Startseite A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
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A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques

  • Md Nur Hossain , Md Mamunur Rasid , I. Abouelfarag , K. El-Rashidy , M. Mamun Miah EMAIL logo und Mohammad Kanan EMAIL logo
Veröffentlicht/Copyright: 22. November 2024
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Open Physics
Aus der Zeitschrift Open Physics Band 22 Heft 1

Abstract

Soliton solutions play a crucial role in modeling stable phenomena across optical communications, fluid dynamics, and plasma physics, owing to their stability and persistence in solving nonlinear equations. This study centers on the extended Sakovich equation, emphasizing the importance of soliton solutions in predicting and controlling localized wave behaviors, which advances nonlinear dynamics and its various applications due to its integrable properties and flexible soliton characteristics. This equation is applicable across diverse fields such as fluid dynamics, nonlinear optics, and plasma physics, where it effectively models nonlinear wave phenomena, including solitons and shock waves. Additionally, it provides crucial insights into wave propagation in biological systems and acoustics, making it a valuable tool for analyzing complex wave dynamics. Additionally, we investigate bifurcation and modulation instability within this equation, employing the improved Sardar subequation method and the ℛ ′ ℛ , 1 ℛ method to derive solitary wave solutions. These methods yield a diverse range of waveforms – hyperbolic, trigonometric, and rational functions – validated rigorously using Mathematica software for accuracy. Graphical representations vividly display various soliton patterns, such as singular, multi-singular, periodic singular, kink, anti-kink, bell-shaped, Kuznetsov–Ma Breather, and parabolic-shaped, highlighting their effectiveness in revealing innovative solutions. Furthermore, a comparative analysis verified the novelty of our derived soliton solutions. This research significantly contributes to advancing soliton solutions for the Sakovich equation, promising diverse applications across scientific disciplines.

Keywords: nonlinear partial differential equations; improved Sardar subequation technique; (ℛ′/ℛ,1/ℛ) technique; the extended Sakovich equation; soliton solutions

1 Introduction

Nonlinear partial differential equations (PDEs) are crucial mathematical tools for modeling the behavior of unknown multivariable functions and their partial derivatives in a nonlinear context. Unlike linear PDEs, which superimposing solutions can solve, nonlinear PDEs exhibit complex phenomena such as singularities, shock waves, and solitons. These equations are essential for modeling various physical phenomena across numerous fields. In fluid dynamics, for instance, they capture turbulence and wave breaking; in quantum mechanics, they describe the particle behavior in nonlinear media; and in general relativity, they characterize the curvature of spacetime, etc. [1,2,3,4,5,6].

Exploring nonlinear PDEs to derive analytical solutions is of fundamental importance in understanding and modeling complex systems across various scientific and engineering disciplines. These equations often describe phenomena where linear approximations fall short, such as the propagation of solitons, shock waves, and other nonlinear effects. Analytical solutions provide exact and detailed insights into these behaviors, revealing intricate dynamics that are otherwise challenging to discern. By uncovering conservation laws and symmetries, these solutions enhance our mathematical toolkit, allowing for more accurate validation of numerical methods and theoretical models. Moreover, the insights gained from analytical solutions extend beyond theoretical interest; they play a crucial role in practical applications such as optimizing optical communication systems, predicting and managing natural wave phenomena like tsunamis, and designing advanced materials with specific properties. Thus, the study of analytical solutions to nonlinear PDEs not only deepens our theoretical understanding but also drives significant advancements in technology and engineering, highlighting their essential role in both scientific inquiry and practical problem-solving [7,8,9,10,11,12,13].

The extended (3 + 1)-dimensional Sakovich equation is a significant advancement in the study of nonlinear PDEs, expanding the original Sakovich equation to three spatial dimensions and one temporal dimension. This equation effectively captures the intricate dynamics of wave phenomena, including the complex interactions and energy dissipation mechanisms present in higher-dimensional systems [8,9]. It incorporates nonlinear terms and higher-order derivatives, which are necessary for accurately describing soliton interactions, wave breaking, and other nonlinear wave behaviors. The extended Sakovich equation finds wide-ranging applications in science and industrial engineering such as in fluid dynamics; it models wave propagation in pipelines and channels, aiding in the prediction and control of turbulence and energy loss. In the case of optical fiber communications, it describes soliton behavior, minimizing signal degradation and enhancing the transmission efficiency. In materials science, the equation is used to study strain waves, facilitating the design of materials with desired properties, such as improved strength or flexibility. It also supports vibration analysis in mechanical systems, like machinery and aerospace components, to ensure stability and prevent failures. Additionally, in renewable energy, the equation helps optimize the interaction between ocean waves and mechanical structures, improving the efficiency of wave energy converters. Overall, it is a useful tool for analyzing and enhancing nonlinear wave dynamics in various industries [10,11]. The original Sakovich equation is provided by Sakovich [12] as follows:

(1) v yy + v zx + 2 v v xy + 6 v 2 v xx + 2 v xx 2 = 0 .

Subsequently, Wazwaz modified this in the following manner [13,14]:

(2) v xt + v xx + v yy + v xy + v yz + v zx + 2 v v xy + 6 v 2 v xx + 2 v xx 2 = 0 .

Recently, Ma et al. [15] extended Eq. (2) by incorporating two additional terms. The first is a linear term v zz , representing the second-order dispersion effect along the z-axis. The second is a nonlinear term v v xx , which accounts for the nonlinear effect induced by v and he second-order dispersion along the x-axis. The modified equation now takes the following form [10,15]:

(3) v xt + v xx + v yy + v zz + v xy + v yz + v zx + 2 v v xy + 6 v 2 v xx + 2 v xx 2 + v v xx = 0 ,

where v = v ( x , t ) represents the function that depends on the three spatial dimensions and the time-based variable.

This equation provides a physical description of wave propagation across three spatial dimensions and one temporal dimension. It models intricate wave phenomena including solitons – stable, localized structures that preserve their shape during movement. By incorporating nonlinear effects and wave interactions, this equation effectively represents realistic scenarios in fluid dynamics and other disciplines. Its applications extend to fields such as hydrodynamics, nonlinear optics, and plasma physics, where understanding the wave behavior is crucial. Ultimately, this equation serves as a foundational mathematical framework for analyzing and comprehending wave dynamics in diverse physical settings [10,11,13,15].

