to Eq. (17) for selected parameter values r 1 = 2.2 , r 2 = 1.2 , r 4 = 1.6 , h 1 = 2.4 , h 2 = 1.4 , a = 0.4 , b = 0.62 , d = 0.5 , l = 3.9 , λ = 0.34 . Figure 1(b) shows the contour of Figure 1(a).

Figure 1 The three-dimensional (3D) (a) and contour diagram (b) representing the IE-type wave.
Figure 1

The three-dimensional (3D) (a) and contour diagram (b) representing the IE-type wave.

3.2 Interaction of M-shaped rational wave with one kink (M1K)

This wave configuration is provided by using [58]

(18) Γ ( x , t ) = h 1 exp ( r 1 ( λ t + x ) + r 2 ) + ( r 3 ( λ t + x ) + r 4 ) 2 + ( r 5 ( λ t + x ) + r 6 ) 2 + r 7 ,

where r i (for 1 ≤ i ≤ 7 ), and h 1 are arbitrary constants, and the parameter λ ≠ 0 is the wave speed. Let us substitute Eq. (18) and its partial derivatives up to the fourth order in Eq. (15) and simplify algebraically. Proceeding as in the previous Section 3.1, we obtain one set of constant values from the resulting system.

Set 1. r 1 = − λ 2 − a 2 d , r 3 = 0 , r 5 = 0 .

Substituting these values in Eq. (18), and then the resulting equation in Eq. (12), we obtain

(19) L 1 M 1 K ( x , t ) = − 12 d h 1 e r 2 λ 2 − a 2 b ( r 4 2 + r 6 2 + r 7 ) e λ 2 − a 2 ( λ t + x ) d + b h 1 e r 2 ,

where a , b , d ≠ 0 are free parameters. The solution L 1 M 1 K ( x , t ) corresponding to Eq. (19) is shown in Figure 2(a) for the parameter values r 2 = 3.2 , r 4 = 0.5 , r 6 = 0.1 , r 7 = 2.3 , h 1 = 4.1 , a = 0.5 , b = 0.68 , d = 0.7 , λ = 0.18 . Meanwhile, Figure 2(b) shows its corresponding contour plot.

Figure 2 The 3D (a) and contour diagram (b) of M1K wave-type.
Figure 2

The 3D (a) and contour diagram (b) of M1K wave-type.

3.3 Interaction of M-shaped rational wave with two kinks (M2K)

This wave configuration is provided by [58]

(20) Γ ( x , t ) = h 1 exp ( r 1 ( λ t + x ) + r 2 ) + h 2 exp ( r 3 ( λ t + x ) + r 4 ) + ( r 5 ( λ t + x ) + r 6 ) 2 + ( r 7 ( λ t + x ) + r 8 ) 2 + r 9 ,

where r i (for 1 ≤ i ≤ 9 ) and h j (for 1 ≤ j ≤ 2 ) are arbitrary constants, and the parameter λ ≠ 0 is the wave speed. Substitute Eq. (20) and its partial derivatives up to the fourth order in Eq. (15) and simplify. Proceeding as before, we obtain a set of constant values.

Set 1. r 1 = − λ 2 − a 2 d , r 3 = − λ 2 − a 2 d , r 5 = 0 , r 7 = 0 , r 8 = 0 .

After inserting these values in Eq. (20) and the outcome in Eq. (12), we readily obtain

(21) L 1 M 2 K ( x , t ) = − 12 d λ 2 − a 2 ( h 1 e r 2 + h 2 e r 4 ) b ( r 6 2 + r 9 ) e λ 2 − a 2 ( λ t + x ) d + h 1 e r 2 + h 2 e r 4 ,

where a , b , d ≠ 0 are free parameters. Figure 3(a) depicts the solution L 1 M 2 K ( x , t ) from Eq. (21) with chosen parameter values r 2 = 2.0 , r 4 = 4.6 , r 6 = 7.2 , r 9 = 3.3 , h 1 = 0.3 , h 2 = 1.9 , a = 0.5 , b = 1.08 , d = 0.1 , λ = 0.11 . Figure 3(b) illustrates its corresponding contour plot.

