Startseite Naturwissenschaften Establishing breather and N-soliton solutions for conformable Klein–Gordon equation
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Establishing breather and N-soliton solutions for conformable Klein–Gordon equation

  • Muhammad Bilal , Javed Iqbal , Rashid Ali EMAIL logo , Fuad A. Awwad und Emad A. A. Ismail
Veröffentlicht/Copyright: 20. Juni 2024
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Open Physics
Aus der Zeitschrift Open Physics Band 22 Heft 1

Abstract

This article develops and investigates the behavior of soliton solutions for the spatiotemporal conformable Klein–Gordon equation (CKGE), a well-known mathematical physics model that accounts for spinless pion and de-Broglie waves. To accomplish this task, we deploy an effective analytical method, namely, the modified extended direct algebraic method (mEDAM). This method first develops a nonlinear ordinary differential equation (NODE) through the use of a wave transformation. With the help of generalized Riccati NODE and balancing nonlinearity with the highest derivative term, it then assumes a finite series-form solution for the resulting NODE, from which four clusters of soliton solutions – generalized rational, trigonometric, exponential, and hyperbolic functions – are derived. Using contour and three-dimensional visuals, the behaviors of the soliton solutions – which are prominently described as dark kink, bright kink, breather, and other N -soliton waves – are examined and analyzed. These results have applications in solid-state physics, nonlinear optics, quantum field theory, and a more thorough knowledge of the dynamics of the CKGE.

Keywords: fractional partial differential equations; homogenous balancing principle; modified extended direct algebraic method; wave transformation; breather waves; generalized Riccati NODE

1 Introduction

The prevalence of nonlinearity across the world emphasizes the necessity of creating nonlinear models, especially those that include fractional partial differential equations (FPDEs) and partial differential equations (PDEs) [1–4]. The wide range of applications in fluid dynamics, acoustics, image processing, vibration, biology, chemistry, physics, and control has attracted a lot of researchers to study nonlinear FPDEs [5–7]. Because of the great potential applications of these nonlinear FPDEs in various fields, researchers have invested a great deal of time and energy in finding analytical and numerical solutions [8–12]. Researchers have created a number of reliable and effective approaches to find and analyze exact solutions [13–17]. These methodologies encompass the first integral method [18], Laplace Adomian decomposition method [19,20], homotopy analysis method [21], natural transform decomposition method [22], modified simple equation method [23], ( G ′ ∕ G )-expansion method [24–26], extended direct algebraic method (mEDAM), [27,28] and numerous others.

Expanding on current methods to address nonlinear FPDEs (NFPDEs) is still a compelling and important area of research. Many NFPDEs, such as the impulsive fractional differential equations [29], the space-time fractional advection-dispersion equation [30–32], the fractional generalized Burgers’ equation [33], the fractional heat and mass-transport equation [34], and others, have been studied and solved as a result of the efforts of numerous researchers. Motivated by the ongoing research on addressing NFPDEs, this study aims to address conformable Klein–Gordon equation (CKGE) with mEDAM. The fractional generalization of the Klein–Gordon equation (KGE) is known as the CKGE. It substitutes fractional order conformable derivatives for the integer ordered derivatives in KGE, a relativistic wave equation related to Schrödinger equation. In mathematical physics, particularly in relativistic quantum mechanics, the spinless pion and de-Broglie waves are well explained by the KGE, a well-known model that was first developed in 1926 by Klein and Gordon as a relativistic equation for the function of a wave of an individual particle with zero spins. Numerous scientific domains, including quantum field theory, solid-state physics, and nonlinear optics, are found in this model. The equation has shown tremendous interest in the fields of condensed matter physics, solitons in a collisionless plasma, nonlinear wave equations, and recurrence of initial states. In a mathematical model, this equation is used in many scientific fields, including quantum field theory, nonlinear optics, and solid-state physics. This model is stated as follows [31]:

(1) D t δ ( D t δ w ( x , t ) ) − D x δ ( D x δ w ( x , t ) ) + σ w ( x , t ) − ρ w 2 ( x , t ) = 0 ,

where 0 < δ ≤ 1 , w ( x , t ) represents the quantum state of the particle and is a harmonic function, σ = ( m c ) 2 , where m is the mass, c is the speed of the spinless particle, and ρ is the nonzero coefficient of nonlinear term. The last two terms describe an appropriate nonlinear function that shows the change in potential energy [35]. A series of mathematical equations, including the Landau-Ginzburg-Higgs, PHI-4 equations, Duffing, and sine-Gordon, can be obtained by choosing various representations of this nonlinear function. These equations may be employed to simulate a range of physical events, from wave propagation to quantum mechanical phenomena [36]. Renowned scientists such as Klein, Schrödinger, and Fock were crucial for discovering the connection among the general relativity theories with certain KGE’s earlier renditions, in keeping with Kragh’s account of the KGE’s roots in the study by Kragh [37]. Some of the most significant equations is this one, which serves as the basis for the officially accepted nonlinear Schrödinger equation [38]. Galehouse [39] used the proper gauge changes to derive the KGE spatially. For the more generalized form of the KGE, Schechter also provides arguments in favor of the theory of scattering hypothesis [40]. The traditional method of generating a comparable equation in a finitely normed Hilbert space containing the first-order time is followed to create the analog theory of the KGE [41]. In this work, Weder also provided justification for the existence and coherence of the wave operators, the invariant principle, and the interconnected links. Strong findings, reminiscent of those given in the Schrödinger instance, can be obtained by adopting similar eigenfunction extensions to a specific field condition to broaden the spectral and dispersion theory of the KGE [42]. Tsukanov examines the mobility of a KGE in an alternative field [43].

The fractional derivative operators D t δ and D t δ in (1) are conformable fractional derivatives (CFDs), which are defined in the upcoming section. CFDs are a helpful tool for explaining the physical aspects of events that display noninteger behaviors, including memory-related events that also exhibit nonlocal as well as local activity. In mathematics, traditional derivatives stand for limited behavior, where a place’s change in magnitude is only dependent on the environment around it. Nevertheless, a lot of real-world systems exhibit nonlocal behavior, in which the system’s past or remote locations affect its present state at any particular moment. Fractional derivatives, or CFDs, are an extension of calculus used to explain these kinds of circumstances. Fractional interactions, for example, are properties of elastic components within the domain of substances that ascertain stresses or tension at a region by virtue of the behavior of the material during deformation and its current environment. This memory effect is accurately captured by recording sessions, which take into account the actions of the entire system throughout an uninterrupted spectrum of historical data. Furthermore, nonlocal structures are frequently observed in the diffusion processes that molecules go through. This causes problems with fractional diffusing because the molecules’ dispersion over a nonlocal region determines the degree of intensity fluctuations at a certain site. To accurately simulate these kinds of occurrences, CFDs offer a mathematical tool that bridges the gap between both local and nonlocal behavior and takes into consideration memory-linked features that are essential to the system’s operation.

Before this research, several researchers have already used various analytical and numerical methods to investigate this equation in integer and fractional orders. For instance, the homotopy perturbation method was employed by Golmankhaneh et al. [44] to study fractional (FKGE). Khan et al. in [25] utilized the ( G ′ ∕ G )-expansion method to construct families of travelling wave solutions for FKGE. Gepreel and Mohamed recently established the analytical approximate solution of the FKGE [45]. By using the homotopy analysis method, Jafari et al. [46] investigated the Cahn-Hilliard and KGE using the fractional sub-equation method. Moreover, Ran and Zhang used the compact difference method for the space FKGE in [47]. The modified extended tanh-method [48], the ( w ∕ g )-expansion [49], the Riccati expansion [50–52], and other techniques [53] are also used to look for exact and numerical solutions to the FKGE.

Nevertheless, the main goal of this study is to construct and examine new families of soliton solutions for CKGE using the enhanced mEDAM. Applications of these findings include quantum field theory, solid-state physics, nonlinear optics, and a deeper understanding of the dynamics of the CKGE. The proposed mEDAM is one of the most straightforward, important, and efficient algebraic procedures. Under the application of wave transformation and assumption of a series form solution, the strategic mEDAM transforms the CKGE into a system of nonlinear algebraic equations. Numerous soliton solutions in the form of generalized rational, trigonometric, exponential, and hyperbolic functions are produced when the resulting problem is solved using the Maple tool. From an academic perspective, soliton solutions for NFPDEs remain significant because they offer more depth and granularity than conventional solutions [54–57]. They are valuable in many technical and scientific fields due to their inherent stability and longevity. They provide effective information transfer and extensive concordance retention for nonlinear systems. To put it succinctly, the new findings of this study demonstrate the inventive character of our research by offering a unique and methodical discovery of numerous new soliton solutions. Soliton research is interested in studying nonlinear FPDEs that occur in quantum field theory, fluid dynamics, and optics. These soliton solutions offer a more profound understanding of the fundamental phenomena of CKGE in the associated scientific domains.

The rest of the article is organized as follows: Section 2 describes the CFD and proposed method’s methodology, Section 3 presents soliton solutions for the targeted biological population models, while Section 4 presents graphs of some soliton solutions and a discussion of our findings. Finally, a brief conclusion has been provided.