Given the significance of the extended Sakovich equation in multiple scientific fields, some researchers have focused their efforts on finding its solutions. Among them, Ali et al. [8] explored this equation’s exact solutions using the exp { − φ ( ξ ) } , Bernoulli sub-ordinary differential equation (ODE), and the G ′ kG ′ + G + r technique. Using the generalized exponential rational function method, Alqahtani and Kaplan [11] unveiled a bell-type soliton solution. Conversely, employing the enhanced direct algebraic method [10], solitary wave solutions of this equation were obtained. The inerrability of the modified Sakovich equation enables exact soliton solutions, which is crucial for understanding nonlinear dynamics in physics. These solitons are stable, localized waves that maintain their shape during propagation, offering precise descriptions of phenomena such as optical pulses and water waves. Despite the greater challenges compared to the classical Sakovich equation, which has received more attention, research on the modified Sakovich equation is still nascent. This work is centered on deriving new exact soliton solutions for the extended Sakovich equation through appropriate methodologies.

Due to the significant fascination and prominence attached to finding exact solutions for nonlinear PDEs, many scholars have explored a variety of mathematical approaches. These methods span a wide range, incorporating techniques such as the modified simple equation technique [16], the extended sinh-Gordon equation expansion method [17], the generalized ( G ′ / G ) method [18], the G ′ G ′ + G + A method [19], the Hirota bilinear formulation [20,21], the Riccati equation method [22], the auxiliary equation method [23], the extended Jacobi elliptic function method [24], the extended trial equation method [25], the exp { − φ ( ξ ) } method [26,27], the functional variable method [28], the multiple exp-function method [29], the lie symmetry method [30], the new auxiliary equation technique [31], the tanh–function method [32], the simple equation method [33], the tanh–coth method [34], the generalized Kudryshov method [35], the unified method [36], the Bernoulli ( G ′ / G ) -expansion method [37,38], the simplest equation method [39], and many others.

Among numerous techniques, the improved Sardar subequation technique’s advantage in discovering exact solutions lies in its capacity to simplify intricate nonlinear equations, making it easier to identify a wide array of soliton solutions efficiently. Specifically tailored for equations with one variable, it yields precise and diverse soliton patterns that might pose challenges for other methodologies. Additionally, the improved Sardar subequation method offers 14 unique solution forms, allowing for the derivation of a greater number of exact solutions compared to other methods [40,41,42,43,44,45].

In contrast, the ℛ ′ ℛ , 1 ℛ method operates with two variables. It provides a comprehensive framework for exploring various transformations and solution strategies, thereby offering insights into the behavior and properties of solutions. Notably, this method encompasses trigonometric, hyperbolic, and rational solutions, making it highly versatile. Overall, the ℛ ′ ℛ , 1 ℛ method serves as a valuable tool for efficiently addressing nonlinear equations and obtaining meaningful results. Besides, the ℛ ′ ℛ , 1 ℛ -expansion method is an influential analytical approach used for tackling nonlinear PDEs [46,47,48,49,50].

To date, no exploration of Eq. (3) has employed these particular methods, representing a significant gap in the current literature. This research aims to fill this void by applying these techniques to obtain exact solutions for this equation, offering new insights into its behavior and characteristics. By leveraging these methodologies, we seek to uncover novel soliton solutions and analyze their properties in depth. The manuscript is organized to systematically present our approach and findings, ensuring a comprehensive exploration of the equation and its implications. The structure of this manuscript is designed to guide the reader through the methodology, results, and analysis, culminating in a thorough comparison with the existing research and a detailed discussion of the outcomes. The manuscript is organized in the following manner: Section 2 provides a short description of methodologies used in this study. Section 3 details how these methodologies are employed to derive the solutions. Section 4 presents a bifurcation analysis. Section 5 explores dynamic representations, showcasing several soliton solutions through 3D, 2D, and density plots, accompanied by detailed discussions. Section 6 compares our findings with those reported in the existing literature. The overall conclusion is presented in Section 7. Finally, the bibliography of references is provided.

2 Methodology

2.1 Improved Sardar subequation method

This section explores the improved Sardar subequation method, which is renowned for providing accurate solutions to various nonlinear PDEs. Our analysis centers on the typical structure of nonlinear PDEs, usually defined by the independent variables x and t , as shown below [51,52]:

(4) L ( v , v x , v y , v z , v xx , v yy , v zz , v xy , v yz , v zx , … … … . ) = 0 ,

where L is a polynomial.

Let us introduce a new variable θ that transforms Eq. (4) into an ODE form as follows:

(5) v ( x , y , z , t ) = U ( θ ) , θ = ( α 1 x + α 2 y + α 3 z − ϑ t ) ,

where α 1 , α 2 , and α 3 are the arbitrary constants and ϑ is the new wave celerity constant.

Introducing Eq. (5) in Eq. (3), Eq. (4) is converted into the following manner:

(6) T ( U , U ′ , U ″ , U ‴ , … … … , … … … . ) = 0 ,

where T is the new polynomial with ordinary derivatives of the functions.

Eq. (6) delivers the following general solution (GS) forms:

(7) U ( θ ) = ∑ i = 0 N a i V i ( θ ) ,

succeeding in:

(8) V ′ ( θ ) = V 4 ( θ ) + ρ V 2 ( θ ) + τ ,

where a i (i = 0, 1, 2, 3, …, n); ρ and τ are unknown constants; N is the balance number.

Depending on the values of ρ and τ, Eq. (7) provides the following sets of solutions:

Case I

If ρ > 0 but τ = 0 , then:

(9) V 1 ± ( θ ) = ± g l ρ cosec h g l ( ρ θ ) ,

(10) V 2 ± ( θ ) = ± − g l ρ sec h g l ( ρ θ ) ,

where cosec h g l ( θ ) = 2 g e θ − l e − θ and sec h g l ( θ ) = 2 g e θ + l e − θ .

Case II

If ρ < 0 but τ = 0 , then:

(11) V 3 ± ( θ ) = ± − g l ρ cosec g l ( − ρ θ ) ,

(12) V 4 ± ( θ ) = ± − g l ρ sec g l ( − ρ θ ) ,

where cosec g l ( θ ) = 2 i g e i θ − l e − i θ and sec g l ( θ ) = 2 g e i θ + l e − i θ .