Figure 3 The 3D (a) and contour diagram (b) of M2K wave-type.
Figure 3

The 3D (a) and contour diagram (b) of M2K wave-type.

3.4 Interaction of multi-lump and periodic solutions (MLP)

This wave configuration is provided by [52]

(22) Γ ( x , t ) = h 3 sinh ( r 1 ( λ t + x ) + r 2 ) + h 4 cosh ( r 3 ( λ t + x ) + r 4 ) + h 1 cos ( r 3 ( λ t + x ) + r 4 ) cosh ( r 1 ( λ t + x ) + r 2 ) + h 2 sin ( r 3 ( λ t + x ) + r 4 ) cosh ( r 1 ( λ t + x ) + r 2 ) + 1 ,

where r i (for 1 ≤ i ≤ 4 ) and h j (for 1 ≤ j ≤ 4 ) are arbitrary constants, and the parameter λ ≠ 0 is the wave speed, once again. We substitute Eq. (22) and its first four partial derivatives into Eq. (15), and we simplify algebraically. In similar fashion as with the previous wave structures, we obtain a single set of parameter values.

Set 1. h 1 = − i h 2 , h 3 = 0 , h 4 = 0 , r 1 = − λ 2 − a 2 2 d , r 3 = − i λ 2 − a 2 2 d .

Substituting these parameter values in Eq. (22) and then the result in Eq. (12), we obtain

(23) L 1 MLP ( x , t ) = 6 d h 2 λ 2 − a 2 sinh λ 2 − a 2 ( λ t + x ) d − r 2 + i r 4 + cosh λ 2 − a 2 ( λ t + x ) d − r 2 + i r 4 b h 2 cosh λ 2 − a 2 ( λ t + x ) 2 d − r 2 sinh λ 2 − a 2 ( λ t + x ) 2 d + i r 4 + cosh λ 2 − a 2 ( λ t + x ) 2 d + i r 4 + i ,

where a , b , d ≠ 0 are free real parameters. Figure 4(a) is obtained from the solution L 1 MLP ( x , t ) , corresponding to Eq. (23) for the parameter values r 2 = 0.2 , r 4 = 4.0 , h 2 = 3.1 , a = 2.3 , b = 0.82 , d = 0.3 , Λ = 0.087 . Figure 4(b) represents its respective contour plot.

Figure 4 The 3D (a) and contour diagram (b) are obtained from the MLP wave-type.
Figure 4

The 3D (a) and contour diagram (b) are obtained from the MLP wave-type.

3.5 Multi waves (MU)

This pattern of waves is derived when [59]

(24) Γ ( x , t ) = h 2 cos ( r 3 ( λ t + x ) + r 4 ) + h 1 cosh ( r 1 ( λ t + x ) + r 2 ) + h 3 cosh ( r 5 ( λ t + x ) + r 6 ) ,

where r i (for 1 ≤ i ≤ 6 ) and h j (for 1 ≤ j ≤ 3 ) are arbitrary constants, and λ ≠ 0 is the wave speed. Let us substitute Eq. (24) and its first four partial derivatives in Eq. (15), and then we reduce algebraically. Proceeding as in the previous Section 3.4, we obtain two sets of parameter values.

Set 1. r 1 = λ 2 − a 2 2 d , r 3 = i λ 2 − a 2 2 d , r 5 = λ 2 − a 2 2 d .

Substituting them in Eq. (24), and then the result in Eq. (12), yields the solution

(25) L 1 MU ( x , t ) = 6 d λ 2 − a 2 ( h 1 sinh ( A 1 + r 2 ) + h 2 sinh ( A 1 − i r 4 ) + h 3 sinh ( A 1 + r 6 ) ) b ( h 1 cosh ( A 1 + r 2 ) + h 2 cosh ( A 1 − i r 4 ) + h 3 cosh ( A 1 + r 6 ) ) ,

where A 1 = λ 2 − a 2 ( λ t + x ) 2 d , and a , b , d ≠ 0 are free parameters. Using the solution L 1 MU ( x , t ) from Eq. (25), Figure 5(a) is obtained by selecting the parameter values r 2 = 2.3 , r 4 = 5.2 , r 6 = 2.1 , h 1 = 6.9 , h 2 = 0.2 , h 3 = 9.1 , a = 4.9 , b = 2.28 , d = 0.3 , λ = 2.9 . The associated contour plot is portrayed as Figure 5(b).