2 Methodology and materials

2.1 The definition of CFD

Several fractional derivative operators such as the Caputo operator, Riemann-Liouville operator, Atangana-Baleanu operator, Caputo-Fabrizio operator, CFD operator, and many more have been introduced by different mathematicians in literature [58–68]. Among these fractional derivative operators, the CFD operator is preferred by academics due to its predominant applications over other derivative operators. For instance, employing Khalil’s CFD and a matrix of Hessian, Lavín-Delgado et al. presented an innovative edge recognition technique that lowers noise and maintains image outlines regardless of dull contrasting situations [69]. Their method improved clarity of vision in edge identification and has potential uses in health care imaging for recognizing diseases, including cervical cancer along with medial cranial arterial ruptured arteries, hence enhancing clinical surveillance and the precision of diagnosis. One another advantage of CFD is that, unlike other derivative operators, it satisfies all derivative properties, particularly the chain rule, which is essential for our method for creating soliton solutions. By taking advantage of these beneficial properties of CFDs over alternative fractional derivative operators, explicit solutions for FPDEs can be derived. Notably, the soliton solutions of Eq. (1) cannot be obtained using alternative fractional derivative formulations because they violate the chain rule [70,71]. As a result, CFDs were introduced into Eq. (1). The study by Sarikaya et al. [72] defines the CFD operator of order α as follows:

(2) D ψ α w ( ψ ) = lim γ → 0 w ( γ ψ 1 − α + ψ ) − w ( ψ ) γ , α ∈ ( 0 , 1 ] .

In this investigation, the following properties of this derivative are utilized:

(3) D ψ α ψ r = r ψ r − α ,

(4) D ψ α ( r 1 ϖ ( ψ ) ± r 2 π ( ψ ) ) = r 1 D ψ α ( ϖ ( ψ ) ) ± r 2 D ψ α ( π ( ψ ) ) ,

(5) D ψ α χ [ τ ( ψ ) ] = χ τ ′ ( τ ( ψ ) ) D ψ α τ ( ψ ) ,

where ϖ ( ψ ) , π ( ψ ) , χ ( ψ ) , and τ ( ψ ) are arbitrary differentiable functions, whereas r , r 1 , and r 2 signify constants.

2.2 The working procedure of mEDAM

The mEDAM is an efficient technique, which is utilized by many researchers to construct travelling wave and soliton solutions for nonlinear FPDEs [73]. The operational procedure of this proposed method is explained in this section [54]. Examine the FPDE of the structure:

(6) P ( w , ∂ t α w , ∂ z 1 β w , ∂ z 2 γ w , w ∂ z 1 β w , … ) = 0 , 0 < α , β , γ ≤ 1 ,

where t , z 1 , z 2 , z 3 , … , z r , and w are functions. The steps to solve Eq. (6) are as follows:

  1. The variable transformation of the form w ( z 1 , z 2 , z 3 , … , z r ) = W ( ψ ) , where ψ is defined in a number of ways, is first applied. This transformation converts (6) into a nonlinear ODE with the following form:

    (7) Q ( W , W ′ W , W ′ , … ) = 0 ,

    where W in (7) has derivatives concerning ψ . To find the integration constant(s), (7) may be integrated one or more times.

  2. Next, we assume that (7) has the following solution:

    (8) W ( ψ ) = ∑ l = − ς ς d l ( Z ( ψ ) ) l ,

    where d l ( l = − ς , … , 0 , … , ς ) are constants to be determined, and Z ( ψ ) is the general solution of the following ODE:

    (9) Z ′ ( ψ ) = ln ( Ω ) ( η + ψ Z ( ψ ) + ν ( Z ( ψ ) ) 2 ) ,

    where Ω ≠ 0 , 1 and the constants η , μ , and ν are used.

  3. The positive integer ς presented in (8) is obtained by establishing the homogeneous balance between the biggest nonlinear term and the highest order derivative in (7).

  4. Next, we integrate (7) to obtain the equation that results from (8), and we put all of the terms of Z ( ψ ) in the same order. Then, all of the coefficients of the ensuing polynomial are set to zero, yielding an algebraic system of equations for d l ( l = − ς , … , 0 , … , ς ) and extra parameters.

  5. We utilize MAPLE to solve this set of algebraic problems.

  6. After figuring out the unknown values, they are added to (10) along with the Z ( ψ ) (solution of Eq. (9)) to yield the analytical soliton solutions to (6). The following families of solutions can be generated by using the general solution of (9).

Family 1: When M < 0 and ν ≠ 0 :

Z 1 ( ψ ) = − μ 2 ν + − M tan Ω 1 2 − M ψ 2 ν , Z 2 ( ψ ) = − μ 2 ν − − M cot Ω 1 2 − M ψ 2 ν , Z 3 ( ψ ) = − μ 2 ν + − M ( tan Ω ( − M ψ ) ± ( sec Ω ( − M ψ ) ) ) 2 ν , Z 4 ( ψ ) = − μ 2 ν − − M ( cot Ω ( − M ψ ) ± ( csc Ω ( − M ψ ) ) ) 2 ν ,

and

Z 5 ( ψ ) = − μ 2 ν + − M tan Ω 1 4 − M ψ − cot Ω 1 4 − M ψ 4 ν .

Family 2: When M > 0 and ν ≠ 0 :

Z 6 ( ψ ) = − μ 2 ν − M tanh Ω 1 2 M ψ 2 ν ,

Z 7 ( ψ ) = − μ 2 ν − M coth Ω 1 2 M ψ 2 ν ,

Z 8 ( ψ ) = − μ 2 ν − M ( tanh Ω ( M ψ ) ± i ( sech Ω ( M ψ ) ) ) 2 ν ,

Z 9 ( ψ ) = − μ 2 ν − M ( coth Ω ( M ψ ) ± ( csch Ω ( M ψ ) ) ) 2 ν ,

and

Z 10 ( ψ ) = − μ 2 ν − M tanh Ω 1 4 M ψ − coth Ω 1 4 M ψ 4 ν .

Family 3: When η ν > 0 and μ = 0 :

Z 11 ( ψ ) = η ν tan Ω ( η ν ψ ) , Z 12 ( ψ ) = − η ν cot Ω ( η ν ψ ) , Z 13 ( ψ ) = η ν ( tan Ω ( 2 η ν ψ ) ± ( sec Ω ( 2 η ν ψ ) ) ) , Z 14 ( ψ ) = − η ν ( cot Ω ( 2 η ν ψ ) ± ( csc Ω ( 2 η ν ψ ) ) ) ,

and

Z 15 ( ψ ) = 1 2 η ν tan Ω 1 2 η ν ψ − cot Ω 1 2 η ν ψ .

Family 4: When η ν > 0 and μ = 0 , then

Z 16 ( ψ ) = − − η ν tanh Ω ( − η ν ψ ) , Z 17 ( ψ ) = − − η ν coth Ω ( − η ν ψ ) , Z 18 ( ψ ) = − − η ν ( tanh Ω ( 2 − η ν ψ ) ± ( i sech Ω ( 2 − η ν ψ ) ) ) , Z 19 ( ψ ) = − − η ν ( coth Ω ( 2 − η ν ψ ) ± ( csch Ω ( 2 − η ν ψ ) ) ) ,

and

Z 20 ( ψ ) = − 1 2 − η ν tanh Ω 1 2 − η ν ψ + coth Ω 1 2 − η ν ψ .

Family 5: When ν = η and μ = 0 :

Z 21 ( ψ ) = tan Ω ( η ψ ) , Z 22 ( ψ ) = − cot Ω ( η ψ ) , Z 23 ( ψ ) = tan Ω ( 2 η ψ ) ± ( sec Ω ( 2 η ψ ) ) , Z 24 ( ψ ) = − cot Ω ( 2 η ψ ) ± ( csc Ω ( 2 η ψ ) ) ,

and

Z 25 ( ψ ) = 1 2 tan Ω 1 2 η ψ − 1 2 cot Ω 1 2 η ψ ,

Family 6: When ν = − η and μ = 0 :

Z 26 ( ψ ) = − tanh Ω ( η ψ ) , Z 27 ( ψ ) = − coth Ω ( η ψ ) , Z 28 ( ψ ) = − tanh Ω ( η a ψ ) ± ( i sech Ω ( 2 η ψ ) ) , Z 29 ( ψ ) = − coth Ω ( 2 η ψ ) ± ( csch Ω ( 2 η ψ ) ) ,

and

Z 30 ( ψ ) = − 1 2 tanh Ω 1 2 η ψ − 1 2 coth Ω 1 2 η ψ .

Family 7: When M = 0 :

Z 31 ( ψ ) = − 2 η ( μ ψ ln ( Ω ) + 2 ) μ 2 ψ ln ( Ω ) .

Family 8: When μ = τ , η = n τ ( n ≠ 0 ) and ν = 0 :

Z 32 ( ψ ) = Ω τ ψ − n .

Family 9: When μ = ν = 0 :

Z 33 ( ψ ) = η ψ ln ( Ω ) .

Family 10: When μ = η = 0 :

Z 34 ( ψ ) = − 1 ν ψ ln ( Ω ) .

Family 11: When η = 0 , μ ≠ 0 and ν ≠ 0 :

Z 35 ( ψ ) = − μ ν ( cosh Ω ( μ ψ ) − sinh Ω ( μ ψ ) + 1 )

and

Z 36 ( ψ ) = − μ ( cosh Ω ( μ ψ ) + sinh Ω ( μ ψ ) ) ν ( cosh Ω ( μ ψ ) + sinh Ω ( μ ψ ) + 1 ) .

Family 12: When μ = τ , ν = n τ ( n ≠ 0 ) , and η = 0 :

Z 37 ( ψ ) = Ω τ ψ 1 − n Ω τ ψ ,

where M = μ 2 − 4 η ν , while the generalized hyperbolic and trigonometric functions are described as follows:

cos Ω ( ψ ) = Ω i ψ + Ω − i ψ 2 , sin Ω ( ψ ) = Ω i ψ − Ω − i ψ 2 i , cot Ω ( ψ ) = cos Ω ( ψ ) sin Ω ( ψ ) , tan Ω ( ψ ) = sin Ω ( ψ ) cos Ω ( ψ ) , csc Ω ( ψ ) = 1 sin Ω ( ψ ) , sec Ω ( ψ ) = 1 cos Ω ( ψ ) .