Case III

If ρ < 0 but τ = ρ 2 4 , then:

(13) V 5 ± ( θ ) = ± − ρ 2 tan h g l − ρ 2 θ ,

(14) V 6 ± ( θ ) = ± − ρ 2 cot h g l − ρ 2 θ ,

(15) V 7 ± ( θ ) = ± − ρ 2 { tan h g l ( − 2 ρ θ ) ± i g l sec h g l ( − 2 ρ θ ) } ,

(16) V 8 ± ( θ ) = ± − ρ 2 { cot h g l ( − 2 ρ θ ) ± g l cosec h g l ( − 2 ρ θ ) } ,

(17) V 9 ± ( θ ) = ± − ρ 8 tan h g l − ρ 8 θ + g l cot h g l − ρ 8 θ ,

where tan h g l ( θ ) = g e θ − l e − θ g e θ + l e − θ and cot h g l ( θ ) = g e θ + l e − θ g e θ − l e − θ .

Case IV

If ρ > 0 but τ = ρ 2 4 , then:

(18) V 10 ± ( θ ) = ± ρ 2 tan g l ρ 2 θ ,

(19) V 11 ± ( θ ) = ± ρ 2 cot g l ρ 2 θ ,

(20) V 12 ± ( θ ) = ± ρ 2 { tan g l ( 2 ρ θ ) ± g l sec g l ( 2 ρ θ ) } ,

(21) V 13 ± ( θ ) = ± − ρ 2 { cot g l ( 2 ρ θ ) ± g l cosec g l ( 2 ρ θ ) } ,

(22) V 14 ± ( θ ) = ± ρ 8 tan g l − ρ 8 θ − g l cot g l ρ 8 θ ,

where tan g l ( θ ) = − i g e i θ − l e − i θ g e i θ + l e − i θ , and cot g l ( θ ) = i g e i θ + l e − i θ g e i θ − l e − i θ .

It is noted that in Eqs. (9)–(22), the parameters g and l are arbitrary nonzero constants. When g and l are both set to 1 or −1, these equations reduce to familiar trigonometric and hyperbolic forms [53]. Specifically, the solution given in Eq. (17) matches the solution from Eq. (14), and the solutions in Eqs. (22) and (21) are equivalent to the solution in Eq. (19). Consequently, we have V 6 ± ( θ ) = V 9 ± ( θ ) and V 11 ± ( θ ) = V 13 ± ( θ ) = V 14 ± ( θ ) .

To derive the exact solution of Eq. (3) using this method, we start by identifying the balance number N through the homogeneous balance method. Subsequently, we substitute N into Eq. (6) and integrate this transformed equation into Eq. (6) via Eq. (8). Therefore, the left-hand side of Eq. (6) is in a polynomial form. By equating coefficients of the corresponding power terms to zero, we establish a set of algebraic equations. By working out these equations, the value of the unknown coefficients is figured out. Switching these coefficients back into Eq. (6) supplies the exact solutions of Eq. (3) in the 14 distinct arrangements outlined in the above section.

2.2 ℛ ′ ℛ , 1 ℛ -expansion method

Similarly, this method transforms nonlinear PDEs into ODEs by introducing a transformation variable, as defined in Eq. (4). By rationalizing the analytical procedure, this approach yields a linear ODE, expressed as follows [54,55]:

(23) ℛ ″ ( θ ) + λ ℛ ( θ ) = η ,

subject to

(24) ℒ 1 ( θ ) = ℛ ′ ( θ ) ℛ ( θ ) ℒ 2 ( θ ) = 1 ℛ ( θ ) ,

subsequent in

(25) ℒ 1 ′ = ‒ ℒ 1 2 + η ℒ 2 − λ ℒ 2 ′ = − ℒ 1 × ℒ 2 .

Depending on λ, this method provides the following solutions:

Case I

When λ > 0 :

(26) ℛ ( θ ) = d 1 sin ( θ λ ) + d 2 cos ( θ λ ) + η / λ ,

which generates ℒ 2 2 = ℒ 1 2 − 2 η ℒ 2 + λ A 1 λ 2 − η 2 λ , where A 1 = d 1 2 + d 2 2 .

Case II

When λ < 0 :

(27) ℛ ( θ ) = d 1 sin h ( θ − λ ) + d 2 cos h ( θ − λ ) + η / λ ,

subsequent in ℒ 2 2 = − λ ℒ 1 2 − 2 η ℒ 2 + λ A 2 λ 2 + η 2 , where A 2 = d 1 2 − d 2 2 .

Case III

When λ = 0 :

(28) ℛ ( θ ) = η θ 2 2 + d 1 θ + d 2 ,

which gives ℒ 2 2 = ℒ 1 2 − 2 η ℒ 2 d 1 2 − 2 η d 2 .

The GSs provided by this method are as follows:

(29) ℛ ( θ ) = b 0 + ∑ j = 1 N b j ℒ 1 j ( θ ) + ∑ j = 1 P c j ℒ 1 j − 1 ( θ ) ℒ 2 ( θ ) ,

where b 0 , b j , and c j (j = 1, 2, 3, …, N) are the arbitrary constants, guaranteeing that b N 2 + c N 2 ≠ 0 , and N is the balance number.

To derive the exact solution using this approach, we first employ the homogeneous balance method to determine the balance number. Once identified, this number is incorporated into the GS format along with Eqs. (24) and (25), which include the transformation variable. This step results in a new polynomial with specific coefficients. By equating the coefficients of each term to zero according to their respective powers in the polynomial, we formulate a system of algebraic equations. Working out these equations provides the unknown coefficient value. Afterward, substituting these into Eq. (29) allows us to extract soliton solutions for Eq. (3) articulated in trigonometric functions (as shown in Eq. (26)), hyperbolic functions (as in Eq. (27)), and rational functions (as given by Eq. (28)).