Figure 5 The 3D (a) and contour diagram (b) of MU wave-type.
Figure 5

The 3D (a) and contour diagram (b) of MU wave-type.

Set 2. h 3 = 0 , r 1 = λ 2 − a 2 2 d , r 3 = i λ 2 − a 2 2 d .

Substituting them in Eq. (24) and then the result in Eq. (12), we obtain

(26) L 2 MU ( x , t ) = 6 d λ 2 − a 2 h 1 sinh λ 2 − a 2 ( λ t + x ) 2 d + r 2 + h 2 sinh λ 2 − a 2 ( λ t + x ) 2 d − i r 4 b h 1 cosh λ 2 − a 2 ( λ t + x ) 2 d + r 2 + h 2 cosh λ 2 − a 2 ( λ t + x ) 2 d − i r 4 ,

where a , b , d ≠ 0 are free parameters. The solution L 2 MU ( x , t ) given by Eq. (26) is represented in Figure 6(a) by selecting the parameter values r 2 = 0.9 , r 4 = 2.4 , h 1 = 1.2 , h 2 = 3.2 , a = 3.6 , b = 0.67 , d = 0.7 , λ = 0.1 . The corresponding contour plot is shown in Figure 6(b).

Figure 6 The 3D (a) and contour diagram (b) of MU wave-type.
Figure 6

The 3D (a) and contour diagram (b) of MU wave-type.

3.6 Periodic lump (PL)

This wave structure is given by [59]

(27) Γ ( x , t ) = ( r 1 ( λ t + x ) + r 2 ) 2 + ( r 3 ( λ t + x ) + r 4 ) 2 + cos ( r 5 ( λ t + x ) + r 6 ) + r 7 ,

where r i (for 1 ≤ i ≤ 7 ) are arbitrary constants, and the parameter λ ≠ 0 is, as usual, the wave speed. Substituting Eq. (27) and its partial derivatives in Eq. (15), and proceeding then as before, we obtain a single set of parameter values.

Set 1. r 1 = − i r 3 , r 2 = 0 , r 4 = 0 , r 5 = i λ 2 − a 2 2 d , r 7 = 0 .

By replacing these parameter values in Eq. (27), and the outcome in Eq. (12), we obtain

(28) L 1 PL ( x , t ) = 6 d λ 2 − a 2 tanh λ 2 − a 2 ( λ t + x ) 2 d − i r 6 b ,

where a , b , d ≠ 0 are free parameters. The solution L 1 LP ( x , t ) corresponding to Eq. (28) is depicted in Figure 7(a) with the chosen parameter values r 6 = 1.2 , a = 3.77 , b = 0.47 , d = 4.8 , λ = 7.41 . Figure 7(b) portrays its corresponding contour.

Figure 7 The 3D (a) and contour diagram (b) obtained from PL type wave.
Figure 7

The 3D (a) and contour diagram (b) obtained from PL type wave.

3.7 Periodic cross kink (PCK)

This type of solution is obtained when [59]

(29) Γ ( x , t ) = h 1 exp ( l ( r 3 ( λ t + x ) + r 4 ) ) + exp ( − l ( r 1 ( λ t + x ) + r 2 ) ) + h 2 cos ( l ( r 5 ( λ t + x ) + r 6 ) ) + h 3 cosh ( l ( r 7 ( λ t + x ) + r 8 ) ) + r 9 ,

where r i (for 1 ≤ i ≤ 9 ), h j (for 1 ≤ j ≤ 3 ), and l ≠ 0 are arbitrary constants, and the parameter λ ≠ 0 is the wave speed. Substitute Eq. (29) and its first four partial derivatives in Eq. (15). Simplifying algebraically and proceeding as previously, we derive two sets of parameter values in this case.

Set 1. r 1 = − λ 2 − a 2 2 d l , r 3 = λ 2 − a 2 2 d l , r 5 = i λ 2 − a 2 2 d l , r 7 = − λ 2 − a 2 2 d l , r 9 = 0 .