Similarly,

cosh Ω ( ψ ) = Ω ψ + Ω − ψ 2 , sinh Ω ( ψ ) = Ω ψ − Ω − ψ 2 , coth Ω ( ψ ) = cosh Ω ( ψ ) sinh Ω ( ψ ) , tanh Ω ( ψ ) = sinh Ω ( ψ ) cosh Ω ( ψ ) , csch Ω ( ψ ) = 1 sinh Ω ( ψ ) , sech Ω ( ψ ) = 1 cosh Ω ( ψ ) .

3 Execution of the mEDAM

In the present section, we use the suggested method mEDAM for constructing soliton solutions for CKGE stated in (1). We offer the subsequent transformation so that this approach may be expanded to solve (1):

(10) w ( x , t ) = W ( ψ ) ; ψ = β x δ δ − α t δ δ ,

where α and β stands for nonzero constants. When this transformation is applied, Eq. (1) converts to the following ordinary equation:

(11) ( α 2 − β 2 ) W ″ + σ W − ρ W 2 = 0 .

Establishing the homogenous balance principle between W ″ and W 2 , we obtain ς = 2 . By substituting ς = 2 in (8), we obtain the following series form solution for (11):

(12) W ( ψ ) = ∑ l = − 2 2 d l ( Z ( ψ ) ) l .

With the help of (9), we insert (12) into (11) which create a polynomial in Z ( ψ ) by collecting the terms with same power of Z ( ψ ) . By equating the coefficients of the polynomial to zero, we obtain a system of nonlinear algebraic equations. We use Maple to solve the system and reach at the following four distinct cases of solutions:

Case 1:

(13) α = σ − β 2 ( ln ( Ω ) ) 2 M ln ( Ω ) − M , β = β , d − 2 = − 6 η 2 σ M ρ , d − 1 = − 6 η μ σ M ρ , d 0 = − 6 σ η ν M ρ , d 1 = 0 , d 2 = 0 .

Case 2:

(14) α = σ + β 2 ( ln ( Ω ) ) 2 M ln ( Ω ) M , β = β , d − 2 = 6 η 2 σ M ρ , d − 1 = 6 η μ σ M ρ , d 0 = σ ( 2 ν η + μ 2 ) M ρ , d 1 = 0 , d 2 = 0 .

Case 3:

(15) α = − σ + β 2 ( ln ( Ω ) ) 2 M ln ( Ω ) M , β = β , d − 2 = 0 , d − 1 = 0 , d 0 = − 6 σ η ν M ρ , d 1 = − 6 μ ν σ M ρ , d 2 = − 6 ν 2 σ M ρ .

Case 4:

(16) α = σ + β 2 ( ln ( Ω ) ) 2 M ln ( Ω ) M , β = β , d − 2 = 0 , d − 1 = 0 , d 0 = σ ( 2 ν η + μ 2 ) M ρ , d 1 = 6 μ ν σ M ρ , d 2 = 6 ν 2 σ M ρ .

Assuming case 1, we obtain the following families of soliton solutions for (1):

Family 1.1: When M < 0 and ν ≠ 0 :

(17) w 1 , 1 ( x , t ) = − 6 η 2 σ M ρ − μ 2 ν + − M tan Ω 1 2 − M ψ 2 ν 2 − 6 η μ σ M ρ − μ 2 ν + − M tan Ω 1 2 − M ψ 2 ν − 6 σ η ν M ρ ,

(18) w 1 , 2 ( x , t ) = − 6 η 2 σ M ρ − 1 2 μ ν − 1 2 − M cot Ω 1 2 − M ψ ν 2 − 6 η μ σ M ρ − 1 2 μ ν − 1 2 − M cot Ω 1 2 − M ψ ν − 6 σ η ν M ρ ,

(19) w 1 , 3 ( x , t ) = − 6 η 2 σ M ρ − 1 2 μ ν + 1 2 − M tan Ω ( − M ψ ) ± ( sec Ω ( − M ψ ) ) ν 2 − 6 η μ σ M ρ − 1 2 μ ν + 1 2 − M tan Ω ( − M ψ ) ± ( sec Ω ( − M ψ ) ) ν − 6 σ η ν M ρ ,

(20) w 1 , 4 ( x , t ) = − 6 η 2 σ M ρ − 1 2 μ ν + 1 2 − M cot Ω ( − M ψ ) ± ( csc Ω ( − M ψ ) ) ν 2 − 6 η μ σ M ρ − 1 2 μ ν + 1 2 − M cot Ω ( − M ψ ) ± ( csc Ω ( − M ψ ) ) ν − 6 σ η ν M ρ ,

and

(21) w 1 , 5 ( x , t ) = − 6 η 2 σ M ρ − 1 2 μ ν + 1 4 − M tan Ω 1 4 − M ψ − cot Ω 1 4 − M ψ ν 2 − 6 η μ σ M ρ − 1 2 μ ν + 1 4 − M tan Ω 1 4 − M ψ − cot Ω 1 4 − M ψ ν − 6 σ ν η M ρ .

Family 1.2: When M > 0 , and ν ≠ 0 :

(22) w 1 , 6 ( x , t ) = − 6 η 2 σ M ρ − 1 2 μ ν − 1 2 M tanh Ω 1 2 − M ψ ν 2 − 6 η μ σ M ρ − 1 2 μ ν − 1 2 M tanh Ω 1 2 − M ψ ν − 6 σ η ν M ρ ,

(23) w 1 , 7 ( x , t ) = − 6 η 2 σ M ρ − 1 2 μ ν − 1 2 M coth Ω 1 2 − M ψ ν 2 − 6 η μ σ M ρ − 1 2 μ ν − 1 2 M coth Ω 1 2 − M ψ ν − 6 σ η ν M ρ ,

(24) w 1 , 8 ( x , t ) = − 6 η 2 σ M ρ − 1 2 μ ν + 1 2 M tanh Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν 2 − 6 η μ σ M ρ − 1 2 μ ν + 1 2 Z tanh Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν − 6 σ η ν M ρ ,

(25) w 1 , 9 ( x , t ) = − 6 η 2 σ M ρ − 1 2 μ ν + 1 2 M coth Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν 2 − 6 η μ σ M ρ − 1 2 μ ν + 1 2 − M tanh Ω ( − M ψ ) ± ( csch Ω ( − M ψ ) ) ν − 6 σ η ν M ρ ,

and

(26) w 1 , 10 ( x , t ) = − 6 η 2 σ M ρ − 1 2 μ ν − 1 4 M tanh Ω 1 4 M ψ − coth Ω 1 4 M ψ ν 2 − 6 η μ σ M ρ − 1 2 μ ν − 1 4 M tanh Ω 1 4 M ψ − coth Ω 1 4 M ψ ν − 6 σ η ν M ρ .

Family 1.3: When η ν > 0 and μ = 0 :

(27) w 1 , 11 ( x , t ) = − 6 σ η ν ( tan Ω ( η ν ψ ) ) 2 M ρ − 6 σ η ν M ρ

(28) w 1 , 12 ( x , t ) = − 6 σ η ν ( cot Ω ( η ν ψ ) ) 2 M ρ − 6 σ η ν M ρ ,

(29) w 1 , 13 ( x , t ) = − 6 σ η ν ( tan Ω ( 2 η ν ψ ) ± ( sec Ω ( 2 η ν ψ ) ) ) 2 M ρ − 6 σ η ν M ρ ,

(30) w 1 , 14 ( x , t ) = − 6 σ η ν 2 ( cot Ω ( 2 η ν ψ ) ± ( csc Ω ( 2 η ν ψ ) ) ) 2 M ρ − 6 σ η ν M ρ ,

and

(31) w 1 , 15 ( x , t ) = − 24 σ η ν tan Ω 1 2 η ν ψ − cot Ω 1 2 η ν ψ 2 M ρ − 6 σ η ν M ρ .

Family 1.4: When η ν < 0 and μ = 0 ;

(32) w 1 , 16 ( x , t ) = 6 σ η ν ( tanh Ω ( − η ν ψ ) ) 2 M ρ − 6 σ η ν M ρ ,

(33) w 1 , 17 ( x , t ) = 6 σ η ν ( coth Ω ( − ν η ψ ) ) 2 M ρ − 6 σ η ν M ρ ,

(34) w 1 , 18 ( x , t ) = 6 σ η ν ( tanh Ω ( 2 − η ν ψ ) ± ( i sech Ω ( 2 − η ν ψ ) ) ) 2 M ρ − 6 σ η ν M ρ ,

(35) w 1 , 19 ( x , t ) = 6 σ η ν ( tanh Ω ( 2 − η ν ψ ) ± ( i sech Ω ( 2 − η ν ψ ) ) ) 2 M ρ − 6 σ η ν M ρ ,

and

(36) w 1 , 20 ( x , t ) = 6 σ η ν ( coth Ω ( 2 − η ν ψ ) ± ( i csch Ω ( 2 − η ν ψ ) ) ) 2 M ρ − 6 σ η ν M ρ .