3 Application

Applying the transformation method as detailed in Section 2, we convert it into an ODE, resulting in

(30) U ″ ( − ϑ α 1 + α 1 α 2 + α 3 α 1 + α 2 α 3 + α 1 2 + α 2 2 + α 3 2 ) + U U ″ ( 2 α 1 α 2 + α 1 2 ) + 6 α 1 2 U 2 U ″ + 2 α 1 4 ( U ″ ) 2 .

Now, we can apply the methodology described in Section 2 in the following steps.

3.1 Improved Sardar subequation method

By applying the homogeneous balance method to Eq. (30), we derive N = 2 , which converted the GS of this method in the following format:

(31) U ( θ ) = a 0 + a 1 V ( θ ) + a 2 V 2 ( θ ) .

Inserting this into Eq. (31) results in

(32) a 0 = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 a 1 = 0 a 2 = − 2 α 1 2 ϑ = 23 α 1 2 + 20 α 1 α 2 + 20 α 2 2 + 24 α 1 α 3 + 24 α 2 α 3 + 24 α 3 2 + 64 α 1 6 ρ 2 − 192 α 1 6 τ 24 α 1 .

Finally, the exact solutions are as follows:

Case I:

(33) U 1 ± ( θ ) = v 1 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 { ± g l ρ cosec h g l ( ρ θ ) } 2 ,

(34) U 2 ± ( θ ) = v 2 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 { ± − g l ρ sec h g l ( ρ θ ) } 2 .

Case II:

(35) U 3 ± ( θ ) = v 3 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 { ± − g l ρ cosec g l ( − ρ θ ) } 2 ,

(36) U 4 ± ( θ ) = v 4 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 { ± − g l ρ sec g l ( − ρ θ ) } 2 .

Case III:

(37) U 5 ± ( θ ) = v 5 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 ± − ρ 2 tan h g l − ρ 2 θ 2 ,

(38) U 6 ± ( θ ) = v 6 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 ± − ρ 2 cot h g l − ρ 2 θ 2 ,

(39) U 7 ± ( θ ) = v 7 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 ± − ρ 2 { tan h g l ( − 2 ρ θ ) ± i g l sec h g l ( − 2 ρ θ ) } 2 ,

(40) U 8 ± ( θ ) = v 8 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 ± − ρ 2 { cot h g l ( − 2 ρ θ ) ± g l cosec h g l ( − 2 ρ θ ) } 2 ,

(41) U 9 ± ( θ ) = v 9 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 ± − ρ 8 tan h g l − ρ 8 θ + cot h g l − ρ 8 θ 2 .

Case IV:

(42) U 10 ± ( θ ) = v 10 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 ± ρ 2 tan g l ρ 2 θ 2 ,

(43) U 11 ± ( θ ) = v 11 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 ± ρ 2 cot g l ρ 2 θ 2 ,

(44) U 12 ± ( θ ) = v 12 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 ± ρ 2 { tan g l ( 2 ρ θ ) ± g l sec g l ( 2 ρ θ ) } 2 ,

(45) U 13 ± ( θ ) = v 13 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 ± ϑ 2 { cot g l ( 2 ρ θ ) ± g l cosec g l ( 2 ρ θ ) } 2 ,

(46) U 14 ± ( θ ) = v 14 ( x , y , z , t ) = − α 1 − 2 α 2 − α 1 3 ρ 12 α 1 − 2 α 1 2 ± ρ 8 tan g l ρ 8 θ − cot g l ρ 8 θ 2 ,

where θ = ( α 1 x + α 2 y + α 3 z − ϑ t ) , and ϑ comes from the last equation of the system of Eq. (32) for all cases.

3.2 R ′ R , 1 R -expansion method

Since the balance number is 2, the GS of this method is formulated as follows:

(47) U ( θ ) = b 0 + b 1 ℒ 1 ( θ ) + c 1 ℒ 2 ( θ ) + b 2 ℒ 1 2 ( θ ) + c 2 ℒ 1 ( θ ) ℒ 2 ( θ ) .

3.2.1 For trigonometric solutions

Applying the described methodology for λ > 0 , we identified a set of algebraic equations, solving which yielded the following values for the unknown coefficients:

(48) b 0 = − α 1 − 2 α 2 − 10 α 1 3 λ 12 α 1 b 1 = 0 b 2 = − α 1 2 c 1 = η α 1 2 c 2 = ± α 1 2 − η 2 + λ 2 A 1 λ ϑ = 23 α 1 2 + 20 α 1 α 2 + 20 α 2 2 + 24 α 1 α 3 + 24 α 2 α 3 + 24 α 3 2 + 4 α 1 6 λ 2 24 α 1 .

Substituting these values into Eq. (47), then into Eq. (30), and finally into Eq. (3) yields the following GS:

(49) U ( θ ) = v ( x , y , z , t ) = − α 1 − 2 α 2 − 10 α 1 λ 12 α 1 + η α 1 2 1 d 1 sin ( θ λ ) + d 2 cos ( θ λ ) + η / λ − λ α 1 2 d 1 cos ( θ λ ) − d 2 sin ( θ λ ) d 1 sin ( θ λ ) + d 2 cos ( θ λ ) + η / λ 2 ± α 1 2 − η 2 + λ 2 A 1 × d 1 cos ( θ λ ) − d 2 sin ( θ λ ) { d 1 sin ( θ λ ) + d 2 cos ( θ λ ) + η / λ } 2 ,

where θ = ( α 1 x + α 2 y + α 3 z − ϑ t ) , and ϑ comes from the last equation of the system of Eq. (48).

Setting both η = d 2 = 0 but d 1 ≠ 0, Eq. (49) yields

(50) U ( θ ) = v ( x , y , z , t ) = − α 1 − 2 α 2 − 10 α 1 λ 12 α 1 − λ α 1 2 { cot ( θ λ ) } 2 ± λ α 1 2 cot ( θ λ ) cosec ( θ λ ) ,

where θ = ( α 1 x + α 2 y + α 3 z − ϑ t ) , and ϑ comes from the last equation of the system of Eq. (48).

Similarly, for η and d 1 equal to zero but d 2 ≠ 0 , Eq. (49) yields

(51) U ( θ ) = v ( x , y , z , t ) = − α 1 − 2 α 2 − 10 α 1 λ 12 α 1 − λ α 1 2 { tan ( θ λ ) } 2 ± λ α 1 2 tan ( θ λ ) sec ( θ λ ) .