Inserting them in Eq. (29) and then the outcome in Eq. (12), we have

(30) L 1 P C K ( x , t ) = 6 d λ 2 − a 2 h 1 exp λ 2 − a 2 ( λ t + x ) 2 d + l ( r 2 + r 4 ) + e λ 2 − a 2 ( λ t + x ) 2 d + A 2 e l r 2 b h 1 exp λ 2 − a 2 ( λ t + x ) 2 d + l ( r 2 + r 4 ) + e λ 2 − a 2 ( λ t + x ) 2 d + A 3 e l r 2 ,

where

(31) A 2 = h 3 sinh λ 2 − a 2 ( λ t + x ) 2 d − l r 8 − i h 2 sin l r 6 + i λ 2 − a 2 ( λ t + x ) 2 d l ,

(32) A 3 = h 3 cosh λ 2 − a 2 ( λ t + x ) 2 d − l r 8 + h 2 cosh λ 2 − a 2 ( λ t + x ) 2 d − i l r 6 ,

and a , b , d ≠ 0 are free parameters. Using the solution L 1 P C K ( x , t ) from Eq. (30), Figure 8(a) is obtained by selecting the parameter values r 2 = 2.3 , r 4 = 1.8 , r 6 = 3.1 , r 8 = 3.1 , h 1 = 7.1 , h 2 = 9.4 , h 3 = 1.7 , a = 1.1 , b = 4.32 , d = 0.3 , l = 0.9 , λ = 0.2 . As expected, its corresponding contour plot is shown in Figure 8(b).

Figure 8 The 3D (a) and contour diagram (b) representing the PCK-type wave.
Figure 8

The 3D (a) and contour diagram (b) representing the PCK-type wave.

Set 2. h 1 = 0 , r 1 = λ 2 − a 2 2 d l , r 5 = i λ 2 − a 2 2 d l , r 7 = λ 2 − a 2 2 d l , r 9 = 0 .

Plugging these parameter values in Eq. (29) and then the result in Eq. (12), we readily reach

(33) L 2 PCK ( x , t ) = 6 d λ 2 − a 2 A 2 − e − λ 2 − a 2 ( λ t + x ) 2 d − l r 2 b e − λ 2 − a 2 ( λ t + x ) 2 d − l r 2 + A 3 .

The solution L 2 PCK ( x , t ) corresponding to Eq. (33) is represented in Figure 9(a) for the parameter values r 2 = 6.3 , r 6 = 1.4 , r 8 = 4.8 , h 2 = 9.1 , h 3 = 4.3 , a = 2.8 , b = 3.16 , d = 0.5 , l = 2.1 , λ = 2.5 . The corresponding contour plot is depicted in Figure 9(b).

Figure 9 The 3D (a) and contour diagram (b) representing the PCK-type wave.
Figure 9

The 3D (a) and contour diagram (b) representing the PCK-type wave.

3.8 Interactions of exponential, trigonometric, and tanh (ETH)

This wave structure is given by [60]

(34) Γ ( x , t ) = h 1 exp ( r 1 ( λ t + x ) + r 2 ) + h 2 exp ( − ( r 1 ( λ t + x ) + r 2 ) ) + h 3 sin ( r 3 ( λ t + x ) + r 4 ) + h 4 cos ( r 5 ( λ t + x ) + r 6 ) + h 5 tanh ( r 7 ( λ t + x ) + r 8 ) ,

where r i (for 1 ≤ i ≤ 8 ) and h j (for 1 ≤ j ≤ 5 ) are arbitrary constants, and the parameter λ ≠ 0 is the wave speed. Substitute Eq. (34) and its first four partial derivatives in Eq. (15), and proceed as before. In this case, we derive a single set of parameter values.

Set 1. h 2 = 0 , r 3 = a 2 − λ 2 d , r 5 = 0 , r 7 = − i λ 2 − a 2 2 2 d .