Family 1.5: When ν = η and μ = 0 ;

(37) w 1 , 21 ( x , t ) = − 6 η 2 σ M ρ ( tan Ω ( η ψ ) ) 2 − 6 η 2 σ M ρ ,

(38) w 1 , 22 ( x , t ) = − 6 η 2 σ M ρ ( cot Ω ( η ψ ) ) 2 − 6 η 2 σ M ρ ,

(39) w 1 , 23 ( x , t ) = − 6 η 2 σ M ρ ( tan Ω ( 2 η ψ ) ± ( sec Ω ( 2 η ψ ) ) ) 2 − 6 η 2 σ M ρ ,

(40) w 1 , 24 ( x , t ) = − 6 η 2 σ M ρ ( − cot Ω ( 2 η ψ ) ± ( csc Ω ( 2 η ψ ) ) ) 2 − 6 η 2 σ M ρ ,

and

(41) w 1 , 25 ( x , t ) = − 6 η 2 σ M ρ 1 2 tan Ω 1 2 η ψ − 1 2 cot Ω 1 2 η ψ 2 − 6 η 2 σ M ρ .

Family 1.6: When ν = − η and μ = 0 :

(42) w 1 , 26 ( x , t ) = − 6 η 2 σ M ρ ( tanh Ω ( η ψ ) ) 2 + 6 η 2 σ M ρ

(43) w 1 , 27 ( x , t ) = − 6 η 2 σ M ρ ( coth Ω ( η ψ ) ) 2 + 6 η 2 σ M ρ ,

(44) w 1 , 28 ( x , t ) = − 6 η 2 σ M ρ ( − tanh Ω ( 2 η ψ ) ± ( i sech Ω ( 2 η ψ ) ) ) 2 + 6 η 2 σ M ρ ,

(45) w 1 , 29 ( x , t ) = − 6 η 2 σ M ρ ( − coth Ω ( 2 η ψ ) ± ( i cech ( 2 η ψ ) ) ) 2 + 6 η 2 σ M ρ ,

and

(46) w 1 , 30 ( x , t ) = − 6 η 2 σ M ρ − 1 2 tanh Ω 1 2 η ψ − 1 2 coth Ω 1 2 η ψ 2 + 6 η 2 σ M ρ .

Family 1.7: When μ = τ , η = n τ ( n ≠ 0 ) and ν = 0 :

(47) w 1 , 31 ( x , t ) = − 6 n σ Ω τ ψ ρ ( Ω τ ψ − n ) 2 ,

where ψ = β x δ δ − ( σ − β 2 ( ln ( Ω ) ) 2 M ln ( Ω ) − M ) t δ δ .

Assuming case 2, we obtain the following families of soliton solutions for (1):

Family 2.1: When M < 0 ν ≠ 0 :

(48) w 2 , 1 ( x , t ) = 6 η 2 σ M ρ − 1 2 μ ν + 1 2 − M tan Ω 1 2 − M ψ ν − 2 + 6 η μ σ M ρ − 1 2 μ ν + 1 2 − M tan Ω 1 2 − M ψ ν − 1 + σ ( μ 2 + 2 ν η ) M ρ ,

(49) w 2 , 2 ( x , t ) = 6 η 2 σ M ρ − 1 2 μ ν − 1 2 − M cot Ω 1 2 − M ψ ν − 2 + 6 η μ σ M ρ − 1 2 μ ν − 1 2 − M cot Ω 1 2 − M ψ ν − 1 + σ ( μ 2 + 2 ν η ) M ρ ,

(50) w 2 , 3 ( x , t ) = 6 η 2 σ M ρ − 1 2 μ ν + 1 2 − M tan Ω ( − M ψ ) ± ( sec Ω ( − M ψ ) ) ν − 2 + 6 η μ σ M ρ − 1 2 μ ν + 1 2 − M tan Ω ( − M ψ ) ± ( sec Ω ( − M ψ ) ) ν − 1 + σ ( μ 2 + 2 ν η ) M ρ ,

(51) w 2 , 4 ( x , t ) = 6 η 2 σ M ρ − 1 2 μ ν + 1 2 − M cot Ω ( − M ψ ) ± ( csc Ω ( − M ψ ) ) ν − 2 + 6 η μ σ M ρ − 1 2 μ ν + 1 2 − M cot Ω ( − M ψ ) ± ( csc Ω ( − M ψ ) ) ν − 1 + σ ( μ 2 + 2 ν η ) M ρ ,

and

(52) w 2 , 5 ( x , t ) = 6 η 2 σ M ρ − 1 2 μ ν + 1 4 − M tan Ω 1 4 − M ψ − cot Ω 1 4 − M ψ ν − 2 + 6 η μ σ M ρ − 1 2 μ ν + 1 4 − M tan Ω 1 4 − M ψ − cot Ω 1 4 − M ψ ν − 1 + σ ( μ 2 + 2 ν η ) M ρ .

Family 2.2: When M > 0 ν ≠ 0 :

(53) w 2 , 6 ( x , t ) = 6 η 2 σ M ρ − 1 2 μ ν − 1 2 M tanh Ω 1 2 − M ψ ν − 2 + 6 η μ σ M ρ − 1 2 μ ν − 1 2 M tanh Ω 1 2 − M ψ ν − 1 + σ ( μ 2 + 2 ν η ) M ρ ,

(54) w 2 , 7 ( x , t ) = 6 η 2 σ M ρ − 1 2 μ ν − 1 2 M coth Ω 1 2 − M ψ ν − 2 + 6 η μ σ M ρ − 1 2 μ ν − 1 2 M coth Ω 1 2 − M ψ ν − 1 + σ ( μ 2 + 2 ν η ) M ρ ,

(55) w 2 , 8 ( x , t ) = 6 η 2 σ M ρ − 1 2 μ ν + 1 2 M tanh Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν − 2 + 6 η μ σ M ρ − 1 2 μ ν + 1 2 Z tanh Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν − 1 + σ ( μ 2 + 2 ν η ) M ρ ,

(56) w 2 , 9 ( x , t ) = 6 η 2 σ M ρ − 1 2 μ ν + 1 2 M coth Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν − 2 + 6 η μ σ M ρ − 1 2 μ ν + 1 2 − M tanh Ω ( − M ψ ) ± ( csch Ω ( − M ψ ) ) ν − 1 + σ ( μ 2 + 2 ν η ) M ρ ,

and

(57) w 2 , 10 ( x , t ) = 6 η 2 σ M ρ − 1 2 μ ν − 1 4 M tanh Ω 1 4 M ψ − coth Ω 1 4 M ψ ν − 2 + 6 η μ σ M ρ − 1 2 μ ν − 1 4 M tanh Ω 1 4 M ψ − coth Ω 1 4 M ψ ν − 1 + σ ( μ 2 + 2 ν η ) M ρ .

Family 2.3: When η ν > 0 and μ = 0 :

(58) w 2 , 11 ( x , t ) = 6 η ν σ ( tan Ω ( ν η ψ ) ) 2 M ρ + 2 σ ν η M ρ ,

(59) w 2 , 12 ( x , t ) = 6 η ν σ ( cot Ω ( ν η ψ ) ) 2 M ρ + 2 σ ν η M ρ ,

(60) w 2 , 13 ( x , t ) = 6 η ν σ M ρ ( tan Ω ( 2 ν η ψ ) ± ( sec Ω ( 2 ν η ψ ) ) ) 2 + 2 σ ν η M ρ ,

(61) w 2 , 14 ( x , t ) = 6 η ν σ M ρ ( − cot Ω ( 2 ν η ψ ) ± ( csc Ω ( 2 ν η ψ ) ) ) 2 + 2 σ ν η M ρ ,

and

(62) w 2 , 15 ( x , t ) = 6 η ν σ M ρ 1 2 tan Ω 1 2 ν η ψ − cot Ω 1 2 ν η ψ 2 + 2 σ ν η M ρ .

Family 2.4: When η ν < 0 and μ = 0 :

(63) w 2 , 16 ( x , t ) = 6 η ν σ ( tanh Ω ( − ν η ψ ) ) 2 M ρ + 2 σ ν η M ρ ,

(64) w 2 , 17 ( x , t ) = 6 η ν σ ( coth Ω ( − ν η ψ ) ) 2 M ρ + 2 σ ν η M ρ ,

(65) w 2 , 18 ( x , t ) = 6 η ν σ M ρ ( tanh Ω ( 2 − ν η ψ ) ± ( i sech Ω ( 2 − ν η ψ ) ) ) 2 + 2 σ ν η M ρ ,

(66) w 2 , 19 ( x , t ) = 6 η ν σ ( coth Ω ( 2 − ν η ψ ) ± ( csch Ω ( 2 − ν η ψ ) ) ) 2 M ρ + 2 σ ν η M ρ ,

and

(67) w 2 , 20 ( x , t ) = 24 η ν σ tanh Ω 1 2 − ν η ψ ± coth Ω 1 2 − ν η ψ 2 M ρ + 2 σ ν η M ρ .

Family 2.5: When ν = η and μ = 0 :

(68) w 2 , 21 ( x , t ) = 6 η 2 σ M ρ ( tan Ω ( η ψ ) ) 2 + 2 σ η 2 M ρ ,

(69) w 2 , 22 ( x , t ) = 6 η 2 σ M ρ ( cot Ω ( η ψ ) ) 2 + 2 σ η 2 M ρ ,

(70) w 2 , 23 ( x , t ) = 6 η 2 σ M ρ ( tan Ω ( 2 η ψ ) ± ( sec Ω ( 2 η ψ ) ) ) 2 + 2 σ η 2 M ρ ,

(71) w 2 , 24 ( x , t ) = 6 η 2 σ M ρ ( − cot Ω ( 2 η ψ ) ± ( csc Ω ( 2 η ψ ) ) ) 2 + 2 σ η 2 M ρ ,

and

(72) w 2 , 25 ( x , t ) = 6 η 2 σ M ρ 1 2 tan Ω 1 2 η ψ − 1 2 cot Ω 1 2 η ψ 2 + 2 σ η 2 M ρ .