3.2.2 For hyperbolic solutions

Using the described methodology for λ < 0 , we established a system of algebraic equations. Solving these equations provided the following values for the unknown coefficients:

(52) b 0 = − α 1 − 2 α 2 − 102 α 1 3 λ 12 α 1 b 1 = 0 b 2 = − α 1 2 c 1 = η α 1 2 c 2 = ± α 1 2 η 2 + λ 2 A 2 − λ ϑ = 23 α 1 2 + 20 α 1 α 2 + 20 α 2 2 + 24 α 1 α 3 + 24 α 2 α 3 + 24 α 3 2 + 4 α 1 6 λ 2 24 α 1 .

By substituting these values into Eq. (47), then into Eq. (30), and finally into Eq. (3), we arrive at the following GS:

(53) U ( θ ) = v ( x , y , z , t ) = − α 1 − 2 α 2 − 10 α 1 λ 12 α 1 + η α 1 2 1 d 1 sin h ( θ − λ ) + d 2 cos h ( θ − λ ) + η / λ + λ α 1 2 d 1 cos h ( θ − λ ) + d 2 sin h ( θ − λ ) d 1 sin h ( θ − λ ) + d 2 cos h ( θ − λ ) + η / λ 2 ± α 1 2 η 2 + λ 2 A 2 − λ − λ d 1 cos h ( θ − λ ) + d 2 sin h ( θ − λ ) d 1 sin h ( θ − λ ) + d 2 cos h ( θ − λ ) + η / λ × 1 d 1 sin h ( θ − λ ) + d 2 cos h ( θ − λ ) + η / λ ,

where θ = ( α 1 x + α 2 y + α 3 z − ϑ t ) , and ϑ comes from the last equation of the system of Eq. (52).

In particular, setting both η = d 2 = 0 but d 1 ≠ 0, Eq. (53) yields

(54) U ( θ ) = v ( x , y , z , t ) = − α 1 − 2 α 2 − 10 α 1 λ 12 α 1 + λ α 1 2 { cot h ( θ − λ ) } 2 ± λ α 1 2 cot h ( θ − λ ) cosec h ( θ − λ ) ,

where θ = ( α 1 x + α 2 y + α 3 z − ϑ t ) , and ϑ comes from the last equation of the system of Eq. (52).

3.2.3 For rational solutions

Similarly, for λ = 0 yields

(55) b 0 = − α 1 − 2 α 2 12 α 1 b 1 = 0 b 2 = − α 1 2 c 1 = η α 1 2 c 2 = ± α 1 2 d 1 2 − 2 η d 2 ϑ = 23 α 1 2 + 20 α 1 α 2 + 20 α 2 2 + 24 α 1 α 3 + 24 α 2 α 3 + 24 α 3 2 24 α 1 .

Putting this value in Eq. (47), then into Eq. (30), and finally into Eq. (3) gives the following GS:

(56) U ( θ ) = v ( x , y , z , t ) = − α 1 − 2 α 2 12 α 1 + η α 1 2 1 η θ 2 2 + d 1 θ + d 2 − α 1 2 2 η θ + d 1 η θ 2 2 + d 1 θ + d 2 2 ± α 1 2 d 1 2 − 2 η d 2 2 η θ + d 1 η θ 2 2 + d 1 θ + d 2 1 η θ 2 2 + d 1 θ + d 2 ,

where θ = ( α 1 x + α 2 y + α 3 z − ϑ t ) , and ϑ comes from the last equation of the system of Eq. (55).

If, setting both η = d 2 = 0 but d 1 ≠ 0, Eq. (56) yields

(57) U ( θ ) = v ( x , y , z , t ) = − α 1 − 2 α 2 12 α 1 − α 1 2 1 θ 2 ± α 1 2 1 θ 2 ,

where θ = ( α 1 x + α 2 y + α 3 z − ϑ t ) , and ϑ comes from the last equation of the system of Eq. (55).

If we take only a positive value, it is transferred to the constant solution.

4 Analysis

4.1 Bifurcation

Consider the planar dynamical system described by the following equations (via Galilean conversion of ODE) [56,57]:

(58) U ′ = P P ′ = − G U 2 − FU − E ,

where U and P are state variables, and G = 3 α 1 2 , F = α 1 2 − 2 α 1 α 2 2 α 1 2 , and E = − v α 1 + α 1 2 + α 1 α 2 + α 2 2 + α 1 α 3 + α 2 α 3 + α 3 2 2 α 1 2 are arbitrary constants. The objective is to analyze this system by determining the Jacobian matrix, the eigenvalues, and the equilibrium points. This analysis will help identify the nature of the equilibrium points, such as whether they are saddle points, center points, or cuspidal points.

Eq. (58) presents the following Hamiltonian functions:

(59) H ( U , P ) = 1 2 P 2 + EU + 1 2 U 2 + U 3 = h .

The first term 1 2 P 2 of Eq. (59) is the kinetic energy, EU + 1 2 U 2 + U 3 is the potential energy of the system, and h is the Hamiltonian constant.

Now, Eq. (58) yields the following value for the Jacobian matrix:

(60) J ( U , P ) = 0 1 − 6 U − F 0 = − 6 U − F .

Therefore, the equilibrium points occur when the system is set to zero. Solving the system (Eq. (58)) yields the equilibrium points ( e 1 , e 2 ) as follows:

(61) e 1 = − F + F 2 − 12 E 6 e 2 = − F − F 2 − 12 E 6 .

Therefore, after solving the characteristic equation | J ( U , P ) − Y I | yields the eigenvalues ( Y 1 , Y 2 ) as follows:

(62) Y 1 = 6 U e + F Y 2 = − 6 U e + F .

So, the equilibrium points are categorized into the following cases:

Saddle point: If the eigenvalues have opposite signs.

Center point: If the eigenvalues are complex conjugates with a negative real part.

Cuspidal point: If the eigenvalues are real and equal.

Case I (saddle point): If F 2 − 12 E > 0 , at any U , the eigenvalues are real and distinct, representing the equilibrium point as the saddle point. This scenario is presented in Figure 1(a), where one point is the saddle point and another one is the center point. The corresponding values of the parameters are given in the caption of the figure.