By replacing them in Eq. (34) and the outcome in Eq. (12), we obtain

(35) L 1 ETH ( x , t ) = 3 d 4 h 3 ( a − λ ) ( a + λ ) cos ( a − λ ) ( a + λ ) ( λ t + x ) d + r 4 − A 4 b − i h 5 tan λ 2 − a 2 ( λ t + x ) 2 2 d + i r 8 + A 5 + h 4 cos ( r 6 ) ,

where

(36) A 4 = 4 d h 1 r 1 e r 1 ( λ t + x ) + r 2 + i 2 h 5 λ 2 − a 2 × sec 2 λ 2 − a 2 ( λ t + x ) 2 2 d + i r 8 ,

(37) A 5 = h 3 sin ( a − λ ) ( a + λ ) ( λ t + x ) d + r 4 + h 1 e r 1 ( λ t + x ) + r 2 ,

and a , b , d ≠ 0 are free parameters. The solution L 1 ETH ( x , t ) given by Eq. (35) is shown in Figure 10(a) for the parameter values r 1 = 2.3 , r 2 = 2.3 , r 4 = 1.8 , r 6 = 3.1 , r 8 = 3.1 , h 1 = 7.1 , h 3 = 9.4 , h 4 = 1.7 , h 5 = 1.7 , a = 1.1 , b = 4.32 , d = 0.3 , λ = 5.2 . Figure 10(b) shows the related contour plot.

Figure 10 The 3D (a) and contour diagram (b) representing the ETH-type wave.
Figure 10

The 3D (a) and contour diagram (b) representing the ETH-type wave.

3.9 Mixed waves (MI)

This pattern of waves is derived by letting [61]

(38) Γ ( x , t ) = h 1 exp ( l ( r 1 ( λ t + x ) + r 2 ) ) + h 2 exp ( − l ( r 1 ( λ t + x ) + r 2 ) ) + h 3 sin ( l ( r 3 ( λ t + x ) + r 4 ) ) + h 4 sinh ( l ( r 5 ( λ t + x ) + r 6 ) ) ,

where r i (for 1 ≤ i ≤ 6 ), h j (for 1 ≤ j ≤ 4 ), and l ≠ 0 are arbitrary constants, and the parameter λ ≠ 0 is the wave speed. Proceeding as before, we obtain two sets of parameter values in this case.

Set 1. r 1 = λ 2 − a 2 2 d l , r 3 = i λ 2 − a 2 2 d l , r 5 = λ 2 − a 2 2 d l .

In this case, we derive the solution

(39) L 1 MI ( x , t ) = 6 d λ 2 − a 2 A 6 e λ 2 − a 2 ( λ t + x ) 2 d + l r 2 + h 1 e λ 2 − a 2 ( λ t + x ) d + 2 l r 2 − h 2 b A 7 e λ 2 − a 2 ( λ t + x ) 2 d + l r 2 + h 1 e λ 2 − a 2 ( λ t + x ) d + 2 l r 2 + h 2 ,

where

(40) A 6 = h 4 cosh λ 2 − a 2 ( λ t + x ) 2 d + l r 6 + i h 3 cosh λ 2 − a 2 ( λ t + x ) 2 d − i l r 4 ,

(41) A 7 = h 4 sinh λ 2 − a 2 ( λ t + x ) 2 d + l r 6 + h 3 sin l r 4 + i λ 2 − a 2 ( λ t + x ) 2 d l ,

and a , b , d ≠ 0 are free parameters. The solution L 1 MI ( x , t ) given by Eq. (39) is depicted in Figure 11(a) by selecting the parameter values r 2 = 7.7 , r 4 = 1.8 , r 6 = 3.1 , h 1 = 7.1 , h 2 = 9.4 , h 3 = 1.7 , h 4 = 4.5 , a = 2.1 , b = 1.26 , d = 1.3 , l = 0.9 , λ = 0.8 . In turn, Figure 11(b) shows the corresponding contour plot.

Figure 11 The 3D (a) and contour diagram (b) representing the MI-type wave.
Figure 11

The 3D (a) and contour diagram (b) representing the MI-type wave.

Set 2. h 2 = 0 , r 1 = λ 2 − a 2 2 d l , r 3 = i λ 2 − a 2 2 d l , r 5 = − λ 2 − a 2 2 d l .