Family 2.6: When ν = − η and μ = 0 :

(73) w 2 , 26 ( x , t ) = 6 η 2 σ M ρ ( tanh Ω ( η ψ ) ) 2 − 2 η 2 σ M ρ ,

(74) w 2 , 27 ( x , t ) = 6 η 2 σ M ρ ( coth Ω ( η ψ ) ) 2 − 2 η 2 σ M ρ ,

(75) w 2 , 28 ( x , t ) = 6 η 2 σ M ρ ( − tanh Ω ( 2 η ψ ) ± ( i sech Ω ( 2 η ψ ) ) ) 2 − 2 η 2 σ M ρ ,

(76) w 2 , 29 ( x , t ) = 6 η 2 σ M ρ ( − coth Ω ( 2 η ψ ) ± ( i cech ( 2 η ψ ) ) ) 2 − 2 η 2 σ M ρ ,

and

(77) w 2 , 30 ( x , t ) = 6 η 2 σ M ρ − 1 2 tanh Ω 1 2 η ψ − 1 2 coth Ω 1 2 η ψ 2 − 2 η 2 σ M ρ .

Family 2.7: When μ = τ , η = n τ ( n ≠ 0 ) and ν = 0 :

(78) w 2 , 31 ( x , t ) = σ ( n 2 + 4 Ω τ ψ n + Ω 2 τ ψ ) ρ ( Ω τ ψ − n ) 2 ,

where ψ = β x δ δ − ( σ + β 2 ( ln ( Ω ) ) 2 M ln ( Ω ) M ) t δ δ .

Assuming case 3, we obtain the following families of soliton solutions for (1):

Family 3.1: When M < 0 ν ≠ 0 :

(79) w 3 , 1 ( x , t ) = − 6 σ η ν M ρ − 6 μ ν σ M ρ × − 1 2 μ ν + 1 2 − M tan Ω 1 2 − M ψ ν − 6 ν 2 σ M ρ − 1 2 μ ν + 1 2 − M tan Ω 1 2 − M ψ ν 2 ,

(80) w 3 , 2 ( x , t ) = − 6 σ η ν M ρ − 6 μ ν σ M ρ × − 1 2 μ ν − 1 2 − M cot Ω 1 2 − M ψ ν − 6 ν 2 σ M ρ − 1 2 μ ν − 1 2 − M cot Ω 1 2 − M ψ ν 2 ,

(81) w 3 , 3 ( x , t ) = − 6 σ η ν M ρ − 6 μ ν σ M ρ − 1 2 μ ν + 1 2 − M tan Ω ( − M ψ ) ± ( sec Ω ( − M ψ ) ) ν − 6 ν 2 σ M ρ − 1 2 μ ν + 1 2 × − M tan Ω ( − M ψ ) ± ( sec Ω ( − M ψ ) ) ν 2 ,

(82) w 3 , 4 ( x , t ) = − 6 σ η ν M ρ − 6 μ ν σ M ρ − 1 2 μ ν + 1 2 − M cot Ω ( − M ψ ) ± ( csc Ω ( − M ψ ) ) ν − 6 ν 2 σ M ρ − 1 2 μ ν + 1 2 × − M cot Ω ( − M ψ ) ± ( csc Ω ( − M ψ ) ) ν 2 ,

and

(83) w 3 , 5 ( x , t ) = − 6 σ η ν M ρ − 6 μ ν σ M ρ − 1 2 μ ν + 1 4 × − M tan Ω 1 4 − M ψ − cot Ω 1 4 − M ψ ν − ν 2 σ M ρ − 1 2 μ ν + 1 4 × − M tan Ω 1 4 − M ψ − cot Ω 1 4 − M ψ ν 2 .

Family 3.2: When M > 0 ν ≠ 0 :

(84) w 3 , 6 ( x , t ) = − 6 σ η ν M ρ − 6 μ ν σ M ρ × − 1 2 μ ν − 1 2 M tanh Ω 1 2 − M ψ ν − 6 ν 2 σ M ρ − 1 2 μ ν − 1 2 M tanh Ω 1 2 − M ψ ν 2 ,

(85) w 3 , 7 ( x , t ) = − 6 σ η ν M ρ − 6 μ ν σ M ρ × − 1 2 μ ν − 1 2 M coth Ω 1 2 − M ψ ν − 6 ν 2 σ M ρ − 1 2 μ ν − 1 2 M coth Ω 1 2 − M ψ ν 2 ,

(86) w 3 , 8 ( x , t ) = − 6 σ η ν M ρ − 6 μ ν σ M ρ × − 1 2 μ ν + 1 2 × M tanh Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν − 6 ν 2 σ M ρ − 1 2 μ ν + 1 2 × M tanh Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν 2 ,

(87) w 3 , 9 ( x , t ) = − 6 σ η ν M ρ − 6 μ ν σ M ρ − 1 2 μ ν + 1 2 × M coth Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν − 6 ν 2 σ M ρ − 1 2 μ ν + 1 2 × M coth Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν 2 ,

and

(88) w 3 , 10 ( x , t ) = − 6 σ η ν M ρ − 6 μ ν σ M ρ − 1 2 μ ν − 1 4 × M tanh Ω 1 4 M ψ − coth Ω 1 4 M ψ ν − 6 ν 2 σ M ρ − 1 2 μ ν − 1 4 × M tanh Ω 1 4 M ψ − coth Ω 1 4 M ψ ν 2 .

Family 3.3: When η ν > 0 and μ = 0 :

(89) w 3 , 11 ( x , t ) = − 6 σ η ν M ρ − 6 ν σ η M ρ ( tan Ω ( η ν ψ ) ) 2

(90) w 3 , 12 ( x , t ) = − 6 σ η ν M ρ − 6 ν σ η M ρ ( cot Ω ( η ν ψ ) ) 2

(91) w 3 , 13 ( x , t ) = − 6 σ η ν M ρ − 6 ν σ η M ρ × ( tan Ω ( 2 η ν ψ ) ± ( sec Ω ( 2 η ν ψ ) ) ) 2 ,

(92) w 3 , 14 ( x , t ) = − 6 σ η ν M ρ − 6 ν σ η M ρ × ( cot Ω ( 2 η ν ψ ) ± ( csc Ω ( 2 η ν ψ ) ) ) 2 ,

and

(93) w 3 , 15 ( x , t ) = − 6 σ η ν M ρ − 3 ν σ η 2 M ρ × tan Ω 1 2 η ν ψ − cot Ω 1 2 η ν ψ 2

Family 3.4: When η ν < 0 and μ = 0 :

(94) w 3 , 16 ( x , t ) = − 6 σ η ν M ρ − 6 ν σ η M ρ ( tanh Ω ( − η ν ψ ) ) 2 ,

(95) w 3 , 17 ( x , t ) = − 6 σ η ν M ρ − 6 ν σ η M ρ ( coth Ω ( − η ν ψ ) ) 2 ,

(96) w 3 , 18 ( x , t ) = − 6 σ η ν M ρ − 6 ν σ η M ρ × ( tanh Ω ( 2 − η ν ψ ) ± ( i sech Ω ( 2 − η ν ψ ) ) ) 2 ,

(97) w 3 , 19 ( x , t ) = − 6 σ η ν M ρ − 6 ν σ η M ρ × ( coth Ω ( 2 − η ν ψ ) ± ( csch Ω ( 2 − η ν ψ ) ) ) 2 ,

and

(98) w 3 , 20 ( x , t ) = − 6 σ η ν M ρ − 3 ν σ η 2 M ρ × tanh Ω 1 2 − η ν ψ + coth Ω 1 2 − η ν ψ 2 .

Family 3.5: When ν = η and μ = 0 :

(99) w 3 , 21 ( x , t ) = − 6 σ η 2 M ρ − 6 σ η 2 ( tan Ω ( η ψ ) ) 2 M ρ ,

(100) w 3 , 22 ( x , t ) = − 6 σ η 2 M ρ − 6 σ η 2 ( cot Ω ( η ψ ) ) 2 M ρ ,

(101) w 3 , 23 ( x , t ) = − 6 σ η 2 M ρ − 6 σ η 2 ( tan Ω ( 2 η ψ ) ± ( sec Ω ( 2 η ψ ) ) ) 2 M ρ ,

(102) w 3 , 24 ( x , t ) = − 6 σ η 2 M ρ − 6 σ η 2 ( − cot Ω ( 2 η ψ ) ± ( csc Ω ( 2 η ψ ) ) ) 2 M ρ ,

and

(103) w 3 , 25 ( x , t ) = − 6 σ η 2 M ρ − 6 σ η 2 1 2 tan Ω 1 2 η ψ − 1 2 cot Ω 1 2 η ψ 2 M ρ .

Family 3.6: When ν = − η and μ = 0 :

(104) w 3 , 26 ( x , t ) = 6 σ η 2 M ρ − 6 σ η 2 ( tanh Ω ( η ψ ) ) 2 M ρ ,

(105) w 3 , 27 ( x , t ) = 6 σ η 2 M ρ − 6 σ η 2 ( coth Ω ( η ψ ) ) 2 M ρ ,

(106) w 3 , 28 ( x , t ) = 6 σ η 2 M ρ − 6 σ η 2 ( − tanh Ω ( 2 η ψ ) ± ( i sech Ω ( 2 η ψ ) ) ) 2 M ρ ,

(107) w 3 , 29 ( x , t ) = 6 σ η 2 M ρ − 6 σ η 2 ( − coth Ω ( 2 η ψ ) ± ( i cech ( 2 η ψ ) ) ) 2 M ρ ,

and

(108) w 3 , 30 ( x , t ) = 6 σ η 2 M ρ − 6 σ η 2 − 1 2 tanh Ω 1 2 η ψ − 1 2 coth Ω 1 2 η ψ 2 M ρ .