Figure 1 Phase diagram of Eq. (1) depicting (a) one saddle point (*) and one center point (○), where α 1 = α 2 = α 3 = 1 , ϑ = 10 {\alpha }_{1}={\alpha }_{2}={\alpha }_{3}=1,\hspace{.5em}{\vartheta }=10 and (b) both are center points (○), where α 1 = α 3 = 1 , α 2 = ‒ 0.5 , ϑ = 1 . {\alpha }_{1}={\alpha }_{3}=1,\hspace{.5em}{\alpha }_{2}=&#x2012;0.5,\hspace{.25em}{\vartheta }=1.
Figure 1

Phase diagram of Eq. (1) depicting (a) one saddle point (*) and one center point (○), where α 1 = α 2 = α 3 = 1 , ϑ = 10 and (b) both are center points (○), where α 1 = α 3 = 1 , α 2 = ‒ 0.5 , ϑ = 1 .

Case II (center point): If F 2 − 12 E < 0 , at any U , the eigenvalues are purely imaginary, representing the equilibrium point as the center point. This scenario is presented in Figure 1(b). The corresponding values of the parameters are given in the caption of the figure.

In Figure 1, both saddle points and center points have specific physical interpretations. A saddle point represents an equilibrium where the system has both stable and unstable directions; perturbations in some directions will diminish, leading to stability, while in others they will increase, causing instability. This indicates regions where the system’s behavior can significantly change, such as transitioning between different wave solutions or experiencing substantial shifts in wave characteristics. On the other hand, a center point represents an equilibrium where nearby trajectories form closed orbits, signifying periodic or quasi-periodic behavior. This means the system tends to display stable, oscillatory solutions, with consistent wave patterns or solitons that remain intact over time. These points highlight how the system’s response evolves with varying parameters, underscoring regions of stability and oscillation within this equation’s dynamics.

4.2 Modulation instability

Modulation instability, marked by the rapid amplification of small disturbances on a continuous wave, reveals the complex interactions within nonlinear systems influenced by dispersion and nonlinear effects. This process often results in the generation of sidebands around the original wave frequency, forming intricate patterns such as solitons and pulses. Understanding modulation instability is crucial in fields like optics and plasma physics for accurately predicting and controlling wave behaviors [58,59,60,61]. Applying these principles to the modified Sakovich equation supplies valuable insights into the stability and dynamics of its solutions, refining models to capture real-world phenomena across diverse scientific disciplines better. Let us assume the steady-state solution with a small perturbation of the following form:

(63) v = Ω + ε ( x , t ) ,

where ε ( x , t ) is the small perturbation.

Now, substituting this in Eq. (1) and linearizing, considering ignoring the higher order term, yields:

(64) ε x t + ( 1 + 12 Ω 2 + 6 Ω ) ε x x + ε y y + ε z z + ( 1 + 2 Ω ) ε x y + ε y z + ε x z = 0 .

Let us assume a perturbation solution of the form:

(65) ε ( x , t ) = ε 0 e i ( κ x + σ t ) ,

where ε 0 is the perturbation amplitude, κ is the wave number, and σ is the frequency of perturbation.

Substituting this in Eq. (64) and performing some simplification yield

(66) σ = − i κ ( 2 + 12 Ω 2 + 8 Ω ) .

The analysis suggests that the given PDE exhibits modulation instability for any non-zero steady-state solution amplitude Ω . This instability results in the growth of perturbations, potentially leading to the formation of localized structures or patterns. The following figure visually demonstrates how small perturbations in the initial wave can amplify over time, leading to instability during wave propagation. Significant changes in amplitude from the initial state to the propagated state indicate that the wave is experiencing modulation instability due to nonlinear effects (Figure 2).

Figure 2 Modulation instability of Eq. (3) with small perturbation ε = 0.01 . \varepsilon =0.01.
Figure 2

Modulation instability of Eq. (3) with small perturbation ε = 0.01 .

5 Illustration of the exact solutions

To illustrate the exact solutions of the extended Sakovich equation, we employed Mathematica, a powerful computational tool known for its advanced capabilities in mathematical computation and visualization. Our presentation encompasses a diverse array of visualizations, including 3D renderings, 2D plots, and density plots, providing a comprehensive view of the solutions’ behaviors and characteristics.

We have carefully chosen four solutions derived from the improved Sardar subequation method for every single case. This method is known for its effectiveness in handling nonlinear differential equations, allowing us to capture a wide range of solution behaviors. Additionally, we chose three solutions which are derived from the ℛ ′ ℛ , 1 ℛ -expansion technique for each case. This method provides a different analytical approach, further enriching our set of solutions and offering varied perspectives on the equation’s dynamics.

The visualizations span the interval −20 ≤ (x, t) ≤ 20, ensuring a broad and detailed view of the solutions across significant ranges of the variables x and t. For each case, we have provided the subsequent values of the parameter in the figure descriptions, allowing for precise replication and verification of the results.

In our 2D plots, we have synthesized multiple graphs into a unified figure, manipulating the parameter t across the variations. This approach enables a comparative analysis of how the solutions evolve over time, offering insights into the temporal dynamics and stability of the solutions. By integrating these multiple visual perspectives, we have supplied a complete and nuanced insight of the extended Sakovich equation’s solutions.

5.1 Visualization through the improved Sardar subequation method

5.2 Visualization via the ℛ ′ ℛ , 1 ℛ approach

5.3 Analysis of figures

This study presents a series of figures depicting numerous soliton solutions achieved via two distinct methods: the improved Sardar subequation method and the ℛ ′ ℛ , 1 ℛ technique. Figures 3–6, using the Sardar subequation technique, showcase diverse soliton behaviors influenced by different parameters. For instance, Figure 3 displays a multi-singular soliton; however, Figure 4 features a V-type and bell-type soliton. Additionally, Figures 5 and 6 illustrate parabolic and periodic singular soliton structures, respectively. These soliton characteristics are mostly governed by different parameters, for instance α 1 , α 2 , α 3 , and ρ , where specific values dictate the type and shape of solitons observed. Similarly, Figures 7–9, employing the ℛ ′ ℛ , 1 ℛ method, highlight different soliton structures, including the Kuznetsov–Ma Breather soliton (Figure 7), complex multi-singular solitons (Figure 8), and kink-like solitons (Figure 9). Here, parameters like λ, α 1 , α 2 and α 3 play significant roles in shaping soliton behavior, with α 1 particularly influential near 1 and less so as it diverges from this value. This comparative approach underscores how distinct mathematical techniques and parameter variations contribute to understanding and visualizing soliton dynamics in the context of the extended Sakovich equation.