In this case, we obtain the solution

(42) L 2 MI ( x , t ) = 6 d λ 2 − a 2 A 8 − h 4 cosh λ 2 − a 2 ( λ t + x ) 2 d − l r 6 b A 9 − h 4 sinh λ 2 − a 2 ( λ t + x ) 2 d − l r 6 ,

where

(43) A 8 = h 1 e λ 2 − a 2 ( λ t + x ) 2 d + l r 2 + i h 3 cosh λ 2 − a 2 ( λ t + x ) 2 d − i l r 4 ,

(44) A 9 = h 1 e λ 2 − a 2 ( λ t + x ) 2 d + l r 2 + h 3 sin l r 4 + i λ 2 − a 2 ( λ t + x ) 2 d l ,

and a , b , d ≠ 0 are free parameters. The solution L 1 MI ( x , t ) corresponding to Eq. (42) is shown in Figure 12(a) for the parameter values r 2 = 1.4 , r 4 = 0.6 , r 6 = 4.6 , h 1 = 4.8 , h 3 = 5.7 , h 4 = 7.8 , a = 8.4 , b = 2.83 , d = 4.2 , l = 3.1 , λ = 0.5 . The corresponding contour plot is shown in Figure 12(b).

Figure 12 The 3D (a) and contour diagram (b) representing the MI-type wave.
Figure 12

The 3D (a) and contour diagram (b) representing the MI-type wave.

3.10 Homoclinic breather (HB)

This pattern of waves is reached when [61]

(45) Γ ( x , t ) = h 1 exp ( l ( r 1 ( λ t + x ) + r 2 ) ) + h 2 exp ( − l ( r 3 ( λ t + x ) + r 4 ) ) + h 3 cos ( l ( r 5 ( λ t + x ) + r 6 ) ) ,

where r i (for 1 ≤ i ≤ 6 ), h j (for 1 ≤ j ≤ 3 ), and l ≠ 0 are arbitrary constants, and the parameter λ ≠ 0 is the wave speed, as usual. Eq. (38) and its first four partial derivatives were substituted in Eq. (15) and we obtain a simplified expression. We collecting similar terms and equated their coefficients to zero, resulting in the following sets of constant values.

Set 1. r 1 = i r 5 , r 3 = − i r 5 , l = i λ 2 − a 2 2 d r 5 .

Inserting them in Eq. (45) and then the outcome in Eq. (12), we obtain

(46) L 1 HB ( x , t ) = − 6 d λ 2 − a 2 − A 10 h 3 sinh λ 2 − a 2 ( r 5 ( λ t + x ) + r 6 ) 2 d r 5 + h 1 e i ( r 2 + r 4 ) λ 2 − a 2 2 d r 5 + h 2 b A 10 h 3 cosh λ 2 − a 2 ( r 5 ( λ t + x ) + r 6 ) 2 d r 5 + h 1 e i ( r 2 + r 4 ) λ 2 − a 2 2 d r 5 + h 2 ,

where

(47) A 10 = exp λ 2 − a 2 ( r 5 ( λ t + x ) + i r 4 ) 2 d r 5 ,

and a , b , d ≠ 0 are free parameters. Based on the solution L 1 Hom ( x , t ) given by Eq. (46), Figure 13(a) is obtained by selecting the parameter values r 2 = 9.2 , r 4 = 0.2 , r 5 = 0.3 , r 6 = 0.4 , h 1 = 3.5 , h 2 = 1.7 , h 3 = 3.6 , a = 2.0 , b = 3.16 , d = 1.3 , λ = 5.4 . The corresponding contour plot is given by Figure 13(b).

Figure 13 The 3D (a) and contour diagram (b) representing the HB-type wave.
Figure 13

The 3D (a) and contour diagram (b) representing the HB-type wave.

Set 2. h 1 = 0 , r 3 = − i r 5 , l = i λ 2 − a 2 2 d r 5 .