Family 3.7: When μ = τ , c = n τ ( n ≠ 0 ) , and η = 0 :

(109) w 3 , 31 ( x , t ) = − 6 τ 2 n σ Ω τ ψ M ρ ( − 1 + n Ω τ ψ ) 2 ,

where ψ = β x δ δ − ( − σ + β 2 ( ln ( Ω ) ) 2 M ln ( Ω ) M ) t δ δ .

Assuming case 4, we obtain the following families of soliton solutions for (1):

Family 4.1: When M < 0 ν ≠ 0 :

(110) w 4 , 1 ( x , t ) = 6 μ ν σ M ρ − 1 2 μ ν + 1 2 − M tan Ω 1 2 − M ψ ν + 6 ν 2 σ M ρ − 1 2 μ ν + 1 2 − M tan Ω 1 2 − M ψ ν 2 + σ ( μ 2 + 2 ν η ) M ρ ,

(111) w 4 , 2 ( x , t ) = 6 μ ν σ M ρ − 1 2 μ ν − 1 2 − M cot Ω 1 2 − M ψ ν + 6 ν 2 σ M ρ − 1 2 μ ν − 1 2 − M cot Ω 1 2 − M ψ ν 2 + σ ( μ 2 + 2 ν η ) M ρ ,

(112) w 4 , 3 ( x , t ) = 6 μ ν σ M ρ − 1 2 μ ν + 1 2 − M tan Ω ( − M ψ ) ± ( sec Ω ( − M ψ ) ) ν + 6 ν 2 σ M ρ − 1 2 μ ν + 1 2 − M tan Ω ( − M ψ ) ± ( sec Ω ( − M ψ ) ) ν 2 + σ ( μ 2 + 2 ν η ) M ρ ,

(113) w 4 , 4 ( x , t ) = 6 μ ν σ M ρ − 1 2 μ ν + 1 2 − M cot Ω ( − M ψ ) ± ( csc Ω ( − M ψ ) ) ν + 6 ν 2 σ M ρ − 1 2 μ ν + 1 2 − M cot Ω ( − M ψ ) ± ( csc Ω ( − M ψ ) ) ν 2 + σ ( μ 2 + 2 ν η ) M ρ ,

and

(114) w 4 , 5 ( x , t ) = 6 μ ν σ M ρ − 1 2 μ ν + 1 4 − M tan Ω 1 4 − M ψ − cot Ω 1 4 − M ψ ν + 6 ν 2 σ M ρ − 1 2 μ ν + 1 ∕ 4 − M tan Ω 1 4 − M ψ − cot Ω 1 4 − M ψ ν 2 + σ ( μ 2 + 2 ν η ) M ρ .

Family 4.2: When M > 0 ν ≠ 0 :

(115) w 4 , 6 ( x , t ) = 6 μ ν σ M ρ − 1 2 μ ν − 1 2 M tanh Ω 1 2 − M ψ ν + 6 ν 2 σ M ρ − 1 2 μ ν − 1 2 M tanh Ω 1 2 − M ψ ν 2 + σ ( μ 2 + 2 ν η ) M ρ ,

(116) w 4 , 7 ( x , t ) = 6 μ ν σ M ρ − 1 2 μ ν − 1 2 M coth Ω 1 2 − M ψ ν + 6 ν 2 σ M ρ − 1 2 μ ν − 1 2 M coth Ω 1 2 − M ψ ν 2 + σ ( μ 2 + 2 ν η ) M ρ ,

(117) w 4 , 8 ( x , t ) = 6 μ ν σ M ρ − 1 2 μ ν + 1 2 × M tanh Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν + 6 ν 2 σ M ρ − 1 2 μ ν + 1 2 × M tanh Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν 2 + σ ( μ 2 + 2 ν η ) M ρ ,

(118) w 4 , 9 ( x , t ) = 6 μ ν σ M ρ − 1 2 μ ν + 1 2 × M coth Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν + 6 ν 2 σ M ρ − 1 2 μ ν + 1 2 × M coth Ω ( − M ψ ) ± ( sech Ω ( − M ψ ) ) ν 2 + σ ( μ 2 + 2 ν η ) M ρ ,

and

(119) w 4 , 10 ( x , t ) = 6 μ ν σ M ρ − 1 2 μ ν − 1 4 × M tanh Ω 1 4 M ψ − coth Ω 1 4 M ψ ν + 6 ν 2 σ M ρ − 1 2 μ ν − 1 4 × M tanh Ω 1 4 M ψ − coth Ω 1 4 M ψ ν 2 + σ ( μ 2 + 2 ν η ) M ρ .

Family 4.3: When η ν > 0 and μ = 0 :

(120) w 4 , 11 ( x , t ) = − 3 σ 2 ( tan Ω ( ν η ψ ) ) 2 − 1 2 σ ρ ,

(121) w 4 , 12 ( x , t ) = − 3 σ 2 ( cot Ω ( ν η ψ ) ) 2 − 1 2 σ ρ ,

(122) w 4 , 13 ( x , t ) = − 3 σ 2 ( tan Ω ( 2 ν η ψ ) ± ( sec Ω ( 2 ν η ψ ) ) ) 2 − 1 2 σ ρ ,

(123) w 4 , 14 ( x , t ) = − 3 σ 2 ( cot Ω ( 2 ν η ψ ) ± ( csc Ω ( 2 ν η ψ ) ) ) 2 − 1 2 σ ρ ,

and

(124) w 4 , 15 ( x , t ) = − 3 σ 2 tan Ω 1 2 ν η ψ − cot Ω 1 2 ν η ψ 2 − 1 2 σ ρ .

Family 4.4: When η ν < 0 and μ = 0 :

(125) w 4 , 16 ( x , t ) = − 3 σ 2 ( tanh Ω ( − ν η ψ ) ) 2 − 1 2 σ ρ ,

(126) w 4 , 17 ( x , t ) = − 3 σ 2 ( ( tanh Ω ( 2 − ν η ψ ) ± ( i sech Ω ( 2 − ν η ψ ) ) ) ) 2 − 1 2 σ ρ ,

(127) w 4 , 18 ( x , t ) = − 3 σ 2 ( tanh Ω ( 2 − ν η ψ ) ± ( i sech Ω ( 2 − ν η ψ ) ) ) 2 − 1 2 σ ρ ,

(128) w 4 , 19 ( x , t ) = − 3 σ 2 ( ( coth Ω ( 2 − ν η ψ ) ± ( csch Ω ( 2 − ν η ψ ) ) ) ) 2 − 1 2 σ ρ ,

and

(129) w 4 , 20 ( x , t ) = − 3 σ 2 tanh Ω 1 2 − ν η ψ + coth Ω 1 2 − ν η ψ 2 − 1 2 σ ρ .

Family 4.5: When ν = η and μ = 0 :

(130) w 4 , 21 ( x , t ) = − 3 σ ( tan Ω ( η ψ ) ) 2 2 ρ − 1 2 σ ρ ,

(131) w 4 , 22 ( x , t ) = − 3 σ ( cot Ω ( η ψ ) ) 2 2 ρ − 1 2 σ ρ ,

(132) w 4 , 23 ( x , t ) = − 3 σ ( ( tan Ω ( 2 η ψ ) ± ( sec Ω ( 2 η ψ ) ) ) ) 2 2 ρ − 1 2 σ ρ ,

(133) w 4 , 24 ( x , t ) = − 3 σ ( ( − cot Ω ( 2 η ψ ) ± ( csc Ω ( 2 η ψ ) ) ) ) 2 2 ρ − 1 2 σ ρ ,

and

(134) w 4 , 25 ( x , t ) = − 3 σ 1 2 tan Ω 1 2 η ψ − 1 2 cot Ω 1 2 η ψ 2 2 ρ − 1 2 σ ρ .

Family 4.6: When ν = − η and μ = 0 :

(135) w 4 , 26 ( x , t ) = − 3 σ ( tanh Ω ( η ψ ) ) 2 2 ρ − 1 2 σ ρ ,

(136) w 4 , 27 ( x , t ) = − 3 σ ( coth Ω ( η ψ ) ) 2 2 ρ − 1 2 σ ρ ,

(137) w 4 , 28 ( x , t ) = − 3 σ ( ( − tanh Ω ( 2 η ψ ) ± ( i sech Ω ( 2 η ψ ) ) ) ) 2 2 ρ − 1 2 σ ρ ,

(138) w 4 , 29 ( x , t ) = − 3 σ ( ( − coth Ω ( 2 η ψ ) ± ( cech ( 2 η ψ ) ) ) ) 2 2 ρ − 1 2 σ ρ ,

and

(139) w 4 , 30 ( x , t ) = − 3 σ − 1 2 tanh Ω 1 2 η ψ − 1 2 coth Ω 1 2 η ψ 2 2 ρ − 1 2 σ ρ .

Family 4.7: When η = 0 , μ ≠ 0 and ν ≠ 0 :

(140) w 4 , 31 ( x , t ) = 2 σ ( ( cosh Ω ( μ ψ ) ) 2 − cosh Ω ( μ ψ ) sinh Ω ( μ ψ ) − 2 cosh Ω ( μ ψ ) + 2 sinh Ω ( μ ψ ) ) ρ ( cosh Ω ( μ ψ ) − sinh Ω ( μ ψ ) + 1 ) 2 ,

and

(141) w 4 , 32 ( x , t ) = 2 σ ( ( cosh Ω ( μ ψ ) ) 2 + cosh Ω ( μ ψ ) sinh Ω ( μ ψ ) − 2 cosh Ω ( μ ψ ) − 2 sinh Ω ( μ ψ ) ) ρ ( cosh Ω ( μ ψ ) + sinh Ω ( μ ψ ) + 1 ) 2 .