Figure 3 Visualizing the solution that consists of multi-singular solitons across different plots, where α 1 = 4.00 , α 2 = α 3 = 1.00 , g = 1.00 {\alpha }_{1}=4.00,\hspace{.5em}{\alpha }_{2}={\alpha }_{3}=1.00,\hspace{.5em}g=1.00 , ℓ = 1.00 , z = t = 1.00 \ell =1.00,\hspace{.5em}z=t=1.00 , and ρ = 0.10 \rho =0.10 . (a) 3D view, (b) 2D view, and (c) density plot.
Figure 3

Visualizing the solution that consists of multi-singular solitons across different plots, where α 1 = 4.00 , α 2 = α 3 = 1.00 , g = 1.00 , ℓ = 1.00 , z = t = 1.00 , and ρ = 0.10 . (a) 3D view, (b) 2D view, and (c) density plot.

Figure 4 Visualizing the solution that consists of V-type and bell-type solitons across different plots, where α 1 = α 2 = α 3 = 0.10 , g = 0.01 {\alpha }_{1}={\alpha }_{2}={\alpha }_{3}=0.10,\hspace{.5em}g=0.01 , ℓ = 0.10 , z = t = 1.00 \ell =0.10,\hspace{.5em}z=t=1.00 , and ρ = ‒ 0.05 \rho =&#x2012;0.05 . (a) 3D view, (b) 2D view, and (c) density plot.
Figure 4

Visualizing the solution that consists of V-type and bell-type solitons across different plots, where α 1 = α 2 = α 3 = 0.10 , g = 0.01 , ℓ = 0.10 , z = t = 1.00 , and ρ = ‒ 0.05 . (a) 3D view, (b) 2D view, and (c) density plot.

Figure 5 Visualizing the solution that consists of parabolic solitons across different plots, where α 1 = 0.10 , α 2 = ‒ 0.10 , α 3 = 0.01 , {\alpha }_{1}=0.10,{\alpha }_{2}=&#x2012;0.10,\hspace{.25em}{\alpha }_{3}=0.01, g = ℓ = 1.00 , z = t = 1.00 g=\ell =1.00,z=t=1.00 , and ρ = ‒ 0.01 \rho =&#x2012;0.01 . (a) 3D view, (b) 2D view, and (c) density plot.
Figure 5

Visualizing the solution that consists of parabolic solitons across different plots, where α 1 = 0.10 , α 2 = ‒ 0.10 , α 3 = 0.01 , g = ℓ = 1.00 , z = t = 1.00 , and ρ = ‒ 0.01 . (a) 3D view, (b) 2D view, and (c) density plot.

Figure 6 Visualizing the solution that consists of periodic singular solitons across different plots, where α 1 = α 2 = 0.10 , α 3 = 1.00 , z = t = 1.00 {\alpha }_{1}={\alpha }_{2}=0.10,\hspace{.25em}{\alpha }_{3}=1.00,\hspace{.5em}z=t=1.00 , and ρ = 0.50 \rho =0.50 . (a) 3D view, (b) 2D view, and (c) density plot.
Figure 6

Visualizing the solution that consists of periodic singular solitons across different plots, where α 1 = α 2 = 0.10 , α 3 = 1.00 , z = t = 1.00 , and ρ = 0.50 . (a) 3D view, (b) 2D view, and (c) density plot.

Figure 7 Visualizing the solution that consists of Kuznetsov–Ma Breather solitons across different plots, where α 1 = 1.00 , α 2 = α 3 = ‒ 1.00 , d 1 = 1.10 {\alpha }_{1}=1.00,\hspace{.5em}{\alpha }_{2}={\alpha }_{3}=&#x2012;1.00,\hspace{.5em}{d}_{1}=1.10 , d 2 = 1.00 λ = ‒ 5.10 , z = t = 1.00 {d}_{2}=1.00\lambda =&#x2012;5.10,\hspace{.5em}z=t=1.00 , and η = 0.10 \eta =0.10 . (a) 3D view, (b) 2D view, and (c) density plot.
Figure 7

Visualizing the solution that consists of Kuznetsov–Ma Breather solitons across different plots, where α 1 = 1.00 , α 2 = α 3 = ‒ 1.00 , d 1 = 1.10 , d 2 = 1.00 λ = ‒ 5.10 , z = t = 1.00 , and η = 0.10 . (a) 3D view, (b) 2D view, and (c) density plot.

Figure 8 Visualizing the solution that consists of complex multi-singular solitons across different plots, where α 1 = 1.00 , α 2 = α 3 = ‒ 1.00 , d 1 = 0.20 {\alpha }_{1}=1.00,\hspace{.5em}{\alpha }_{2}={\alpha }_{3}=&#x2012;1.00,\hspace{.5em}{d}_{1}=0.20 , d 2 = 0.10 , λ = ‒ 5.00 , z = t = 1.00 , {d}_{2}=0.10,\hspace{.25em}\lambda =&#x2012;5.00,\hspace{.25em}z=t=1.00, and η = 1.00 \eta =1.00 . (a) 3D view, (b) 2D view, and (c) density plot.
Figure 8

Visualizing the solution that consists of complex multi-singular solitons across different plots, where α 1 = 1.00 , α 2 = α 3 = ‒ 1.00 , d 1 = 0.20 , d 2 = 0.10 , λ = ‒ 5.00 , z = t = 1.00 , and η = 1.00 . (a) 3D view, (b) 2D view, and (c) density plot.