In this case, we derive the exact solution

(48) L 2 HB ( x , t ) = − 6 d λ 2 − a 2 h 2 − h 3 exp λ 2 − a 2 ( r 5 ( λ t + x ) + i r 4 ) 2 d r 5 sinh λ 2 − a 2 ( r 5 ( λ t + x ) + r 6 ) 2 d r 5 b h 2 + h 3 exp λ 2 − a 2 ( r 5 ( λ t + x ) + i r 4 ) 2 d r 5 cosh λ 2 − a 2 ( r 5 ( λ t + x ) + r 6 ) 2 d r 5 ,

where a , b , d ≠ 0 are free parameters. The solution L 2 Hom ( x , t ) given by Eq. (48) is shown in Figure 14(a) for the parameter values r 4 = 0.2 , r 5 = 0.3 , r 6 = 0.4 , h 2 = 1.7 , h 3 = 4.2 , a = 5.5 , b = 3.7 , d = 2.5 , λ = 9.9 . The respective contour plot is given by Figure 14(b).

Figure 14 The 3D (a) and contour diagram (b) representing the HB-type wave.
Figure 14

The 3D (a) and contour diagram (b) representing the HB-type wave.

4 Discussion

Beforehand, it is important to point out that the functions obtained in Section 3 were substituted into the mathematical model using Mathematica software. The results confirm that, indeed, the functions are solutions of our mathematical model. The 3D graphs and their corresponding contour plots for the solutions derived in the previous section were obtained using Mathematica. The graphs are displayed in Figures 1–14 in order to help in visualizing the characteristic of the solutions. Figure 1(a) shows an anti-kink wave obtained from an interaction between two exponential wave functions. Figure 2(a) shows an M-shaped wave with single kink obtained from an interaction between one M-shaped and one kink wave functions. In Figure 3(a), an M-shaped wave with double kinks is portrayed arising from the wave function of M-shaped and two kinks interactions. Figure 4(a) depicts multiple PL waves obtained from the interaction of multi-lump and periodic wave functions. In turn, Figure 5(a) shows a solitary wave derived from a multi wave function, and Figure 6(a) show multiple PL waves obtained from multi wave functions. Figure 7(a) shows periodic bright lump waves obtained from PL wave structures. Figures 8(a) and 9(a) show PCK waves. In Figure 10(a), we have three dark lump strip waves obtained from the interactions of ETH functions. In Figures 11(a) and 12(a), we obtained solitary waves solutions derived from mixed wave functions. In Figure 13(a), we witnessed three bright breather waves from the HB function, whereas in Figure 14(a) we obtained a single layer of bright breather from the same HB wave function. In each case, Figures 1(b) through 14(b) show the corresponding contour plots. These plots assist in the visualization of the wave profiles by displaying changes in both shape and intensity in different spatial areas at different times. Moreover, concrete physical interpretations of these solutions could be provided by scientists in the context of atomic physics.

These wave phenomena obtained from an atomic chain model each represent distinct physical behaviors. The anti-kink wave transitions between two states in a smooth, sharp manner, but the M-shaped wave with a single kink has a peak structure and a sharp transition point, suggesting localized energy concentration. The M-shaped wave with multiple kinks increases complexity with two abrupt transitions, resulting in steeper atomic movement. Periodic multiple lump waves result from multi-lump and periodic wave interactions, with recurrent localized disruptions indicating oscillating energy. Solitary waves travel without changing form and indicate steady energy transmission across the chain, whereas multiple PL waves cause these localized disruptions to repeat on a regular basis. Periodic bright lump waves identify places with significant energy concentration. PCK waves are the outcome of multidimensional kink interactions that generate crossing wave structures. Dark lump strip waves have a lesser intensity resulting in localized amplitude depressions. Solitary wave solutions formed from mixed wave functions exhibit stable, non-dispersive wave forms. Bright breather waves and their single-layer counterparts show localized, periodic energy variations as they surface in and out over time. These waves together demonstrate complicated energy transfer and atomic displacement dynamics in nonlinear atomic systems.