Family 4.8: When μ = τ , ν = n τ ( n ≠ 0 ) and η = 0 :

(142) w 4 , 33 ( x , t ) = σ ( 1 + 4 n Ω τ ψ + n 2 Ω 2 τ ψ ) ρ ( − 1 + n Ω τ ψ ) 2 ,

where ψ = β x δ δ − ( σ + β 2 ( ln ( Ω ) ) 2 M ln ( Ω ) M ) t δ δ .

4 Discussion and graphs

In this section, we graphically depict a number of wave patterns that were seen in the system that is being studied. By using the mEDAM, we were able to identify and display wave patterns in contour and 3D graphs. Predominantly, breather, dark kink, periodic, bright kinks, and other N -soliton structures are found in the soliton solutions constructed for CKGE. They continue travelling at the same pace and in the identical direction. They clarify the behavior of scalar fields in relativistic quantum physics when applied to the CKGE. Similar to vacuum state transitions, kink solitons, stable borders traversing scalar field zones, signify abrupt phase or amplitude shifts. Understanding the basic characteristics of relativistic quantum mechanical phenomena and how they relate to particles and vacuum features is possible through studying solitons in the CKGE. Breather solitons, which are localized disturbances going through energy state transitions, exhibit periodic amplitude oscillations. Stable, recurrent structures with periodic solitons include standing waves. In the field description, dark kinks show localized depressive issues that could be indicative of strong soliton-like qualities. Bright kinks, on the other hand, show peaks or humps in the field’s persona, which could be signs of temporary or meta-stable excitations. Kink solitons, stable boundaries across scalar field zones, represent abrupt phase or amplitude changes, much like vacuum state transitions do. Multi-soliton solutions, or N -solitons, explain the interaction between localized disturbances that result in bounded states, fusion, and dispersion. Studying solitons in the CKGE allows us to comprehend the fundamental properties of relativistic quantum mechanical systems and their relationship in particles and vacuum features. In a nutshell, the novelty of the current study lies in the provision of the systematic and unique unearthing of the new plethora of soliton solutions that showcase the novel and innovative nature of our investigation. Investigations into nonlinear FPDEs that arise in quantum field theory, fluid dynamics, and optics are of importance to solitons. These soliton solutions present a deeper insight into the CKGE underlying phenomena in the related scientific fields (Figures 1, 2, 3, 4, 5, 6, 7, 8).

Figure 1 The three-dimensional and contour graphics of the singular bright kink soliton solution w 1 , 1 {w}_{1,1} stated in (17) are visualized for μ = 1 \mu =1 ; ν = 1 \nu =1 ; η = 2 \eta =2 ; δ = 1 \delta =1 ; β = 1 \beta =1 ; σ = 2 \sigma =2 ; Ω = e \Omega =e ; ρ = 6 \rho =6 .
Figure 1

The three-dimensional and contour graphics of the singular bright kink soliton solution w 1 , 1 stated in (17) are visualized for μ = 1 ; ν = 1 ; η = 2 ; δ = 1 ; β = 1 ; σ = 2 ; Ω = e ; ρ = 6 .

Figure 2 The three-dimensional and contour graphics of the dark soliton solution w 1 , 7 {w}_{1,7} stated in (23) are visualized for μ = 10 \mu =10 ; ν = 8 \nu =8 ; η = 2 \eta =2 ; δ = 1 \delta =1 ; β = 10 \beta =10 ; σ = 20 \sigma =20 ; Ω = e \Omega =e ; ρ = 10 \rho =10 .
Figure 2

The three-dimensional and contour graphics of the dark soliton solution w 1 , 7 stated in (23) are visualized for μ = 10 ; ν = 8 ; η = 2 ; δ = 1 ; β = 10 ; σ = 20 ; Ω = e ; ρ = 10 .

Figure 3 The three-dimensional and contour graphics of the dark soliton solution w 1 , 31 {w}_{1,31} stated in (47) are visualized for μ = 2 \mu =2 ; ν = 0 \nu =0 ; η = 6 \eta =6 ; δ = 0.9 \delta =0.9 ; β = 1 \beta =1 ; σ = 20 \sigma =20 ; Ω = 3 \Omega =3 ; ρ = 10 \rho =10 ; τ = 2 \tau =2 ; n = 3 n=3 .
Figure 3

The three-dimensional and contour graphics of the dark soliton solution w 1 , 31 stated in (47) are visualized for μ = 2 ; ν = 0 ; η = 6 ; δ = 0.9 ; β = 1 ; σ = 20 ; Ω = 3 ; ρ = 10 ; τ = 2 ; n = 3 .

Figure 4 The three-dimensional and contour graphics of the breather dark soliton solution interacts with breather soliton solution w 2 , 5 {w}_{2,5} stated in (52) are visualized for μ = 1 \mu =1 ; ν = 1 \nu =1 ; η = 2 \eta =2 ; δ = 1 \delta =1 ; β = 5 \beta =5 ; σ = 1 \sigma =1 ; Ω = e \Omega =e ; ρ = 6 \rho =6 .
Figure 4

The three-dimensional and contour graphics of the breather dark soliton solution interacts with breather soliton solution w 2 , 5 stated in (52) are visualized for μ = 1 ; ν = 1 ; η = 2 ; δ = 1 ; β = 5 ; σ = 1 ; Ω = e ; ρ = 6 .

Figure 5 The three-dimensional and contour graphics of the breather dark soliton solution w 2 , 17 {w}_{2,17} stated in (64) are visualized for μ = 0 \mu =0 ; ν = − 2 \nu =-2 ; η = 8 \eta =8 ; δ = 0.9 \delta =0.9 ; β = 30 \beta =30 ; σ = 2 \sigma =2 ; Ω = e \Omega =e ; ρ = 6 \rho =6 .
Figure 5

The three-dimensional and contour graphics of the breather dark soliton solution w 2 , 17 stated in (64) are visualized for μ = 0 ; ν = − 2 ; η = 8 ; δ = 0.9 ; β = 30 ; σ = 2 ; Ω = e ; ρ = 6 .

Figure 6 The three-dimensional and contour graphics of the breather dark soliton solution w 3 , 31 {w}_{3,31} stated in (109) are visualized for μ = 5 \mu =5 ; ν = 25 \nu =25 ; η = 0 \eta =0 ; δ = 0.5 \delta =0.5 ; β = 50 \beta =50 ; σ = 1 \sigma =1 ; Ω = e \Omega =e ; ρ = 5 \rho =5 ; τ = 5 \tau =5 ; n = 5 n=5 .
Figure 6

The three-dimensional and contour graphics of the breather dark soliton solution w 3 , 31 stated in (109) are visualized for μ = 5 ; ν = 25 ; η = 0 ; δ = 0.5 ; β = 50 ; σ = 1 ; Ω = e ; ρ = 5 ; τ = 5 ; n = 5 .

Figure 7 The three-dimensional and contour graphics of the dark soliton solution w 4 , 25 {w}_{4,25} stated in (134) are visualized for μ = 0 \mu =0 ; ν = 7 \nu =7 ; η = 7 \eta =7 ; δ = 1 \delta =1 ; β = 4 \beta =4 ; σ = 3 \sigma =3 ; Ω = e \Omega =e ; ρ = 5 \rho =5 .
Figure 7

The three-dimensional and contour graphics of the dark soliton solution w 4 , 25 stated in (134) are visualized for μ = 0 ; ν = 7 ; η = 7 ; δ = 1 ; β = 4 ; σ = 3 ; Ω = e ; ρ = 5 .

Figure 8 The three-dimensional and contour graphics of the bell-shaped or dark kink soliton solution w 4 , 31 {w}_{4,31} stated in (140) are visualized for μ = 2 \mu =2 ; ν = 5 \nu =5 ; η = 0 \eta =0 ; δ = 1 \delta =1 ; β = 5 \beta =5 ; σ = 1 \sigma =1 ; Ω = e \Omega =e ; ρ = 6 \rho =6 .
Figure 8

The three-dimensional and contour graphics of the bell-shaped or dark kink soliton solution w 4 , 31 stated in (140) are visualized for μ = 2 ; ν = 5 ; η = 0 ; δ = 1 ; β = 5 ; σ = 1 ; Ω = e ; ρ = 6 .

5 Conclusion

We have used the upgraded mEDAM to comprehensively study the transmission of solitons in the CKGE, an established model in the fields of solid-state physics, quantum field theory, and nonlinear optics. Several dark kink, bright kink, breather, and other N -soliton solutions, including generalized hyperbolic, trigonometric, exponential, and rational functions, have been found by translating the suggested model into NODEs and assuming closed-form solutions. Crucial insights into the dynamics of propagating soliton processes can be gained from contour and three-dimensional graphs, which graphically represent the propagation behavior of certain soliton solutions. These graphs directly relate to domains related to the models under consideration. The unusual outcomes of applying the mEDAM to the model demonstrate its originality, as they have not been examined in academic literature previously. These findings improve our knowledge of nonlinear dynamics and temporal evolution processes, which have important ramifications for our comprehension of related physical phenomena. The effectiveness and consistency of the methods used in this work also show how widely relevant they are to nonlinear issues in many different scientific fields. Even though the mEDAM has made a substantial contribution to our knowledge of soliton dynamics and how they impact the targeted model, it is crucial to recognize the technique’s limits, especially in cases where the largest derivative and the nonlinear term are not uniformly balanced. Despite this drawback, the study makes clear how many concerns about nonlinear behaviors and soliton dynamics remain unresolved and provides fresh avenues for future research in the area. To put it briefly, what makes our study unusual is that we have discovered a variety of new soliton solutions in a methodical and distinctive way, demonstrating the creative and inventive nature of our research. Solitons are interested in studying nonlinear FPDEs that occur in fluid dynamics, optics, and quantum field theory. In the associated scientific domains, these soliton solutions offer a more profound understanding of the CKGE underlying phenomena. The authors also aim to make the mEDAM efficient in addressing nonlinear FPDEs with variable coefficients and stochastic behaviors for the construction and analysis of soliton solutions.