Figure 9 Visualizing the solution that consists of dark-type kink solitons across different plots, where α 1 = α 2 = α 3 = 1.00 , d 1 = 0.20 {\alpha }_{1}={\alpha }_{2}={\alpha }_{3}=1.00,\hspace{.5em}{d}_{1}=0.20 , d 2 = 0.20 z = t = 1.00 {d}_{2}=0.20z=t=1.00 , and η = 1.00 \eta =1.00 . (a) 3D view, (b) 2D view, and (c) density plot.
Figure 9

Visualizing the solution that consists of dark-type kink solitons across different plots, where α 1 = α 2 = α 3 = 1.00 , d 1 = 0.20 , d 2 = 0.20 z = t = 1.00 , and η = 1.00 . (a) 3D view, (b) 2D view, and (c) density plot.

The solutions derived from Eq. (3) display a variety of forms, including the multi-singular soliton, V-type and bell-type soliton, parabolic soliton, periodic singular soliton, Kuznetsov–Ma Breather soliton, complex multi-singular soliton, and kink-like soliton. These diverse forms of solitons find broad applications across scientific and engineering disciplines. The multi-singular soliton plays a key role in studying wave interactions in nonlinear systems, applicable from optics to plasma physics. V-type and bell solitons are essential in optical communications for their ability to maintain stable waveforms, ensuring reliable data transmission over long distances. Parabolic solitons are significant in fluid dynamics, where they model shallow water waves and predict surface disturbances in natural water bodies. In nonlinear optics, periodic singular solitons are crucial for examining periodic disturbances in electromagnetic fields. Kuznetsov–Ma Breather solitons find practical use in plasma physics and oceanography, providing insights into periodic oscillations and wave behaviors. Complex multi-singular solitons are employed in condensed matter physics to explore intricate interactions in materials science. Kink-like solitons are fundamental in field theory and nonlinear optics, offering insights into phase transitions and dynamics of domain walls in various materials. These solitons contribute significantly to advancing our understanding of nonlinear phenomena and driving innovations across diverse scientific and technological domains.

6 Comparison

This section underscores the originality and scientific insights of our study by judging our results with those reported by Ali et al., Arnous et al., and Alqahtani and Kaplan [8,10,11]. The difference is keenly divided into two parts, examining both the resemblances and variations between our research methodologies. By clarifying these aspects, we aim to offer a comprehensive view of the new understanding provided to the field.

6.1 Equalities

  1. All of the studies focus on the extended Sakovich equation, an essential model used to investigate solitary wave solutions.

  2. All of the studies aim to derive exact solutions through analytical methods.

6.2 Variations and uniqueness

  1. Other studies have employed various methods, such as the exp (−ψ(η)) expansion technique, the G ′ k G ′ + G + r expansion technique, the Bernoulli sub-ODE method [8], the generalized exponential rational function method [11], and the advanced direct algebraic equation method [10], all of which derive solutions using one variable. In contrast, our study used two effective techniques: the improved Sardar subequation technique (single variable expansion method) and the ℛ ′ ℛ , 1 ℛ -expansion technique (double variable expansion method).

  2. Although previous research primarily concentrated on identifying various types of solitons, such as multiple singular, singular, dark, bright, and periodic solitons, our study broadens this scope to include a more diverse range of solitons. This includes singular, multi-singular, periodic singular, kink, anti-kink, bell-shaped, Kuznetsov–Ma Breather, and parabolic-shaped solitons. Among these, the periodic singular, bell-shaped, Kuznetsov–Ma Breather, and parabolic-shaped solitons introduced in our research represent significant and novel contributions within this equation.

  3. Furthermore, this study performs bifurcation and modulation instability analyses on the equation. The bifurcation analysis provides insights into the phase portrait, while the modulation instability analysis reveals stable dispersion during wave propagation. These analyses offer unique contributions, as they are not typically included in similar studies.

7 Conclusions

This study focused on the extended Sakovich equation, emphasizing the crucial role of soliton solutions in predicting and controlling localized wave behaviors, thereby advancing nonlinear dynamics and related applications. Effective bifurcation analysis confirmed the saddle point and center point behaviors within the equation’s phase portrait. Modulation instability analyses revealed a stable dispersion pattern for this equation. The improved Sardar subequation method and the ℛ ′ ℛ , 1 ℛ -expansion method produced a diverse spectrum of waveforms – hyperbolic, trigonometric, and rational functions – that were rigorously validated using Mathematica software for accuracy. Graphical representations vividly illustrated various soliton patterns, including singular, multi-singular, periodic singular, kink, anti-kink, bell-shaped, Kuznetsov–Ma Breather, and parabolic-shaped solitons, highlighting their effectiveness in revealing innovative solutions. The comparative analysis confirmed the novelty of our findings, underscoring a significant contribution to advancing soliton solutions for the Sakovich equation. While this study offers valuable insights into soliton dynamics within the Sakovich equation, it has some limitations. Its applicability is generally confined to idealized conditions, and extending its use to non-integrable systems presents challenges. Additionally, practical applications are limited, and deriving analytical solutions can be difficult. Moreover, experimental validation of theoretical results is often lacking. Future research should address these limitations to enhance practical applicability and validate theoretical findings. Overall, this research advances the soliton theory by demonstrating its broad applicability in fields like fluid dynamics, nonlinear optics, and plasma physics, effectively modeling phenomena such as solitons and shock waves. The study also offers key insights into wave propagation in biological systems and acoustics. The versatility and robustness of soliton solutions provide promising opportunities for future research in nonlinear dynamics, enhancing our ability to model and control complex wave behavior.

Acknowledgments

The authors would like to acknowledge the Deanship of Graduate Studies and Scientific Research, Taif University for funding this work.

  1. Funding information: The authors would like to acknowledge the Deanship of Graduate Studies and Scientific Research, Taif University for funding this work.

  2. Author contributions: 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 analyzed during this study are included in this published article.

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Received: 2024-07-03
Revised: 2024-10-10
Accepted: 2024-10-31
Published Online: 2024-11-22

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

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

Artikel in diesem Heft

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https://doi.org/10.1515/phys-2024-0096
Schlagwörter für diesen Artikel
nonlinear partial differential equations; improved Sardar subequation technique; (ℛ′/ℛ,1/ℛ) technique; the extended Sakovich equation; soliton solutions
Creative Commons
BY 4.0

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
  56. An electro-optic half subtractor from a silicon-based hybrid surface plasmon polariton waveguide
  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
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  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
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  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
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Heruntergeladen am 19.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/phys-2024-0096/html
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