5 Conclusion

This work investigated a diverse spectrum of solitary and soliton solutions resulting from various symmetry wave functions in a nonlinear atomic chain model. The Hirota bilinear transformation approach was used to unveil a wide range of complicated wave phenomena, including anti-kink waves, M-shaped interacting with kink waves, and PL waves. The combination of exponential, trigonometric, and hyperbolic functions has demonstrated its efficiency in revealing various wave forms such as bright breather waves, periodic bright lump waves, and dark lump strip waves, all of which are required to understand localized energy dynamics in these systems. The graphical illustrations, which included 3D and contour plots, contributed to a better understanding of waveform behavior by showing how solitary and periodic wave structures change spatially and temporally. These findings may shed light on the mechanics of energy transfer and atomic displacement in nonlinear atomic chains, highlighting both stability and the possibility for complex wave interactions. Finally, this study not only adds to the theoretical framework of wave dynamics in nonlinear systems, but it also has potential applications in fields like condensed matter physics and material science, where understanding wave interactions in atomic structures is critical for developing new technologies. The discoveries pave the way for further research into more complicated wave phenomena and their practical consequences in advanced scientific and industrial applications.

As a follow-up to this work, various avenues of research are still open. For instance, the problem of investigating the physical model which considers long-range interactions is an interesting topic of research. In that case, the continuum-limit approximation leads to a fully fractional partial differential equation. In that case, the fractional derivatives appear in the spatial variables. For such system, the derivation of solitary wave solutions and soliton-like solutions is a task that merits attention. In addition, it is worthwhile to point out that the mathematical model studied in the present work is a one-dimensional system in space. This model corresponds to the continuum-limit of a system of particles distributed on a straight line. The case when the atoms are distributed on a plane or inside a three-dimensional volume was not tackled in the present investigation. This is a limitation of our work and, at the same time, a potential direction for future work in the area. Moreover, the elucidation of real-life applications of the results obtained in the present work is also an important problem which needs future attention. Finally, as one of the reviewers pointed out, the exact solutions obtained in this work possess a lot of parameters, and the choice of parameter values to guarantee that a particular solution has a specific wave pattern is certainly a complex task. Such analysis could be the topic of a future report in the context of nonlinear physics.

Acknowledgments

The authors wish to thank the anonymous reviewers and the editor in charge of handling this manuscript for their comments and criticisms. Their suggestions helped in improving substantially the quality of the present report.

  1. Funding information: The authors wish to thank the financial support of the University of Guadalajara, Mexico, through the program PROSNI.

  2. Author contributions: Conceptualization: N.A. and J.E.M.D.; methodology: B.C., M.Z.B., N.A., J.E.M.D., and S.M.; software: B.C., M.Z.B., N.A., J.E.M.D., and S.M.; validation: B.C., M.Z.B., N.A., J.E.M.D., and S.M.; formal analysis: B.C., M.Z.B., N.A., J.E.M.D., and S.M.; investigation: B.C., M.Z.B., N.A., J.E.M.D., and S.M.; resources, B.C., M.Z.B., N.A., J.E.M.D., and S.M.; data curation, B.C., M.Z.B., N.A., J.E.M.D., and S.M.; writing - original draft preparation, B.C., M.Z.B., N.A., J.E.M.D., and S.M.; writing - review and editing: B.C., M.Z.B., N.A., J.E.M.D. and S.M.; visualization: B.C., M.Z.B., N.A., J.E.M.D., and S.M.; supervision: N.A. and J.E.M.D.; project administration: 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: The authors state no conflict of interest.

  4. Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2025-01-21
Revised: 2025-02-24
Accepted: 2025-03-29
Published Online: 2025-05-09

© 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-2025-0150
Keywords for this article
nonlinear atomic chain model; solitary wave solutions; soliton solutions; Hirota bilinear transformation method
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Single-step fabrication of Ag2S/poly-2-mercaptoaniline nanoribbon photocathodes for green hydrogen generation from artificial and natural red-sea water
  3. Abundant new interaction solutions and nonlinear dynamics for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation
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  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. Review Article
  110. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  111. Erratum
  112. Erratum to “On Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model (Open Physics 2021;19:679–682)”
  113. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  114. Possible explanation for the neutron lifetime puzzle
  115. Special Issue on Nanomaterial utilization and structural optimization - Part III
  116. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  117. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  118. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  119. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  120. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
  121. Novel exact solitons to the fractional modified mixed-Korteweg--de Vries model with a stability analysis
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