Acknowledgments

Researchers Supporting Project number (RSPD2024R1060), King Saud University, Riyadh, Saudi Arabia.

  1. Funding information: This project was funded by King Saud University, Riyadh, Saudi Arabia.

  2. Author contributions: M.B. and J.I. conceptualized, analyzed data, and drafted the manuscript. R.A. supervised, designed methodologies, and oversaw experimentation. F.A.A managed administration and reviewed the manuscript. E.A.A.I. conducted experiments, analyzed data, and contributed to writing and visualizations. 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: The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-04-05
Revised: 2024-05-15
Accepted: 2024-05-21
Published Online: 2024-06-20

© 2024 the author(s), published by De Gruyter

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

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  61. Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
  62. Linear instability of the vertical throughflow in a porous layer saturated by a power-law fluid with variable gravity effect
  63. Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
  64. A new benchmark for camouflaged object detection: RGB-D camouflaged object detection dataset
  65. Effect of electron temperature and concentration on production of hydroxyl radical and nitric oxide in atmospheric pressure low-temperature helium plasma jet: Swarm analysis and global model investigation
  66. Double diffusion convection of Maxwell–Cattaneo fluids in a vertical slot
  67. Thermal analysis of extended surfaces using deep neural networks
  68. Steady-state thermodynamic process in multilayered heterogeneous cylinder
  69. Multiresponse optimisation and process capability analysis of chemical vapour jet machining for the acrylonitrile butadiene styrene polymer: Unveiling the morphology
  70. Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques
  71. Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain
  72. The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model
  73. Research on optimization of combustor liner structure based on arc-shaped slot hole
  74. Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
  75. Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
  76. Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
  77. Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
  78. Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
  79. Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
  80. A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
  81. New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
  82. Physical aspects of quantile residual lifetime sequence
  83. Synthesis, structure, I–V characteristics, and optical properties of chromium oxide thin films for optoelectronic applications
  84. Smart mathematically filtered UV spectroscopic methods for quality assurance of rosuvastatin and valsartan from formulation
  85. A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
  86. Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
  87. A novel tetra hybrid bio-nanofluid model with stenosed artery
  88. Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
  89. Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
  91. New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
  92. Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
  93. Inflation and the principle of equivalence
  94. The use of a supercontinuum light source for the characterization of passive fiber optic components
  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
  97. Photon balance in the fiber laser model
  98. Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
  99. Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
  100. Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
  101. Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
  102. Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
  103. On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
  104. Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
  105. Review Article
  106. Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
  107. Special Issue on Predicting pattern alterations in nature - Part II
  108. A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
  109. On the existence and numerical simulation of Cholera epidemic model
  110. Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
  111. Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
  112. Prediction of COVID-19 spread with models in different patterns: A case study of Russia
  113. Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
  114. Accumulation process in the environment for a generalized mass transport system
  115. Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
  116. Special Issue on Nanomaterial utilization and structural optimization - Part II
  117. Numerical study on flow and heat transfer performance of a spiral-wound heat exchanger for natural gas
  118. Study of ultrasonic influence on heat transfer and resistance performance of round tube with twisted belt
  119. Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
  120. Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
  121. Special Issue on Nanofluids: Synthesis, Characterization, and Applications
  122. Exact solutions of a class of generalized nanofluidic models
  123. Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
  124. Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
  125. Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
  126. Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate
  127. Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction
  128. Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
  129. Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
  130. Comparative study of couple stress fluid flow using OHAM and NIM
  131. Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon
  132. Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
  133. Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
  134. Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
  135. Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
  136. Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
  137. Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
  138. Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
  139. Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
  140. Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
  141. Pulsed excitation of a quantum oscillator: A model accounting for damping
  142. Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
  143. Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
  144. Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
  145. Effect of the gravitational field strength on the rate of chemical reactions
  146. The kinetic relativity theory – hiding in plain sight
  147. Special Issue on Advanced Energy Materials - Part III
  148. Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
https://doi.org/10.1515/phys-2024-0044
Schlagwörter für diesen Artikel
fractional partial differential equations; homogenous balancing principle; modified extended direct algebraic method; wave transformation; breather waves; generalized Riccati NODE
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Artikel in diesem Heft

  1. Regular Articles
  2. Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
  3. Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
  4. High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
  5. Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
  6. Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
  7. Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
  8. The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
  9. Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
  10. A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
  11. Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
  12. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
  14. Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
  15. Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
  16. Bayesian estimation of equipment reliability with normal-type life distribution based on multiple batch tests
  17. Chaotic control problem of BEC system based on Hartree–Fock mean field theory
  18. Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
  21. Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
  22. Microscopic seepage simulation of gas and water in shale pores and slits based on VOF
  23. Model of conversion of flow from confined to unconfined aquifers with stochastic approach
  24. Study of fractional variable-order lymphatic filariasis infection model
  25. Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves
  26. Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
  27. Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions
  28. Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model
  29. Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
  30. Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
  31. Super-resolution reconstruction method of the optical synthetic aperture image using generative adversarial network
  32. A two-stage framework for predicting the remaining useful life of bearings
  33. Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
  34. Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects
  35. Exact solutions to vorticity of the fractional nonuniform Poiseuille flows
  36. In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
  37. Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems
  38. Substituent effect on the electronic and optical properties of newly designed pyrrole derivatives using density functional theory
  39. A comparative analysis of shielding effectiveness in glass and concrete containers
  40. Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation
  41. Analytical and numerical investigation for viscoelastic fluid with heat transfer analysis during rollover-web coating phenomena
  42. Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials
  43. Analysis of nonlinear fractional-order Fisher equation using two reliable techniques
  44. Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
  45. Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink
  46. Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
  47. Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
  48. Fractional view analytical analysis of generalized regularized long wave equation
  49. Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources
  50. Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
  51. Estimation of monotone α-quantile of past lifetime function with application
  52. Numerical simulation for the slip impacts on the radiative nanofluid flow over a stretched surface with nonuniform heat generation and viscous dissipation
  53. Study of fractional telegraph equation via Shehu homotopy perturbation method
  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
  55. Establishing breather and N-soliton solutions for conformable Klein–Gordon equation
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  57. CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube
  58. Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation
  59. A short report on a probability-based interpretation of quantum mechanics
  60. Study on cavitation and pulsation characteristics of a novel rotor-radial groove hydrodynamic cavitation reactor
  61. Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
  62. Linear instability of the vertical throughflow in a porous layer saturated by a power-law fluid with variable gravity effect
  63. Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
  64. A new benchmark for camouflaged object detection: RGB-D camouflaged object detection dataset
  65. Effect of electron temperature and concentration on production of hydroxyl radical and nitric oxide in atmospheric pressure low-temperature helium plasma jet: Swarm analysis and global model investigation
  66. Double diffusion convection of Maxwell–Cattaneo fluids in a vertical slot
  67. Thermal analysis of extended surfaces using deep neural networks
  68. Steady-state thermodynamic process in multilayered heterogeneous cylinder
  69. Multiresponse optimisation and process capability analysis of chemical vapour jet machining for the acrylonitrile butadiene styrene polymer: Unveiling the morphology
  70. Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques
  71. Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain
  72. The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model
  73. Research on optimization of combustor liner structure based on arc-shaped slot hole
  74. Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
  75. Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
  76. Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
  77. Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
  78. Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
  79. Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
  80. A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
  81. New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
  82. Physical aspects of quantile residual lifetime sequence
  83. Synthesis, structure, I–V characteristics, and optical properties of chromium oxide thin films for optoelectronic applications
  84. Smart mathematically filtered UV spectroscopic methods for quality assurance of rosuvastatin and valsartan from formulation
  85. A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
  86. Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
  87. A novel tetra hybrid bio-nanofluid model with stenosed artery
  88. Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
  89. Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
  91. New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
  92. Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
  93. Inflation and the principle of equivalence
  94. The use of a supercontinuum light source for the characterization of passive fiber optic components
  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
  97. Photon balance in the fiber laser model
  98. Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
  99. Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
  100. Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
  101. Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
  102. Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
  103. On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
  104. Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
  105. Review Article
  106. Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
  107. Special Issue on Predicting pattern alterations in nature - Part II
  108. A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
  109. On the existence and numerical simulation of Cholera epidemic model
  110. Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
  111. Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
  112. Prediction of COVID-19 spread with models in different patterns: A case study of Russia
  113. Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
  114. Accumulation process in the environment for a generalized mass transport system
  115. Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
  116. Special Issue on Nanomaterial utilization and structural optimization - Part II
  117. Numerical study on flow and heat transfer performance of a spiral-wound heat exchanger for natural gas
  118. Study of ultrasonic influence on heat transfer and resistance performance of round tube with twisted belt
  119. Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
  120. Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
  121. Special Issue on Nanofluids: Synthesis, Characterization, and Applications
  122. Exact solutions of a class of generalized nanofluidic models
  123. Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
  124. Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
  125. Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
  126. Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate
  127. Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction
  128. Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
  129. Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
  130. Comparative study of couple stress fluid flow using OHAM and NIM
  131. Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon
  132. Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
  133. Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
  134. Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
  135. Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
  136. Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
  137. Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
  138. Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
  139. Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
  140. Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
  141. Pulsed excitation of a quantum oscillator: A model accounting for damping
  142. Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
  143. Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
  144. Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
  145. Effect of the gravitational field strength on the rate of chemical reactions
  146. The kinetic relativity theory – hiding in plain sight
  147. Special Issue on Advanced Energy Materials - Part III
  148. Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
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